of a row or a column. The result of a sequence of such operations is a new
matrix $N=M_{IJ}$ considered below.

######  Theorem 3.5.

Let $I=(i_{1},\dots,i_{k})$ and $J=(j_{1},\dots,j_{l})$, where $1\leqslant
i_{s}\leqslant n$ and $1\leqslant j_{t}\leqslant m$ for any $s=1,\dots,k$ and
$t=1,\dots,l$. Let $M=\big{(}M^{i}_{j}\big{)}$ be an $n\times m$ matrix over
${\mathfrak{R}}$ and $M_{IJ}$ be $k\times l$ matrix with entries
$(M_{IJ})^{s}_{t}=M^{i_{s}}_{j_{t}}$. Let $\widehat{q}$ and $\widehat{p}$ be
$n\times n$ and $m\times m$ parameter matrices and let $\widehat{p}_{II}$ and
$\widehat{q}_{JJ}$ be $k\times k$ and $l\times l$ matrices with entries
$(\widehat{q}_{II})_{su}=q_{i_{s}i_{u}}$, $s,u=1,\dots,k$,
$(\widehat{p}_{JJ})_{tv}=q_{j_{t}j_{v}}$, $t,v=1,\dots,l$. They are also
parameter matrices. If $M$ is a $(\widehat{q},\widehat{p})$-Manin matrix then
$M_{IJ}$ is a $(\widehat{q}_{II},\widehat{p}_{JJ})$-Manin matrix.

###### Proof.

By substituting $i\to i_{s}$, $j\to i_{u}$, $k\to j_{t}$, $l\to j_{v}$ to
(3.19) we obtain the relations (3.19) for the matrix $M_{IJ}$ with
coefficients defined by the parameter matrices $\widehat{q}_{II}$ and
$\widehat{p}_{JJ}$. ∎

For a non-zero complex number $q$ let us denote by $q^{[n]}$ the $n\times n$
parameter matrix with entries
$\big{(}q^{[n]}\big{)}_{ij}=q^{\mathop{\mathrm{sgn}}\nolimits(j-i)}$. Then
$P_{q^{[n]}}=P_{n}^{q}$ and the $\big{(}q^{[n]},q^{[m]}\big{)}$-Manin matrices
are exactly the $n\times m$ $q$-Manin matrices. A permutation of rows or
columns of a such matrix $M$ gives a $\big{(}\sigma
q^{[n]}\sigma^{-1},q^{[m]}\big{)}$\- or $\big{(}q^{[n]},\tau
q^{[m]}\tau^{-1}\big{)}$-Manin matrix respectively. In general these are not
$q$-Manin matrices any more (see Section 3.2), but they are related with the
quadratic algebras isomorphic to ${\mathfrak{X}}_{A_{n}^{q}}({\mathbb{C}})$
and ${\mathfrak{X}}_{A_{m}^{q}}({\mathbb{C}})$, so that they have the same
properties permuted in some sense. For instance, properties of the
$q$-determinant of ${}_{\sigma}M$ are similar to ones of the $q$-determinant
of a $q$-Manin matrix.

Let $x^{1},\dots,x^{n}$ be the generators of
${\mathfrak{X}}_{A_{n}^{q}}({\mathbb{C}})$. Then the $n\times n$ diagonal
matrix

$\displaystyle M=\begin{pmatrix}x^{1}&0&\cdots&0\\\ 0&x^{2}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&x^{n}\end{pmatrix}$ (3.22)

is an $(A_{n}^{q},A_{n})$-Manin matrix, i.e., a
$\big{(}q^{[n]},1^{[n]}\big{)}$-Manin matrix. More generally, for any
$p\in{\mathbb{C}}\backslash\\{0\\}$ the matrix (3.22) is a
$\big{(}(pq)^{[n]},p^{[n]}\big{)}$-Manin matrix.

An analogue of the $q$-determinant for a $(\widehat{q},\widehat{p})$-Manin
matrix depends on $\widehat{q}$, but not on $\widehat{p}$. We call it
$\widehat{q}$-determinant, it is defined for an $n\times n$ matrix $M$ as
follows [25]:

$\displaystyle\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)$
$\displaystyle=\sum_{\sigma\in{\mathbf{S}}_{n}}(-1)^{\sigma}\prod_{\begin{subarray}{c}i<j\\\
\sigma(i)>\sigma(j)\end{subarray}}q_{ij}^{-1}\cdot
M^{\sigma^{-1}(1)}_{1}M^{\sigma^{-1}(2)}_{2}\cdots M^{\sigma^{-1}(n)}_{n}$
$\displaystyle=\sum_{\sigma\in{\mathbf{S}}_{n}}(-1)^{\sigma}\prod_{\begin{subarray}{c}i<j\\\
\sigma^{-1}(i)>\sigma^{-1}(j)\end{subarray}}q_{ij}^{-1}\cdot
M^{\sigma(1)}_{1}M^{\sigma(2)}_{2}\cdots M^{\sigma(n)}_{n}.$ (3.23)

The products in this formula runs over all inversions of the permutations
$\sigma$ and $\sigma^{-1}$ respectively. Hence for $\widehat{q}=q^{[n]}$ the
formula (3.23) gives the $q$-determinant (3.15). The $\widehat{q}$-determinant
is a “generalised” determinant for the square
$(\widehat{q},\widehat{p})$-Manin matrices. In particular, $q$-determinant is
a “generalised” determinant for the square
$\big{(}q^{[n]},\widehat{p}\big{)}$-Manin matrices.

######  Lemma 3.6.

Let $\psi_{1},\dots,\psi_{n}$ satisfy (3.17). Then

$\displaystyle\psi_{i_{1}}\psi_{i_{2}}\cdots\psi_{i_{n}}=0,\qquad\text{if}\quad
i_{k}=i_{l}\quad\text{for some}\quad k\neq l;$ (3.24)
$\displaystyle\psi_{\sigma(1)}\psi_{\sigma(2)}\cdots\psi_{\sigma(n)}=(-1)^{\sigma}\psi_{1}\psi_{2}\cdots\psi_{n}\\!\\!\\!\\!\prod_{\begin{subarray}{c}i<j\\\
\sigma^{-1}(i)>\sigma^{-1}(j)\end{subarray}}\\!\\!\\!\\!q_{ij}^{-1},\qquad\text{for
any}\quad\sigma\in{\mathbf{S}}_{n}.$ (3.25)

###### Proof.

The relations (3.17) allows us to permute the factors in the left hand side of
(3.24). In particular, one can place the factors $\psi_{i_{k}}$ and
$\psi_{i_{l}}$ to neighbour sites. It $i_{k}=i_{l}$ then
$\psi_{i_{k}}\psi_{i_{l}}=0$.

Let us rewrite the formula (3.25) in terms of Appendix A. Consider the root
system for the reflection group ${\mathbf{S}}_{n}$. Denote $q_{\alpha}=q_{ij}$
for the root $\alpha=e_{i}-e_{j}$. Then due to (A.2) the formula (3.25) takes
the form

$\displaystyle\psi_{\sigma(1)}\cdots\psi_{\sigma(n)}$
$\displaystyle=(-1)^{\sigma}\psi_{1}\cdots\psi_{n}\prod_{\alpha\in
R^{+}_{\sigma^{-1}}}q_{\alpha}^{-1}.$ (3.26)

We prove the formula (3.26) by using induction on the length
$\ell=\ell(\sigma)$. Let
$\sigma=\sigma_{i_{1}}\cdots\sigma_{i_{\ell-1}}\sigma_{i_{\ell}}$ be a reduced
expression and $\tau=\sigma_{i_{1}}\cdots\sigma_{i_{\ell-1}}$. Then
$\psi_{\sigma(1)}\cdots\psi_{\sigma(n)}=\psi_{\tau\sigma_{i_{\ell}}(1)}\cdots\psi_{\tau\sigma_{i_{\ell}}(n)}=-q^{-1}_{\tau(\alpha_{i_{\ell}})}\psi_{\tau(1)}\cdots\psi_{\tau(n)}$.
Note that $\ell(\tau)=\ell-1$. From (A.4) we obtain
$R^{+}_{\sigma^{-1}}=R^{+}_{\tau^{-1}}\sqcup\\{\tau(\alpha_{i_{\ell}})\\}$.
Together with the induction assumption this implies the formula (3.26). ∎

Lemma 3.6 implies that for any $n\times n$ matrix $M$ such that
$[M^{i}_{j},\psi_{k}]=0$ we have

$\displaystyle\phi_{1}\phi_{2}\cdots\phi_{n}=\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)\psi_{1}\psi_{2}\cdots\psi_{n},$
(3.27)

where $\phi_{j}=\sum_{i=1}^{n}\psi_{i}M^{i}_{j}$. A permutation of rows or
columns corresponds to a permutation of $\psi_{1},\dots,\psi_{n}$ or
$\phi_{1},\dots,\phi_{n}$ respectively:

$\displaystyle\phi_{j}=\sum_{i=1}^{n}(^{\tau}M)^{i}_{j}\psi_{\tau^{-1}(i)},\qquad\phi_{\tau^{-1}(j)}=\sum_{i=1}^{n}(_{\tau}M)^{i}_{j}\psi_{i}.$
(3.28)

######  Theorem 3.7.

Let $\widehat{q}=(q_{ij})$ and $\widehat{p}=(p_{ij})$ be $n\times n$ parameter
matrices and $\tau\in{\mathbf{S}}_{n}$. Let $M$ be a
$(\widehat{q},\widehat{p})$-Manin matrix over ${\mathfrak{R}}$. Then the
“generalised” determinants of the
$(\tau\widehat{q}\tau^{-1},\widehat{p})$-Manin matrix ${}^{\tau}M$ and of the
$(\widehat{q},\tau\widehat{p}\tau^{-1})$-Manin matrix ${}_{\tau}M$ have the
form

$\displaystyle\mathop{\mathrm{det}}\nolimits_{\tau\widehat{q}\tau^{-1}}(^{\tau}M)=(-1)^{\tau}\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)\prod_{\begin{subarray}{c}i<j\\\
\tau(i)>\tau(j)\end{subarray}}q_{ij},$ (3.29)
$\displaystyle\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(_{\tau}M)=(-1)^{\tau}\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)\prod_{\begin{subarray}{c}i<j\\\
\tau(i)>\tau(j)\end{subarray}}p_{ij}^{-1}.$ (3.30)

More generally, the formula (3.29) is valid for any $n\times n$ matrix $M$.

###### Proof.

Let $\psi_{i}$ be generators of the algebra
$\Xi_{A_{\widehat{q}}}({\mathbb{C}})$. Then Proposition 2.17 implies that the
elements
$\phi_{j}=\sum_{i=1}^{n}M^{i}_{j}\psi_{i}\in\Xi_{A_{\widehat{q}}}({\mathfrak{R}})$
satisfy the commutation relations (3.17) with the parameter matrix
$\widehat{p}$. Due to the formula (3.20) the elements
$\psi^{\prime}_{i}=\psi_{\tau^{-1}(i)}$ and
$\phi^{\prime}_{j}=\phi_{\tau^{-1}(j)}$ satisfy the commutation relations
(3.17) with the parameter matrices $\sigma\widehat{q}\sigma^{-1}$ and
$\sigma\widehat{p}\sigma^{-1}$ respectively. From the formulae (3.27) and
(3.28) we obtain

$\displaystyle\phi_{1}\cdots\phi_{n}=\psi^{\prime}_{1}\cdots\psi^{\prime}_{n}\mathop{\mathrm{det}}\nolimits_{\tau\widehat{q}\tau^{-1}}(^{\tau}M),\qquad\phi^{\prime}_{1}\cdots\phi^{\prime}_{n}=\psi_{1}\cdots\psi_{n}\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(_{\tau}M).$
(3.31)

The formula (3.25) for $\phi_{j}$ and $\psi_{i}$ with $\sigma=\tau^{-1}$ takes
the form

$\displaystyle\psi^{\prime}_{1}\cdots\psi^{\prime}_{n}=(-1)^{\tau}\psi_{1}\cdots\psi_{n}\prod_{\begin{subarray}{c}i<j\\\
\tau(i)>\tau(j)\end{subarray}}q_{ij}^{-1},\qquad\phi^{\prime}_{1}\cdots\phi^{\prime}_{n}=(-1)^{\tau}\phi_{1}\cdots\phi_{n}\prod_{\begin{subarray}{c}i<j\\\
\tau(i)>\tau(j)\end{subarray}}p_{ij}^{-1}.$

Substitution of these formulae and the formula (3.27) to (3.31) gives (3.29)
and (3.30). We did not use the commutation relations of $\phi_{j}$ for the
proof of the formula (3.29), so it is valid for any matrix $M$. ∎

Let us consider the case when two rows or two columns coincide. We write some
conditions leading to vanishing of the $\widehat{q}$-determinant.

######  Corollary 3.8.

Let $M$ be a $k\times k$ matrix over ${\mathfrak{R}}$. Let $\widehat{q}$ and
$\widehat{p}$ be $k\times k$ parameter matrices.

  * •

Let $i\neq j$. If $M^{i}_{l}=M^{j}_{l}$ and $q_{il}=q_{jl}$ $\forall\,l$ $($in
particular, $q_{ij}=1)$ then
$\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)=0$.

  * •

If two columns of a $(\widehat{q},\widehat{p})$-Manin matrix $M$ coincide
$($that is $M^{l}_{i}=M^{l}_{j}$ $\;\forall\,l$ for some $i\neq j)$ then
$\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)=0$.

###### Proof.

Let $\sigma_{ij}\in{\mathbf{S}}_{k}$ be the transposition of $i$ and $j$.
Suppose $i<j$ (without loss of generality). The conditions $q_{il}=q_{jl}$
imply $\sigma_{ij}\widehat{q}\sigma_{ij}=\widehat{q}$ and

$\prod\limits_{\begin{subarray}{c}s<t\\\
\sigma_{ij}(s)>\sigma_{ij}(t)\end{subarray}}q_{\rm
st}=q_{ij}\prod\limits_{s=i+1}^{j-1}(q_{sj}q_{is})=1,$

so the substitution $\tau=\sigma_{ij}$ to (3.29) gives
$\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)=-\mathop{\mathrm{det}}\nolimits_{\widehat{q}}(M)$.
For generic $\widehat{p}$ one can similarly prove the second statement by
using the formula (3.30) with $\tau=\sigma_{ij}$. For arbitrary $\widehat{p}$
it follows from the relations (3.27) and $\phi_{i}\phi_{j}=\phi_{i}^{2}=0$. ∎

######  Corollary 3.9.

Let $M$ be an $n\times m$ matrix over ${\mathfrak{R}}$. Let $\widehat{q}$ and
$\widehat{p}$ be $n\times n$ and $m\times m$ parameter matrices. Let
$I=(i_{1},\dots,i_{k})$ and $J=(j_{1},\dots,j_{k})$, where $1\leqslant
i_{s}\leqslant n$ and $1\leqslant j_{s}\leqslant m$ for all $s=1,\dots,k$. Let
$\tau\in{\mathbf{S}}_{k}$, $K=(i_{\tau(1)},\dots,i_{\tau(k)})$ and
$L=(j_{\tau(1)},\dots,j_{\tau(k)})$.

  * •

We have

$\mathop{\mathrm{det}}\nolimits_{\widehat{q}_{KK}}(M_{KJ})=(-1)^{\tau}\mathop{\mathrm{det}}\nolimits_{\widehat{q}_{II}}(M_{IJ})\prod_{\begin{subarray}{c}s<t\\\
\tau^{-1}(s)>\tau^{-1}(t)\end{subarray}}q_{i_{s}i_{t}}.$

  * •

If $i_{s}=i_{t}$ for some $s\neq t$ then
$\mathop{\mathrm{det}}\nolimits_{\widehat{q}_{II}}(M_{IJ})=0$.

  * •

If $M$ is a $(\widehat{q},\widehat{p})$-Manin matrix then

$\mathop{\mathrm{det}}\nolimits_{\widehat{q}_{II}}(M_{IL})=(-1)^{\tau}\mathop{\mathrm{det}}\nolimits_{\widehat{q}_{II}}(M_{IJ})\prod_{\begin{subarray}{c}s<t\\\
\tau^{-1}(s)>\tau^{-1}(t)\end{subarray}}p_{j_{s}j_{t}}^{-1}.$

  * •

If $M$ is a $(\widehat{q},\widehat{p})$-Manin matrix and $j_{s}=j_{t}$ for
some $s\neq t$ then
$\mathop{\mathrm{det}}\nolimits_{\widehat{q}_{II}}(M_{IJ})=0$.

###### Proof.

Note that $\tau^{-1}\widehat{q}_{II}\tau=\widehat{q}_{KK}$,
$M_{KJ}=\leftidx{{}^{\tau^{-1}}}(M_{IJ})$ and
$M_{IL}=\leftidx{{}_{\tau^{-1}}}(M_{IJ})$, so the statements follow from
Theorems 3.5, 3.7 and Corollary 3.8. ∎

######  Remark 3.10.

The formula (3.25) follows from
$\psi_{j}\psi_{i}=-q_{ij}^{-1}\psi_{i}\psi_{j}$, $i<j$. As consequence, we did
not need the relations $\phi_{i}^{2}=0$ to prove the formula (3.30). The
relations $\psi_{j}\psi_{i}=-p_{ij}^{-1}\psi_{i}\psi_{j}$, $i<j$, define the
algebra $\Xi_{\widetilde{A}_{\widehat{p}}}({\mathbb{C}})$, where
$\widetilde{A}_{\widehat{p}}=\frac{1-\widetilde{P}_{\widehat{p}}}{2}$ and
$\big{(}\widetilde{P}_{\widehat{p}}\big{)}^{kl}_{ij}=(-1)^{\delta_{ij}}p_{ij}\delta^{k}_{j}\delta^{l}_{i}$.
Hence the formula (3.30) is valid for any
$\big{(}A_{\widehat{q}},\widetilde{A}_{\widehat{p}}\big{)}$-Manin matrix $M$.
As consequence, the second statement of Corollary 3.8 is valid for these
matrices if $\widehat{p}$ is generic. However they are not valid for some
$\widehat{p}$, so it is necessary to require $M$ to be a
$(\widehat{q},\widehat{p})$-Manin matrix. Moreover the third and forth
statements of Corollary 3.9 are not valid for
$\big{(}A_{\widehat{q}},\widetilde{A}_{\widehat{p}}\big{)}$-Manin matrices
even if $\widehat{p}$ is generic since we used Theorem 3.5. For example, let
$M=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)$, $I=(1,2)$,
$J=(1,1)$. Then the $q$-determinant of the matrix
$M_{IJ}=\left(\begin{smallmatrix}a&a\\\ c&c\end{smallmatrix}\right)$ is
$\mathop{\mathrm{det}}\nolimits_{q}(M_{IJ})=ac-q^{-1}ca$. It vanishes if $M$
is a $\big{(}q^{[2]},p^{[2]}\big{)}$-Manin matrix, but the cross relation
$ad-q^{-1}pda+pbc-q^{-1}cb=0$ is not enough for
$\mathop{\mathrm{det}}\nolimits_{q}(M_{IJ})=0$. The cross relation for the
matrix $\left(\begin{smallmatrix}a&a\\\ c&c\end{smallmatrix}\right)$ itself
gives $ac-qp^{-1}ca+pac-q^{-1}ca=0$. This imply
$\mathop{\mathrm{det}}\nolimits_{q}(M)=0$ unless $p=-1$.

######  Remark 3.11.

Let $n=k+l$, $q_{ij}=-1$ for $i,j=k+1,\dots,n$, $i\neq j$, and $q_{ij}=1$ for
other $i$, $j$. By factorizing the algebra
${\mathfrak{X}}_{A_{\widehat{q}}}({\mathbb{C}})$ over the relations
$x_{i}^{2}=0$, $i=k+1,\dots,n$, and introducing a ${\mathbb{Z}}_{2}$-grading
we obtain the free super-commutative quadratic algebra with $k$ even and $l$
odd generators. However the approach of Section 2 applied to this algebra does
not give super-Manin matrices considered in [25, 30]. The reason is that we
suppose commutativity of $x^{i}$ with entries of $M$, which should be replaced
by super-commutativity in the super-case. We will consider Manin matrices for
quadratic super-algebras in future works.

###  3.4 A 4-parametric quadratic algebra

Consider the algebra with generators $x$, $y$, $z$ and relations

$\displaystyle axy-a^{-1}yx=\kappa z^{2},\qquad byz-b^{-1}zy=\kappa
x^{2},\qquad czx-c^{-1}xz=\kappa y^{2},$ (3.32)

where $a,b,c\in{\mathbb{C}}\backslash\\{0\\}$, $\kappa\in{\mathbb{C}}$.

Let $x^{1}=x$, $x^{2}=y$, $x^{3}=z$, $a_{12}=a$, $a_{23}=b$, $a_{31}=c$,
$a_{ij}=a_{ji}^{-1}$, $a_{ii}=1$. Let $\varepsilon_{ijk}$ be the totally
antisymmetric tensor such that $\varepsilon_{123}=1$. Then (3.32) is
equivalent to the system

$\displaystyle
a_{ij}x^{i}x^{j}-a_{ji}x^{j}x^{i}=\kappa\sum_{k=1}^{3}\varepsilon_{ijk}x^{k}x^{k},\qquad
i,j=1,2,3.$ (3.33)

These relations can be written as
$x^{i}x^{j}=\sum_{k,l=1}^{3}P^{ij}_{kl}x^{k}x^{l}$, where

$\displaystyle P^{ij}_{kl}=a_{ji}^{2}\delta^{j}_{k}\delta^{i}_{l}+\kappa
a_{ji}\delta_{kl}\varepsilon_{ijk}.$ (3.34)

The operator
$P\in\mathop{\mathrm{End}}\nolimits\big{(}{\mathbb{C}}^{3}\otimes{\mathbb{C}}^{3}\big{)}$
with the entries (3.34) satisfies $P^{2}=1$. Hence the operator
$A^{a,b,c}_{\kappa}:=\frac{1-P}{2}$ is an idempotent and the relations (3.32)
define the algebra ${\mathfrak{X}}_{A^{a,b,c}_{\kappa}}({\mathbb{C}})$. By
setting $\kappa=0$ we obtain the quadratic algebra
${\mathfrak{X}}_{A_{\widehat{q}}}({\mathbb{C}})$ with the parameters
$q_{ij}=a_{ij}^{2}$, so the algebra
${\mathfrak{X}}_{A^{a,b,c}_{\kappa}}({\mathbb{C}})$ is a generalisation of the
$3$-dimensional case of the algebra
${\mathfrak{X}}_{A_{\widehat{q}}}({\mathbb{C}})$ considered in Section 3.3
(this is not a $\kappa$-deformation in general, see Remark 6.5).

Let us consider some examples of Manin matrices by taking
$A=A^{a,b,c}_{\kappa}$ as one of the idempotents. A $2\times 3$ matrix
$M=\left(\begin{smallmatrix}\alpha_{1}&\alpha_{2}&\alpha_{3}\\\
\beta_{1}&\beta_{2}&\beta_{3}\end{smallmatrix}\right)$ is an
$\big{(}A^{q}_{2},A^{a,b,c}_{\kappa}\big{)}$-Manin matrix iff

$\displaystyle
q(a_{ji}\alpha_{i}\beta_{j}+a_{ij}\alpha_{j}\beta_{i})=a_{ji}\beta_{i}\alpha_{j}+a_{ij}\beta_{j}\alpha_{i},$
(3.35) $\displaystyle
q(a_{ij}\alpha_{k}\beta_{k}+\kappa\alpha_{i}\beta_{j})=a_{ij}\beta_{k}\alpha_{k}+\kappa\beta_{i}\alpha_{j}$
(3.36)

for all cyclic permutation $(i,j,k)$ of $(1,2,3)$. The relation (3.35) is
exactly the cross relation (3.19) for the parameters
$q_{ij}=q^{\mathop{\mathrm{sgn}}\nolimits(j-i)}$ and $p_{ij}=a_{ij}^{2}$,
while the relation (3.36) is a generalisation of the $q$-commutation (3.18).

A $3\times 2$ matrix $M=\left(\begin{smallmatrix}\alpha^{1}&\beta^{1}\\\
\alpha^{2}&\beta^{2}\\\ \alpha^{3}&\beta^{3}\end{smallmatrix}\right)$ is an
$\big{(}A^{a,b,c}_{\kappa},A^{q}_{2}\big{)}$-Manin matrix iff the relations
(3.33) are satisfied by the substitutions $x^{i}=\alpha^{i}$ and
$x^{i}=\beta^{i}$ and

$\displaystyle
a_{ij}\big{(}\alpha^{i}\beta^{j}+q\beta^{i}\alpha^{j}\big{)}-a_{ji}\big{(}\alpha^{j}\beta^{i}+q\beta^{j}\alpha^{i}\big{)}=\kappa\big{(}\alpha^{k}\beta^{k}+q\beta^{k}\alpha^{k}\big{)}$

for all cyclic permutations $(i,j,k)$ of $(1,2,3)$.

##  4 Lax operators

Lax operators are different square matrices and endomorphisms of vector spaces
arisen in the theories of integrable systems and quantum groups. We will
consider Lax operators satisfying $RLL$-relations with some $R$-matrices
(solutions of the Yang–Baxter equation). Different $R$-matrices give different
types of Lax operators. Since many quantum groups can be defined by
$RLL$-relations the Lax operators of a certain type are related with the
representation theory of the corresponding quantum group. Here we consider
connections between Manin matrices associated with some quadratic algebras and
the Lax operators associated with the quantum groups
$U_{q}(\mathfrak{gl}_{n})$, $Y(\mathfrak{gl}_{n})$. Notice also that a
connection between the $q$-Manin matrices and Lax operators associated the
affine quantum group $U_{q}\big{(}\widehat{\mathfrak{gl}}_{n}\big{)}$ was
described in [6].

###  4.1 Lax operators of $\boldsymbol{U_{q}(\mathfrak{gl}_{n})}$ type and
$\boldsymbol{q}$-Manin matrices

A relationship between Lax operator and $q$-Manin matrices was first described
by Manin, see [24]. We investigate this relationship by applying a
decomposition of the corresponding $R$-matrix.

Let us first write the relations for a transposed $q$-Manin matrix. Recall
that the matrices $P_{n}$ defined by (3.1) permute the factors
$\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
in the following way:

$\displaystyle P_{n}TP_{m}=T^{(21)},\qquad
P_{n}M^{(1)}N^{(2)}P_{m}=M^{(2)}N^{(1)},$ (4.1)

where
$T\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m}\otimes{\mathbb{C}}^{m},{\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$,
$M,N\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$.
Let us note that

$\displaystyle\big{(}P_{n}^{q}\big{)}^{(21)}=P_{n}^{q^{-1}},\qquad\big{(}A_{n}^{q}\big{)}^{(21)}=A_{n}^{q^{-1}},\qquad\big{(}S_{n}^{q}\big{)}^{(21)}=S_{n}^{q^{-1}}.$
(4.2)

Note also that transposition gives the same:

$\displaystyle\big{(}P_{n}^{q}\big{)}^{\top}=P_{n}^{q^{-1}},\qquad\big{(}A_{n}^{q}\big{)}^{\top}=A_{n}^{q^{-1}},\qquad\big{(}S_{n}^{q}\big{)}^{\top}=S_{n}^{q^{-1}}.$
(4.3)

######  Lemma 4.1.

Let
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$.
The transposed matrix $M^{\top}$ is a $q$-Manin matrix iff the matrix $M$
satisfies one of the following equivalent relations:

$\displaystyle S^{q^{-1}}_{n}M^{(1)}M^{(2)}A_{m}^{q^{-1}}=0,$ (4.4)
$\displaystyle S_{n}^{q}M^{(2)}M^{(1)}A_{m}^{q}=0.$ (4.5)

###### Proof.

The relation (3.13) for the $m\times n$ matrix $M^{\top}$ has the form
$A_{m}^{q}\big{(}M^{\top}\big{)}^{(1)}\big{(}M^{\top}\big{)}^{(2)}S_{n}^{q}=0$.
If we transpose the both hand sides and take into account
$\big{(}M^{(1)}M^{(2)}\big{)}^{\top}=\big{(}M^{\top}\big{)}^{(1)}\big{(}M^{\top}\big{)}^{(2)}$
and (4.3) we obtain (4.4). Due to (4.2) the permutation of tensor factors
yields (4.5). ∎

Suppose that $q^{2}\neq-1$. Consider the $R$-matrix

$\displaystyle R^{q}=R^{q}_{n}=q^{-1}\sum_{i=1}^{n}E_{i}{}^{i}\otimes
E_{i}{}^{i}+\sum_{i\neq j}E_{i}{}^{i}\otimes
E_{j}{}^{j}+\big{(}q^{-1}-q\big{)}\sum_{i>j}E_{i}{}^{j}\otimes E_{j}{}^{i}.$

It satisfies the Yang–Baxter equation

$\displaystyle(R^{q})^{(12)}(R^{q})^{(13)}(R^{q})^{(23)}=(R^{q})^{(23)}(R^{q})^{(13)}(R^{q})^{(12)}.$
(4.6)

A Lax operator of $U_{q}(\mathfrak{gl}_{n})$ type is an $n\times n$ matrix
$L\in{\mathfrak{R}}\otimes\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n})$
satisfying the $RLL$-relation

$\displaystyle R^{q}L^{(1)}L^{(2)}=L^{(2)}L^{(1)}R^{q}.$

More generally, consider an $n\times m$ matrix
$L\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
satisfying

$\displaystyle R^{q}_{n}L^{(1)}L^{(2)}=L^{(2)}L^{(1)}R^{q}_{m}.$ (4.7)

######  Remark 4.2.

The commutation relations for the quantum group $U_{q}(\mathfrak{gl}_{n})$ can
be written as three matrix relations
$R^{q}_{n}L_{\pm}^{(1)}L_{\pm}^{(2)}=L_{\pm}^{(2)}L_{\pm}^{(1)}R^{q}_{n}$,
$R^{q}_{n}L_{+}^{(1)}L_{-}^{(2)}=L_{-}^{(2)}L_{+}^{(1)}R^{q}_{n}$ for some
matrices $L_{+},L_{-}\in
U_{q}(\mathfrak{gl}_{n})\otimes\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n})$
[9, 33].

By multiplying the relation (4.7) by $P_{n}$ from the left and by taking into
account (4.1) we obtain the equivalent relation

$\displaystyle\widehat{R}^{q}_{n}L^{(1)}L^{(2)}=L^{(1)}L^{(2)}\widehat{R}^{q}_{n},$
(4.8)

where

$\displaystyle\widehat{R}^{q}=\widehat{R}^{q}_{n}:=P_{n}R^{q}_{n}=q^{-1}\sum_{i=1}^{n}E_{i}{}^{i}\otimes
E_{i}{}^{i}+\sum_{i\neq j}E_{i}{}^{j}\otimes
E_{j}{}^{i}+\big{(}q^{-1}-q\big{)}\sum_{i<j}E_{i}{}^{i}\otimes E_{j}{}^{j}.$
(4.9)

The matrix (4.9) satisfies the braid relation

$\displaystyle\big{(}\widehat{R}^{q}\big{)}^{(23)}\big{(}\widehat{R}^{q}\big{)}^{(12)}\big{(}\widehat{R}^{q}\big{)}^{(23)}=\big{(}\widehat{R}^{q}\big{)}^{(12)}\big{(}\widehat{R}^{q}\big{)}^{(23)}\big{(}\widehat{R}^{q}\big{)}^{(12)},$
(4.10)

which is obtained by multiplying left and right hand sides of (3.3) and (4.6).

######  Lemma 4.3 ([33]).

The matrix (4.9) can be decomposed as

$\displaystyle\widehat{R}^{q}=q^{-1}\widehat{R}_{+}^{q}-q\widehat{R}_{-}^{q},$
(4.11)

where

$\displaystyle\widehat{R}_{+}^{q}$
$\displaystyle=\widehat{R}_{n+}^{q}:=\frac{q+\widehat{R}^{q}_{n}}{q+q^{-1}}$
$\displaystyle=\sum_{i=1}^{n}\\!E_{i}{}^{i}\otimes
E_{i}{}^{i}\\!+\\!\frac{1}{q\\!+\\!q^{-1}}\sum_{i\neq j}\\!E_{i}{}^{j}\otimes
E_{j}{}^{i}\\!+\\!\frac{1}{q\\!+\\!q^{-1}}\sum_{i<j}\\!\big{(}q^{-1}E_{i}{}^{i}\otimes
E_{j}{}^{j}\\!+\\!qE_{j}{}^{j}\otimes E_{i}{}^{i}\big{)},$ (4.12)
$\displaystyle\widehat{R}_{-}^{q}$
$\displaystyle=\widehat{R}^{q}_{n-}:=\frac{q^{-1}-\widehat{R}^{q}_{n}}{q+q^{-1}}$
$\displaystyle=-\frac{1}{q+q^{-1}}\sum_{i\neq j}E_{i}{}^{j}\otimes
E_{j}{}^{i}+\frac{1}{q+q^{-1}}\sum_{i<j}\big{(}qE_{i}{}^{i}\otimes
E_{j}{}^{j}+q^{-1}E_{j}{}^{j}\otimes E_{i}{}^{i}\big{)}$ (4.13)

are orthogonal idempotents:

$\displaystyle\widehat{R}_{+}^{q}+\widehat{R}_{-}^{q}=1,\qquad\big{(}\widehat{R}_{+}^{q}\big{)}^{2}=\widehat{R}_{+}^{q},\qquad\big{(}\widehat{R}_{-}^{q}\big{)}^{2}=\widehat{R}_{-}^{q},\qquad\widehat{R}_{+}^{q}\widehat{R}_{-}^{q}=\widehat{R}_{-}^{q}\widehat{R}_{+}^{q}=0.$
(4.14)

The matrix (4.9) satisfies the Hecke relation:

$\displaystyle\big{(}\widehat{R}^{q}-q^{-1}\big{)}\big{(}\widehat{R}^{q}+q\big{)}=0.$
(4.15)

###### Proof.

The formulae (4.11), (4.12), (4.13) and the first of (4.14) are obtained
directly from the definitions of
$\widehat{R}_{+}^{q}=\frac{q+\widehat{R}^{q}_{n}}{q+q^{-1}}$ and
$\widehat{R}_{-}^{q}=\frac{q^{-1}-\widehat{R}^{q}_{n}}{q+q^{-1}}$. Further

$\displaystyle\big{(}q+q^{-1}\big{)}^{2}\big{(}\widehat{R}_{-}^{q}\big{)}^{2}=\bigg{(}{-}\sum_{i\neq
j}E_{i}{}^{j}\otimes E_{j}{}^{i}+\sum_{i<j}\big{(}qE_{i}{}^{i}\otimes
E_{j}{}^{j}+q^{-1}E_{j}{}^{j}\otimes E_{i}{}^{i}\big{)}\bigg{)}^{2}$
$\displaystyle\hphantom{\big{(}q+q^{-1}\big{)}^{2}\big{(}\widehat{R}_{-}^{q}\big{)}^{2}}{}=\sum_{i\neq
j}E_{i}{}^{i}\otimes E_{j}{}^{j}-\sum_{i<j}\big{(}qE_{i}{}^{j}\otimes
E_{j}{}^{j}+q^{-1}E_{j}{}^{i}\otimes E_{i}{}^{i}\big{)}$
$\displaystyle\hphantom{(q+q^{-1})^{2}\big{(}\widehat{R}_{-}^{q}\big{)}^{2}=}{}-\\!\sum_{i<j}\big{(}q^{-1}E_{i}{}^{j}\otimes
E_{j}{}^{j}+qE_{j}{}^{i}\otimes
E_{i}{}^{i}\big{)}+\\!\sum_{i<j}\big{(}q^{2}E_{i}{}^{i}\otimes
E_{j}{}^{j}+q^{-2}E_{j}{}^{j}\otimes E_{i}{}^{i}\big{)}$
$\displaystyle\hphantom{\big{(}q+q^{-1}\big{)}^{2}\big{(}\widehat{R}_{-}^{q}\big{)}^{2}}{}=-(q\\!+\\!q^{-1})\sum_{i\neq
j}\\!E_{i}{}^{j}\otimes
E_{j}{}^{i}+\\!\sum_{i<j}\\!\big{(}\big{(}q^{2}+1\big{)}E_{i}{}^{i}\otimes
E_{j}{}^{j}\\!+\\!\big{(}1\\!+\\!q^{-2}\big{)}E_{j}{}^{j}\otimes
E_{i}{}^{i}\big{)}$
$\displaystyle\hphantom{\big{(}q+q^{-1}\big{)}^{2}\big{(}\widehat{R}_{-}^{q}\big{)}^{2}}{}=\big{(}q+q^{-1}\big{)}\bigg{(}{-}\sum_{i\neq
j}E_{i}{}^{j}\otimes E_{j}{}^{i}+\sum_{i<j}\big{(}qE_{i}{}^{i}\otimes
E_{j}{}^{j}+q^{-1}E_{j}{}^{j}\otimes E_{i}{}^{i}\big{)}\bigg{)}$
$\displaystyle\hphantom{\big{(}q+q^{-1}\big{)}^{2}\big{(}\widehat{R}_{-}^{q}\big{)}^{2}}{}=\big{(}q+q^{-1}\big{)}^{2}\widehat{R}_{-}^{q}.$

Thus we obtain $\big{(}\widehat{R}_{-}^{q}\big{)}^{2}=\widehat{R}_{-}^{q}$.
Other relations (4.14) follows from
$\widehat{R}_{+}^{q}=1-\widehat{R}_{-}^{q}$. Then the Hecke relation (4.15) is
a consequence of $\widehat{R}_{-}^{q}\widehat{R}_{+}^{q}=0$. ∎

We see that

$\displaystyle\widehat{R}_{n-}^{q}=\frac{P_{n}P^{q}_{n}-P_{n}}{q+q^{-1}}=-\frac{2}{q+q^{-1}}P_{n}A_{n}^{q}.$
(4.16)

This means that the idempotents $A_{n}^{q}$ and $\widehat{R}_{n-}^{q}$ are
left-equivalent. Thus an $n\times m$ $q$-Manin matrix is exactly the Manin
matrix for the pair $\big{(}\widehat{R}_{n-}^{q},\widehat{R}_{m-}^{q}\big{)}$.
Due to (4.2) we obtain

$\displaystyle\widehat{R}_{n-}^{q}=-\frac{2}{q+q^{-1}}A_{n}^{q^{-1}}P_{n},$
(4.17)

so idempotent $\widehat{R}_{n-}^{q}$ is right-equivalent to $A_{n}^{q^{-1}}$.

Note that the relation
$\big{(}\widehat{R}_{-}^{q}\big{)}^{2}=\widehat{R}_{-}^{q}$ implies

$\displaystyle\bigg{(}{-}\frac{2}{q+q^{-1}}\bigg{)}^{2}P_{n}A_{n}^{q}P_{n}A_{n}^{q}=-\frac{2}{q+q^{-1}}P_{n}A_{n}^{q},$

so that

$\displaystyle
A_{n}^{q^{-1}}A_{n}^{q}=-\frac{q+q^{-1}}{2}P_{n}A_{n}^{q},\qquad
A_{n}^{q}A_{n}^{q^{-1}}=-\frac{q+q^{-1}}{2}A_{n}^{q}P_{n}.$ (4.18)

######  Theorem 4.4.

Let
$L\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$,
then the following statements are equivalent:

  * •

$L$ satisfies (4.7), that is
$R^{q}_{n}L^{(1)}L^{(2)}=L^{(2)}L^{(1)}R^{q}_{m}$.

  * •

$L$ satisfies (4.8), that is
$\widehat{R}^{q}_{n}L^{(1)}L^{(2)}=L^{(1)}L^{(2)}\widehat{R}^{q}_{m}$.

  * •

$L$ satisfies the relation

$\displaystyle\widehat{R}_{n+}^{q}L^{(1)}L^{(2)}=L^{(1)}L^{(2)}\widehat{R}_{m+}^{q}.$
(4.19)

  * •

$L$ satisfies the relation

$\displaystyle\widehat{R}_{n-}^{q}L^{(1)}L^{(2)}=L^{(1)}L^{(2)}\widehat{R}_{m-}^{q}.$
(4.20)

  * •

$L$ satisfies the relation

$\displaystyle A_{n}^{q}L^{(1)}L^{(2)}=L^{(2)}L^{(1)}A_{m}^{q}.$ (4.21)

  * •

$L$ satisfies the relations

$\displaystyle A_{n}^{q}L^{(1)}L^{(2)}S_{m}^{q}=0,\qquad
S_{n}^{q}L^{(2)}L^{(1)}A_{m}^{q}=0.$ (4.22)

  * •

The matrices $L$ and $L^{\top}$ are both $q$-Manin matrices.

###### Proof.

By adding $qL^{(1)}L^{(2)}$ to the both hand sides of (4.8) and by dividing by
$q+q^{-1}$ we obtain the equivalent relation (4.19). The equivalence of the
relations (4.8) and (4.20) is proved similarly. Further, by using (4.16) one
establishes the equivalence of (4.20) and (4.21). The relations (4.22) are
obtained from (4.21) by multiplying by $S_{m}^{q}$ from the right and by
multiplying by $S_{n}^{q}$ from the left respectively. Conversely, suppose
that $L$ satisfies the relations (4.22). By virtue of the formulae (4.2) the
second of the relations (4.22) can be written in the form
$S^{q^{-1}}_{n}L^{(1)}L^{(2)}A_{m}^{q^{-1}}=0$. Thus we have

$\displaystyle A_{n}^{q}L^{(1)}L^{(2)}A_{m}^{q}=A_{n}^{q}L^{(1)}L^{(2)},\qquad
L^{(1)}L^{(2)}A_{m}^{q^{-1}}=A_{n}^{q^{-1}}L^{(1)}L^{(2)}A_{m}^{q^{-1}}.$

These relations implies

$\displaystyle
A_{n}^{q}L^{(1)}L^{(2)}A_{m}^{q}A_{m}^{q^{-1}}=A_{n}^{q}L^{(1)}L^{(2)}A_{m}^{q^{-1}}=A_{n}^{q}A_{n}^{q^{-1}}L^{(1)}L^{(2)}A_{m}^{q^{-1}}.$

By using (4.18) one yields

$\displaystyle
A_{n}^{q}L^{(1)}L^{(2)}A_{m}^{q}P_{m}=A_{n}^{q}P_{n}L^{(1)}L^{(2)}A_{m}^{q^{-1}}.$

Multiplication by $P_{m}$ from the right gives

$\displaystyle
A_{n}^{q}L^{(1)}L^{(2)}A_{m}^{q}=A_{n}^{q}L^{(2)}L^{(1)}A_{m}^{q}.$

By taking into account (4.22) we obtain (4.21) in the following way:

$\displaystyle A_{n}^{q}L^{(1)}L^{(2)}$
$\displaystyle=A_{n}^{q}L^{(1)}L^{(2)}\big{(}A_{m}^{q}+S_{m}^{q}\big{)}=A_{n}^{q}L^{(1)}L^{(2)}A_{m}^{q}=A_{n}^{q}L^{(2)}L^{(1)}A_{m}^{q}$
$\displaystyle=\big{(}A_{n}^{q}+S_{n}^{q}\big{)}L^{(2)}L^{(1)}A_{m}^{q}=L^{(2)}L^{(1)}A_{m}^{q}.$

Finally, by virtue of Lemma 4.1 the relations (4.22) mean exactly that $L$ and
$L^{\top}$ are $q$-Manin matrices. ∎

Theorem 4.4 implies that a Lax operator of $U_{q}(\mathfrak{gl}_{n})$ type is
a particular case of $q$-Manin matrix. Some properties of these Lax operators
can be generalised to the case of $q$-Manin matrices. The $q$-determinant
(3.15) arose as a natural generalisation of the determinant for the Lax
operators of $U_{q}(\mathfrak{gl}_{n})$ type, its properties were generalised
for the case of $q$-Manin matrices in [6].

Note that the fact that the $RLL$-relation (4.7) is equivalent to the claim
that $L$ and $L^{\top}$ are both $q$-Manin matrices can be proved in the same
way as Proposition 3.1 (see [5, 24]). The approach considered here explains
this fact in terms of left equivalence of idempotents, which will be applied
in Section 6.2. This allows to explain why the Newton identities for the
$q$-Manin matrices proved in [6] differs from the Newton identities for
$L$-operators and $\widehat{R}^{q}_{n-}$-Manin matrices deduced in [15, 16,
17, 32].

######  Remark 4.5.

Replacement of $\widehat{R}^{q}_{n}$ by the idempotent $A^{q}_{n}$ in the
equation (4.8) leads to another relations for the entries of the matrix $L$.
Though $L$ is also a $q$-Manin matrix in this case, the matrix $L^{\top}$ is
not. For $n=2$ it is described in details in [12].

Decomposition of the operator $\widehat{R}^{q}$ into dual idempotents
described in Lemma 4.3 gives a general idea how to connect Lax operators with
Manin matrices. It can be applied to some general class of $R$-matrices.

###  4.2 Lax operators of Yangian type as Manin operators

The decomposition method described in Section 4.1 is generalised here to the
case of the rational $R$-matrix. This gives an interpretation of the
corresponding Lax matrices as a class of Manin operators.

Let $h\colon Y(\mathfrak{gl}_{n})\to{\mathfrak{R}}$ be a homomorphism from the
Yangian to some algebra ${\mathfrak{R}}$. It is defined by the image of the
matrix $T(u)$. It has the form
$L(u)=1+\sum_{r=1}^{\infty}\sum_{i,j=1}^{n}\ell^{(r)}_{ij}E_{i}{}^{j}u^{-r}$,
where $\ell^{(r)}_{ij}=h(t^{(r)}_{ij})$, and satisfies the $RLL$-relation

$\displaystyle
R(u_{1}-u_{2})L^{(1)}(u_{1})L^{(2)}(u_{2})=L^{(2)}(u_{2})L^{(1)}(u_{1})R(u_{1}-u_{2}),$
(4.23)

where $R(u)=R_{n}(u)=u-P_{n}$. Conversely, any $n\times n$ matrix over
${\mathfrak{R}}$ which has this form and satisfies (4.23) defines a
homomorphism $Y(\mathfrak{gl}_{n})\to{\mathfrak{R}}$. These are Lax operators
of Yangian type.

We consider a more general matrix
$L(u)\in{\mathfrak{R}}((u^{-1}))\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
satisfying the $RLL$-relation

$\displaystyle
R_{n}(u_{1}-u_{2})L^{(1)}(u_{1})L^{(2)}(u_{2})=L^{(2)}(u_{2})L^{(1)}(u_{1})R_{m}(u_{1}-u_{2}).$
(4.24)

It could be the Lax operator of the $\mathfrak{gl}_{n}$ $XXX$-model (which is
an image of $T(u)$ under some representation of $Y(\mathfrak{gl}_{n})$
multiplied by a polynomial), the Lax operator of the $n$ particle Toda chain
etc.

Note that the matrix $R(u)$ satisfies the Yang–Baxter equation

$\displaystyle
R^{(12)}(u_{12})R^{(13)}(u_{13})R^{(23)}(u_{23})=R^{(23)}(u_{23})R^{(13)}(u_{13})R^{(12)}(u_{12})$
(4.25)

and

$\displaystyle R^{(21)}(u_{21})R^{(12)}(u_{12})=1-u_{12}^{2},$ (4.26)

where $u_{ij}=u_{i}-u_{j}$.

Define an operator on
${\mathbb{C}}^{n}\big{(}\big{(}u^{-1}\big{)}\big{)}^{\otimes
2}={\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{(}\big{(}u_{1}^{-1},u_{2}^{-1}\big{)}\big{)}$
by the formula

$\displaystyle\widehat{R}=\widehat{R}_{n}:=P_{u_{1},u_{2}}P_{n}R_{n}(u_{12})=P_{u_{1},u_{2}}(u_{12}P_{n}-1),$
(4.27)

where $P_{u_{1},u_{2}}$ is the operator permuting $u_{1}$ and $u_{2}$, that is
$(P_{u_{1},u_{2}}f)(u_{1},u_{2})=f(u_{2},u_{1})$. The relation (4.24) is
equivalent to

$\displaystyle\widehat{R}_{n}L^{(1)}(u_{1})L^{(2)}(u_{2})=L^{(1)}(u_{1})L^{(2)}(u_{2})\widehat{R}_{m}.$
(4.28)

The relation (4.26) takes the from

$\displaystyle\widehat{R}_{n}\widehat{R}_{n}=1-u_{12}^{2}.$ (4.29)

######  Remark 4.6.

The operator (4.27) satisfies the braid relation

$\displaystyle\widehat{R}^{(23)}\widehat{R}^{(12)}\widehat{R}^{(23)}=\widehat{R}^{(12)}\widehat{R}^{(23)}\widehat{R}^{(12)}.$
(4.30)

It is obtained from the Yang–Baxter equation (4.25) via multiplication by
$P_{u_{2},u_{3}}P_{u_{1},u_{2}}P_{u_{2},u_{3}}=P_{u_{1},u_{2}}P_{u_{2},u_{3}}P_{u_{1},u_{2}}$
and (3.3) from the left. Due to the formula (4.29) this operator gives a
representation of the group ${\mathbf{S}}_{k}$ for an arbitrary $k$ after some
renormalization. Namely, the normalised rational $R$-matrix
$\overline{R}(u)=(1-u)^{-1}R(u)$ satisfies
$\overline{R}^{(21)}(u_{21})\overline{R}^{(12)}(u_{12})=1$ and the same Yang
Baxter equation (4.25). Hence the operator
$\widetilde{R}=P_{u_{1},u_{2}}P_{n}\widetilde{R}(u_{12})$ satisfies
$\widetilde{R}^{2}=1$ and the braid relation (4.30). This implies that the map
$\sigma_{a}\mapsto\widetilde{R}^{(a,a+1)}$ gives a representation of
${\mathbf{S}}_{k}$ on the space $({\mathbb{C}}^{n})^{\otimes
k}(u_{1},\dots,u_{k})$.

######  Proposition 4.7.

The operators

$\displaystyle\widehat{R}^{+}=\widehat{R}_{n+}:=\frac{1-u_{12}+\widehat{R}_{n}}{2},\qquad\widehat{R}^{-}=\widehat{R}_{n-}:=\frac{1+u_{12}-\widehat{R}_{n}}{2}$

are orthogonal idempotents dual to each other:

$\displaystyle\widehat{R}^{+}+\widehat{R}^{-}=1,\qquad\big{(}\widehat{R}^{+}\big{)}^{2}=\widehat{R}^{+},\qquad\big{(}\widehat{R}^{-}\big{)}^{2}=\widehat{R}^{-},\qquad\widehat{R}^{+}\widehat{R}^{-}=\widehat{R}^{-}\widehat{R}^{+}=0.$
(4.31)

###### Proof.

The first of (4.31) is obvious. By using (4.29) and
$\widehat{R}f(u_{1},u_{2})=f(u_{2},u_{1})\widehat{R}$ we obtain
$\big{(}\widehat{R}+1-u_{12}\big{)}\big{(}\widehat{R}-1-u_{12}\big{)}=0$. This
implies the rest of (4.31). ∎

Now consider

$\displaystyle\widehat{A}_{n}:=\widehat{R}_{n-}\cdot\frac{1}{u_{12}}=\frac{1+u_{12}^{-1}-P_{u_{1},u_{2}}P_{n}R(u_{12})u_{12}^{-1}}{2}=\frac{1+u_{12}^{-1}+P_{u_{1},u_{2}}\big{(}u_{12}^{-1}-P_{n}\big{)}}{2}.$
(4.32)

######  Lemma 4.8.

The operator $\widehat{A}_{n}$ is an idempotent acting on the space
${\mathbb{C}}^{n}\big{[}u_{1},u_{1}^{-1}\big{]}\otimes{\mathbb{C}}^{n}\big{[}u_{2},u_{2}^{-1}\big{]}$.
It preserves the subspaces
${\mathbb{C}}^{n}[u_{1}]\otimes{\mathbb{C}}^{n}[u_{2}]$ and
${\mathbb{C}}^{n}\big{[}u_{1}^{-1}\big{]}\otimes{\mathbb{C}}^{n}\big{[}u_{2}^{-1}\big{]}$.

###### Proof.

Let us rewrite (4.32) in the form

$\displaystyle\widehat{A}_{n}=\frac{1-P_{u_{1},u_{2}}P_{n}}{2}+\frac{1}{2(u_{1}-u_{2})}(1-P_{u_{1},u_{2}}).$

The first term is obviously preserves all tree spaces. Let us prove that so
does the second term. By acting by $(1-P_{u_{1},u_{2}})$ on a Laurent
polynomial
$p(u_{1},u_{2})\in{\mathbb{C}}\big{[}u_{1},u_{1}^{-1},u_{2},u_{2}^{-1}\big{]}$
we obtain the Laurent polynomial
$q(u_{1},u_{2})=p(u_{1},u_{2})-p(u_{2},u_{1})$. Since $q(u_{1},u_{1})=0$ we
have the formula $q(u_{1},u_{2})=(u_{1}-u_{2})r(u_{1},u_{2})$ for some
$r(u_{1},u_{2})\in{\mathbb{C}}\big{[}u_{1},u_{1}^{-1},u_{2},u_{2}^{-1}\big{]}$.
Hence
$\frac{1}{2(u_{1}-u_{2})}(1-P_{u_{1},u_{2}})p(u_{1},u_{2})=\frac{1}{2}r(u_{1},u_{2})$
is also a Laurent polynomial. If $p(u_{1},u_{2})\in{\mathbb{C}}[u_{1},u_{2}]$
then $r(u_{1},u_{2})\in{\mathbb{C}}[u_{1},u_{2}]$. Analogously, for a Laurent
polynomial $p(u_{1},u_{2})\in{\mathbb{C}}\big{[}u_{1}^{-1},u_{2}^{-1}\big{]}$
we obtain
$q(u_{1},u_{2})=p(u_{1},u_{2})-p(u_{2},u_{1})=\big{(}u_{1}^{-1}-u_{2}^{-1}\big{)}r(u_{1},u_{2})$
for some $r(u_{1},u_{2})\in{\mathbb{C}}\big{[}u_{1}^{-1},u_{2}^{-1}\big{]}$
and hence
$\frac{1}{2(u_{1}-u_{2})}(1-P_{u_{1},u_{2}})p(u_{1},u_{2})=-\frac{u_{1}^{-1}u_{2}^{-1}}{2}r(u_{1},u_{2})\in{\mathbb{C}}\big{[}u_{1}^{-1},u_{2}^{-1}\big{]}$.

By multiplying $2\widehat{A}_{n}=1+u_{12}^{-1}+u_{12}^{-1}\widehat{R}$ with
itself we obtain

$\displaystyle 4\widehat{A}_{n}^{2}$
$\displaystyle=\big{(}1+u_{12}^{-1}\big{)}^{2}+\big{(}1+u_{12}^{-1}\big{)}u_{12}^{-1}\widehat{R}+u_{12}^{-1}\widehat{R}\big{(}1+u_{12}^{-1}\big{)}+u_{12}^{-1}\widehat{R}u_{12}^{-1}\widehat{R}$
$\displaystyle=1+2u_{12}^{-1}+u_{12}^{-2}+\big{(}1+u_{12}^{-1}+1-u_{12}^{-1}\big{)}u_{12}^{-1}\widehat{R}-u_{12}^{-2}\widehat{R}^{2}.$

By taking into account (4.29) we obtain
$4\widehat{A}_{n}^{2}=2+2u_{12}^{-1}+2u_{12}^{-1}\widehat{R}=4\widehat{A}_{n}$.
∎

The basis of
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{[}u_{1},u_{1}^{-1},u_{2},u_{2}^{-1}\big{]}$
is $\big{(}e_{i}\otimes e_{j}u_{1}^{k}u_{2}^{l}\big{)}$, where
$k,l\in{\mathbb{Z}}$, $i,j=1,\dots,n$, and the completion with respect to this
basis is the space
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{[}\big{[}u_{1},u_{1}^{-1},u_{2},u_{2}^{-1}\big{]}\big{]}$.
Since $\widehat{A}_{n}$ preserves the space
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{[}u_{1},u_{1}^{-1},u_{2},u_{2}^{-1}\big{]}$
its matrix in this basis satisfies the column finiteness condition introduced
in Section 2.7. However it does not satisfies the row finiteness condition
since it can not be extended to the completed tensor product space
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{[}\big{[}u_{1},u_{1}^{-1},u_{2},u_{2}^{-1}\big{]}\big{]}$.
Explicitly these can be seen from the formulae

$\displaystyle 2\widehat{A}_{n}\big{(}e_{i}\otimes
e_{j}u_{1}^{k}u_{2}^{l}\big{)}=e_{i}\otimes
e_{j}u_{1}^{k}u_{2}^{l}-e_{j}\otimes e_{i}u_{1}^{k}u_{2}^{l}+e_{i}\otimes
e_{j}\frac{u_{1}^{k}u_{2}^{l}-u_{1}^{l}u_{2}^{k}}{u_{1}-u_{2}},$ (4.33)
$\displaystyle\frac{u_{1}^{k}u_{2}^{l}-u_{1}^{l}u_{2}^{k}}{u_{1}-u_{2}}=\begin{cases}-\sum_{m=k}^{l-1}u_{1}^{m}u_{2}^{k+l-1-m},&k<l,\\\
0,&k=l,\\\ \sum_{m=l}^{k-1}u_{1}^{m}u_{2}^{k+l-1-m},&k>l.\end{cases}$ (4.34)

For instance, for any $k\geqslant 1$ the vector
$\widehat{A}_{n}\big{(}e_{i}\otimes e_{j}u_{1}^{k}u_{2}^{1-k}\big{)}$ has a
non-zero coefficient at the term $e_{i}\otimes e_{j}u_{1}^{0}u_{2}^{0}$.

Consider the topology of the space $V={\mathbb{C}}^{n}\big{[}u,u^{-1}\big{]}$
defined by the neighbourhoods of $0$ of the form
$V_{r}=\big{\\{}\sum_{k=-r}^{N}t_{k}u^{k}\mid
N\geqslant-r,t_{k}\in{\mathbb{C}}^{n}\big{\\}}$. The completion of $V$ with
respect to this topology is the space
${\mathbb{C}}^{n}\big{(}\big{(}u^{-1}\big{)}\big{)}$. The corresponding
completion of the space ${\mathfrak{R}}\otimes V$ is
${\mathfrak{R}}\mathbin{\widehat{\otimes}}V={\mathfrak{R}}\big{(}\big{(}u^{-1}\big{)}\big{)}\otimes{\mathbb{C}}^{n}$.
The neighbourhoods $V_{r}\otimes V_{s}$ defines the topology of $V\otimes V$
which gives the completion
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{(}\big{(}u_{1}^{-1},u_{2}^{-1}\big{)}\big{)}$.
The operator $\widehat{A}_{n}\in\mathop{\mathrm{End}}\nolimits(V\otimes V)$ is
continuous with respect to this topology. In particular, it means that
$\widehat{A}_{n}$ is extended to the completion
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{(}\big{(}u_{1}^{-1},u_{2}^{-1}\big{)}\big{)}$.

Thus an $\big{(}\widehat{A}_{n},\widehat{A}_{m}\big{)}$-Manin operator is an
element $M$ of the space
$\mathop{\mathrm{Hom}}\nolimits\big{(}{\mathbb{C}}^{m}\big{[}\big{[}u,u^{-1}\big{]}\big{]},{\mathfrak{R}}\mathbin{\widehat{\otimes}}V\big{)}=\mathop{\mathrm{Hom}}\nolimits\big{(}{\mathbb{C}}^{m}\big{[}\big{[}u,u^{-1}\big{]}\big{]},{\mathfrak{R}}\big{(}\big{(}u^{-1}\big{)}\big{)}\otimes{\mathbb{C}}^{n}\big{)}$,
satisfying

$\displaystyle\widehat{A}_{n}M^{(1)}M^{(2)}=\widehat{A}_{n}M^{(1)}M^{(2)}\widehat{A}_{m}.$

Note also that the operators
$e^{a(\partial_{u_{1}}+\partial_{u_{2}})}=e^{a\partial_{u}}\otimes
e^{a\partial_{u}}$ commute with $\widehat{A}_{n}$. Hence if $M$ is an
$\big{(}\widehat{A}_{n},\widehat{A}_{m}\big{)}$-Manin operator then
$e^{a\partial_{u}}Me^{b\partial_{u}}$ is also an
$\big{(}\widehat{A}_{n},\widehat{A}_{m}\big{)}$-Manin operator for any
$a,b\in{\mathbb{C}}$ (this follows from Proposition 2.22 generalised to the
infinite-dimensional case).

######  Theorem 4.9.

Let
$L(u)\in{\mathfrak{R}}\big{(}\big{(}u^{-1}\big{)}\big{)}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$.
The matrix $L(u)$ is an $\big{(}\widehat{A}_{n},\widehat{A}_{m}\big{)}$-Manin
operator iff it satisfies $RLL$-relation (4.24).

###### Proof.

Remind first that the relation (4.24) is equivalent to (4.28). Let us multiply

$\displaystyle\widehat{A}_{n}L^{(1)}(u_{1})L^{(2)}(u_{2})=\widehat{A}_{n}L^{(1)}(u_{1})L^{(2)}(u_{2})\widehat{A}_{m}$

by $4u_{12}$ from the right and substitute
$2u_{12}\widehat{A}_{n}=1+u_{12}+\widehat{R}_{n}$,
$2\widehat{A}_{m}=1+u_{12}^{-1}+u_{12}^{-1}\widehat{R}_{m}$. This gives the
equivalent relation

$\displaystyle\big{(}u_{12}-u_{12}^{-1}\big{)}L^{(1)}(u_{1})L^{(2)}(u_{2})+u_{12}^{-1}\widehat{R}_{n}L^{(1)}(u_{1})L^{(2)}(u_{2})\widehat{R}_{m}$
$\displaystyle\qquad{}=\big{(}1+u_{12}^{-1}\big{)}L^{(1)}(u_{1})L^{(2)}(u_{2})\widehat{R}_{m}-\big{(}1+u_{12}^{-1}\big{)}\widehat{R}_{n}L^{(1)}(u_{1})L^{(2)}(u_{2}).$
(4.35)

The left hand side of (4.35) does not change at the conjugation $T\mapsto
u_{12}Tu_{12}^{-1}$ while the right hand side changes the sign. This means
that (4.35) is valid iff the both hand sides vanish. Due to (4.29) the
vanishing of each hand side of (4.35) is equivalent to (4.28). ∎

Let $L(u)$ be an $\big{(}\widehat{A}_{n},\widehat{A}_{m}\big{)}$-Manin
operator. Then $M=L(u+a)e^{b\partial_{u}}$ is also an
$\big{(}\widehat{A}_{n},\widehat{A}_{m}\big{)}$-Manin operator for any
$a,b\in{\mathbb{C}}$. In particular, the $A_{n}$-Manin matrix
$M=L(u)e^{-\partial_{u}}$ considered in Section 3.1 is an
$\widehat{A}_{n}$-Manin operator.

######  Remark 4.10.

One can renormalise the matrix
$L(u)\in{\mathfrak{R}}\big{(}\big{(}u^{-1}\big{)}\big{)}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
by multiplying it by a function in $u$. Such renormalization does not violate
the $RLL$-relation (4.24). Hence one can suppose that
$L(u)\in{\mathfrak{R}}[[u^{-1}]]\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
by multiplying $L(u)$ by $u^{k}$ for big enough $k$ if needed. A matrix
$L(u)\in{\mathfrak{R}}\big{[}\big{[}u^{-1}\big{]}\big{]}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
satisfying $RLL$-relation is exactly an
$\big{(}\big{(}\widehat{A}_{n}\big{)}^{-}_{\rm
res},\big{(}\widehat{A}_{m}\big{)}^{-}_{\rm res}\big{)}$-Manin operator, where
$\big{(}\widehat{A}_{n}\big{)}^{-}_{\rm res}$ is the restriction of
$\widehat{A}_{n}$ to
${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}\big{[}u_{1}^{-1},u_{2}^{-1}\big{]}$.
We see from the formulae (4.33), (4.34) that
$\big{(}\widehat{A}_{n}\big{)}^{-}_{\rm res}$ satisfies the row finiteness
condition. In particular, we have the right quantum algebra
${\mathfrak{U}}_{(\widehat{A}_{n})^{-}_{\rm res}}$ and Theorem 4.9 implies
that the Yangian $Y(\mathfrak{gl}_{n})$ is a factor algebra of
${\mathfrak{U}}_{(\widehat{A}_{n})^{-}_{\rm res}}$.

######  Remark 4.11.

The facts described in this Section works also for a matrix
$L(u)\in{\mathfrak{R}}((u))\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
since we can consider another completion of
${\mathfrak{R}}\big{[}u,u^{-1}\big{]}$. Again, one can suppose that
$L(u)\in{\mathfrak{R}}[[u]]\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
making a renormalization if needed. A matrix
$L(u)\in{\mathfrak{R}}[[u]]\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
satisfying $RLL$-relation is an $\big{(}\big{(}\widehat{A}_{n}\big{)}^{+}_{\rm
res},\big{(}\widehat{A}_{m}\big{)}^{+}_{\rm res}\big{)}$-Manin operator, where
$\big{(}\widehat{A}_{n}\big{)}^{+}_{\rm res}$ is the restriction of
$\widehat{A}_{n}$ to ${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}[u_{1},u_{2}]$.

##  5 Minors of Manin matrices

The notion of minor (determinant of a submatrix) is an important tool in the
classical matrix theory. It can be interpreted in terms of the Grassmann
algebra as some coefficients. This gives minors of $(A_{n},A_{m})$-Manin
matrices which we defined in Section 3.1 as column determinants of
submatrices. The same can be done for $q$\- and
$(\widehat{q},\widehat{p})$-Manin matrices by considering the quadratic
algebras $\Xi_{A}({\mathbb{C}})$ for $A=A_{n}^{q}$ and $A_{\widehat{q}}$
respectively. There is a dual notion of minors corresponding to the quadratic
algebras ${\mathfrak{X}}_{A}({\mathbb{C}})$. In the case of the polynomial
algebras these dual minors are written via permanent.

For a general $(A,B)$-Manin matrix $M$ we define two types of minors
corresponding to the homomorphisms
$f_{M}\colon{\mathfrak{X}}_{A}({\mathbb{C}})\to{\mathfrak{X}}_{B}({\mathfrak{R}})$
and $f^{M}\colon\Xi_{B}({\mathbb{C}})\to\Xi_{A}({\mathfrak{R}})$. In fact
these minors give the graded components of these homomorphisms. As usual
minors they have a good behaviour at the multiplication of Manin matrices and
under permutations of rows and columns. In future works we hope to find more
properties of these minors by generalising the properties of the usual minors.

###  5.1 The $\boldsymbol{q}$-minors and permanents

First we remind that the $q$-determinant (3.15) is the coefficient of
proportionality for the product of the $q$-Grassmann variables [6] (see also
the formula (3.27) for the $\widehat{q}$-version). Namely, let
$\psi_{1},\dots,\psi_{n}$ be the generators of
$\Xi_{A_{n}^{q}}({\mathbb{C}})$, $M$ be an $n\times n$ matrix over an algebra
${\mathfrak{R}}$ and $\phi_{j}=\sum_{i=1}^{n}\psi_{i}M^{i}_{j}$, where
$j=1,\dots,n$, then

$\displaystyle\phi_{1}\phi_{2}\cdots\phi_{n}=\mathop{\mathrm{det}}\nolimits_{q}(M)\psi_{1}\psi_{2}\cdots\psi_{n}.$

More generally, let $M$ be $n\times m$ matrix over ${\mathfrak{R}}$ and
$\phi_{j}=\sum_{i=1}^{n}\psi_{i}M^{i}_{j}$, where $j=1,\dots,m$. For two
$k$-tuples of indices $I=(i_{1},\dots,i_{k})$ and $J=(j_{1},\dots,j_{k})$ we
denote by $M_{IJ}$ the $k\times k$ matrix over ${\mathfrak{R}}$ with entries

$\displaystyle(M_{IJ})_{ab}=M^{i_{a}}_{j_{b}},\qquad a,b=1,\dots,k,$

where we suppose $1\leqslant i_{a}\leqslant n$ and $1\leqslant j_{b}\leqslant
m$. If $i_{1}<\dots<i_{k}$ and $j_{1}<\dots<j_{k}$ then $M_{IJ}$ is a $k\times
k$ submatrix of $M$ and $\mathop{\mathrm{det}}\nolimits_{q}(M_{IJ})$ is a
$q$-analogue of minor defined in Section 3.1. The $q$-determinants
$\mathop{\mathrm{det}}\nolimits_{q}(M_{IJ})$ are the coefficients in the
decomposition

$\displaystyle\phi_{j_{1}}\phi_{j_{2}}\cdots\phi_{j_{k}}=\sum_{I=(i_{1}<\dots<i_{k})}\mathop{\mathrm{det}}\nolimits_{q}(M_{IJ})\psi_{i_{1}}\psi_{i_{2}}\cdots\psi_{i_{k}},$
(5.1)

where the sum is taken over $k$-tuples $I=(i_{1},\dots,i_{k})$ such that
$1\leqslant i_{1}<\dots<i_{k}\leqslant n$.

If $M$ is a $q$-Manin matrix then $\phi_{1},\dots,\phi_{m}$ are also
$q$-anticommuting and hence the elements
$\phi_{j_{1}}\phi_{j_{2}}\cdots\phi_{j_{k}}$ with $1\leqslant
j_{1}<\dots<j_{k}\leqslant m$ span the subalgebra of
$\Xi_{A_{n}^{q}}({\mathfrak{R}})$ generated by $\phi_{1},\dots,\phi_{m}$. This
means that the $q$-determinants of submatrices of $M$ are enough to describe
the decompositions (5.1) in this case.

A dual notion to column determinant is row permanent. Recall that the
permanent of an $n\times n$ matrix $M$ is

$\displaystyle\mathop{\mathrm{perm}}\nolimits(M)=\sum_{\sigma\in{\mathbf{S}}_{n}}M^{1}_{\sigma(1)}\cdots
M^{n}_{\sigma(n)}.$ (5.2)

This is the same expression as for the determinant but without the factors
$(-1)^{\sigma}$. If $M$ is over non-commutative algebra the factors in this
expression should be placed in a certain way. The formula (5.2) defines the
row permanent of $M$ (see [4, 5]). It is invariant under a permutation of
columns:
$\mathop{\mathrm{perm}}\nolimits(_{\tau}M)=\mathop{\mathrm{perm}}\nolimits(M)$.
Contrary to the determinant it does not change sign, so, in particular, the
permanent of a matrix with coinciding rows can be non-zero. If $M$ is an
$A_{n}$-Manin matrix, then the permanent is invariant under a permutation of
rows:
$\mathop{\mathrm{perm}}\nolimits(^{\tau}M)=\mathop{\mathrm{perm}}\nolimits(M)$.

Now consider the analogues decompositions of products of
$y^{i}=\sum_{j=1}^{m}M^{i}_{j}x^{j}$, where $x^{1},\dots,x^{m}$ are generators
of ${\mathfrak{X}}_{A_{m}}({\mathbb{C}})$ (for simplicity we consider the case
$q=1$). Define an action of the group ${\mathbf{S}}_{k}$ on $k$-tuples by the
formula
$\sigma(j_{1},\dots,j_{k})=\big{(}j_{\sigma^{-1}(1)},\dots,j_{\sigma^{-1}(k)}\big{)}$.
For any tuple $I=(i_{1},\dots,i_{k})$ we have (cf. [5])

$\displaystyle y^{i_{1}}\cdots y^{i_{k}}$
$\displaystyle=\sum_{l_{1},\dots,l_{k}=1}^{m}M^{i_{1}}_{l_{1}}\cdots
M^{i_{k}}_{l_{k}}x^{l_{1}}\cdots
x^{l_{k}}=\sum_{J=(j_{1}\leqslant\cdots\leqslant j_{k})}x^{j_{1}}\cdots
x^{j_{k}}\sum_{(l_{1},\dots,l_{k})\in{\mathbf{S}}_{k}J}M^{i_{1}}_{l_{1}}\cdots
M^{i_{k}}_{l_{k}}$ $\displaystyle=\sum_{J=(j_{1}\leqslant\cdots\leqslant
j_{k})}x^{j_{1}}\cdots
x^{j_{k}}\frac{1}{|({\mathbf{S}}_{k})_{J}|}\mathop{\mathrm{perm}}\nolimits(M_{IJ}),$

where $|({\mathbf{S}}_{k})_{J}|$ is the order of the stabiliser subgroup
$({\mathbf{S}}_{k})_{J}=\\{\sigma\in{\mathbf{S}}_{k}\mid\sigma J=J\\}$. For a
fixed $J=(j_{1},\dots,j_{k})$ let $\nu_{j}=|\\{a\mid j_{a}=j\\}|$, then
$|({\mathbf{S}}_{k})_{J}|=\nu_{1}!\nu_{2}!\cdots\nu_{m}!$. The normalised
permanents
$\frac{1}{|({\mathbf{S}}_{k})_{J}|}\mathop{\mathrm{perm}}\nolimits(M_{IJ})=\frac{1}{\nu_{1}!\nu_{2}!\cdots\nu_{m}!}\mathop{\mathrm{perm}}\nolimits(M_{IJ})$
are analogues of the minors for the polynomial algebra
${\mathfrak{X}}_{A_{m}}({\mathbb{C}})={\mathbb{C}}\big{[}x^{1},\dots,x^{m}\big{]}$.

For a general idempotent $A$ we will define minors of two types and they will
be coefficients of decomposition of $y^{i_{1}}\cdots y^{i_{k}}$ and
$\phi_{i_{1}}\cdots\phi_{i_{k}}$ into sums of $x^{j_{1}}\cdots x^{j_{k}}$ and
$\psi_{j_{1}}\cdots\psi_{j_{k}}$ respectively (up to a factor).

###  5.2 Dual quadratic algebras and their pairings

Consider the question of decomposition coefficients in general settings. Let
$V$ and $W$ be vector spaces with a non-degenerate pairing
$\langle\cdot,\cdot\rangle\colon V\times W\to{\mathbb{C}}$. Let them have dual
bases $(v^{i})$ and $(w_{i})$, $\langle v^{i},w_{j}\rangle=\delta^{i}_{j}$.
Let ${\mathfrak{R}}$ be an algebra, then ${\mathfrak{R}}\otimes V$ is a free
${\mathfrak{R}}$-module with the basis $(v^{i})$. Consider a decomposition
$\alpha=\sum_{i}\alpha_{i}v^{i}$ of an element $\alpha\in{\mathfrak{R}}\otimes
V$ in the basis $(v^{i})$. The coefficients $\alpha_{i}\in{\mathfrak{R}}$ can
be calculated via the pairing: $\alpha_{i}=\langle\alpha,w_{i}\rangle$, where
$\langle r\otimes v,w\rangle:=r\cdot\langle v,w\rangle$ $\forall
r\in{\mathfrak{R}},\,v\in V$, $w\in W$.

More generally, let $A\in\mathop{\mathrm{End}}\nolimits(V)$ and
$A^{*}\in\mathop{\mathrm{End}}\nolimits(W)$ be adjoint operators: $\langle
Av,w\rangle=\langle v,A^{*}w\rangle$ $\forall v\in V,\,w\in W$. They act on
basis vectors as $Av^{i}=\sum_{j}A^{i}_{j}v^{j}$,
$A^{*}w_{j}=\sum_{i}A_{j}^{i}w_{i}$, $A^{i}_{j}\in{\mathbb{C}}$ (the both sums
are finite). Suppose they are idempotents:
$\sum_{j}A^{i}_{j}A^{j}_{k}=A^{i}_{k}$. Consider the subspaces
$\overline{V}=AV$ and $\overline{W}=A^{*}W$. These subspaces are spanned by
the vectors $\overline{v}^{i}=Av^{i}$ and $\overline{w}_{i}=A^{*}w_{i}$
respectively. We have
$\langle\overline{v}^{i},\overline{w}_{j}\rangle=A^{i}_{j}$. A decomposition
$\alpha=\sum_{i}\alpha_{i}\overline{v}^{i}$ of an element
$\alpha\in{\mathfrak{R}}\otimes\overline{V}$ is not unique since
$\overline{v}^{i}$ are not linearly independent in general. However we can fix
the coefficients $\alpha_{i}$ by imposing some “symmetry” conditions.

######  Proposition 5.1.

  1. $1.$

The restriction of the non-degenerate pairing $\langle\cdot,\cdot\rangle\colon
V\times W\to{\mathbb{C}}$ to $\overline{V}\times\overline{W}$ is also non-
degenerate.

  2. $2.$

For any $\alpha\in{\mathfrak{R}}\otimes\overline{V}$ there is a unique
$($finite$)$ sequence $(\alpha_{i})$ such that
$\alpha=\sum_{i}\alpha_{i}\overline{v}^{i}$ and
$\sum_{i}A_{j}^{i}\alpha_{i}=\alpha_{j}$. The coefficients
$\alpha_{i}\in{\mathfrak{R}}$ fixed by these conditions can be found by the
formula $\alpha_{i}=\langle\alpha,\overline{w}_{i}\rangle$.

###### Proof.

(1) If $w\in\overline{W}$ and $\langle v,w\rangle=0$ for all
$v\in\overline{V}$, then for any $v\in V$ we have $\langle v,w\rangle=\langle
v,A^{*}w\rangle=\langle Av,w\rangle=0$ and hence $w=0$. (2) Since
$\alpha\in{\mathfrak{R}}\otimes\overline{V}$ we have $A\alpha=\alpha$, where
the action of $A$ is extended on ${\mathfrak{R}}\otimes V$ by the formula
$A(r\otimes v)=r\otimes Av$. Let
$\alpha_{i}=\langle\alpha,\overline{w}_{i}\rangle$, then
$\alpha_{i}=\langle\alpha,A^{*}w_{i}\rangle=\langle
A\alpha,w_{i}\rangle=\langle\alpha,w_{i}\rangle$. Hence
$\sum_{i}A_{j}^{i}\alpha_{i}=\sum_{i}A_{j}^{i}\langle\alpha,w_{i}\rangle=\langle\alpha,\overline{w}_{j}\rangle=\alpha_{j}$.
Moreover,
$\langle\alpha-\sum_{i}\alpha_{i}\overline{v}^{i},w_{j}\rangle=\alpha_{j}-\sum_{i}A_{j}^{i}\alpha_{i}=0$,
so that $\alpha-\sum_{i}\alpha_{i}\overline{v}^{i}=0$. To show the uniqueness
suppose that $\alpha=\sum_{i}\alpha_{i}\overline{v}^{i}$ for some
$\alpha_{i}\in{\mathfrak{R}}$ such that
$\sum_{i}A_{j}^{i}\alpha_{i}=\alpha_{j}$, then we have
$\langle\alpha,\overline{w}_{j}\rangle=\sum_{i}\alpha_{i}A^{i}_{j}=\alpha_{j}$.
∎

Let
$A\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$
be an idempotent. Consider the corresponding quadratic algebra
${\mathfrak{X}}_{A}({\mathbb{C}})$. We need to define a vector space
${{\mathfrak{X}}}^{*}_{A}({\mathbb{C}})$ and a non-degenerate pairing with
${\mathfrak{X}}_{A}({\mathbb{C}})$. This implies that
${{\mathfrak{X}}}^{*}_{A}({\mathbb{C}})$ is a graded vector space and the
pairing respects the grading.

Define the algebra ${{\mathfrak{X}}}^{*}_{A}({\mathbb{C}})$ as the graded
algebra generated by the elements $x_{1},\dots,x_{n}$ with the “dual”
quadratic commutation relations: $\sum_{i,j=1}^{n}x_{i}x_{j}A^{ij}_{kl}=0$,
that is

$\displaystyle(X^{*}\otimes X^{*})A=0,$ (5.3)

where $X^{*}=\sum_{i=1}^{n}x_{i}e^{i}$. We have the algebra isomorphisms
${{\mathfrak{X}}}^{*}_{A}({\mathbb{C}})=\Xi_{1-A}({\mathbb{C}})={\mathfrak{X}}_{A^{\top}}({\mathbb{C}})$.
Let us introduce a pairing
$\langle\cdot,\cdot\rangle\colon{{\mathfrak{X}}}_{A}({\mathbb{C}})\times{{\mathfrak{X}}}^{*}_{A}({\mathbb{C}})\to{\mathbb{C}}$
respecting the grading, i.e., the product $\langle x^{i_{1}}\cdots
x^{i_{l}},x_{j_{1}}\cdots x_{j_{k}}\rangle$ vanishes if $k\neq l$. The pairing
for the elements of the degree $0$ and $1$ are defined as follows: $\langle
1,1\rangle=1$ and $\langle x^{i},x_{j}\rangle=\delta^{i}_{j}$. For the higher
degree elements the pairing has the form

$\displaystyle\langle x^{i_{1}}\cdots x^{i_{k}},x_{j_{1}}\cdots
x_{j_{k}}\rangle=S^{i_{1}\dots i_{k}}_{j_{1}\dots j_{k}},$ (5.4)

for some $S^{i_{1}\dots i_{k}}_{j_{1}\dots j_{k}}\in{\mathbb{C}}$. The
commutation relations (2.3) and (5.3) implies that

$\displaystyle\sum_{i_{a},i_{a+1}=1}^{n}A^{lm}_{i_{a}i_{a+1}}S^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}=0,\qquad\sum_{j_{s},j_{s+1}=1}^{n}S^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}A^{j_{a}j_{a+1}}_{lm}=0,$ (5.5)

for all $a=1,\dots,k-1$ and $l,m=1,\dots,n$. Under these conditions the
formula (5.4) correctly defines a pairing
$\langle\cdot,\cdot\rangle\colon{\mathfrak{X}}_{A}({\mathbb{C}})\times{\mathfrak{X}}^{*}_{A}({\mathbb{C}})\to{\mathbb{C}}$
that respects the grading.

######  Remark 5.2.

In contrast to the situation described in Remark 2.32 one can not correctly
define an operation on quadratic algebras by mapping an algebra
${\mathfrak{X}}_{A}({\mathbb{C}})$ to ${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$.
It happens since the equality
${\mathfrak{X}}_{A}({\mathbb{C}})={\mathfrak{X}}_{A^{\prime}}({\mathbb{C}})$
does not imply an isomorphism of ${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$ and
${\mathfrak{X}}^{*}_{A^{\prime}}({\mathbb{C}})$ (see Section 5.6 for details).

To write the formulae (5.4), (5.5) in a matrix form we introduce some
conventions. Let $V$ and $W$ be vector spaces and
$W^{*}=\mathop{\mathrm{Hom}}\nolimits(W,{\mathbb{C}})$. The product $\pi\xi$
of elements $\pi\in V$ and $\xi\in W^{*}$ is usually identified with the
linear operator $W\to V$ acting as $(\pi\xi)(w)=\xi(w)\cdot\pi$, $w\in W$. For
example,
$e_{i}e^{j}=E_{i}{}^{j}\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n})$.
More generally, $e_{i_{1}\dots i_{k}}e^{j_{1}\dots
j_{k}}=E_{i_{1}}{}^{j_{1}}\otimes\cdots\otimes E_{i_{k}}{}^{j_{k}}$, where
$e^{j_{1}\dots j_{k}}=e^{j_{1}}\otimes\cdots\otimes e^{j_{k}}$ and
$e_{i_{1}\dots i_{k}}=e_{i_{1}}\otimes\cdots\otimes e_{i_{k}}$.

Let us introduce the notation $\langle\alpha,\beta\rangle$ for
$\alpha\in{\mathfrak{X}}_{A}({\mathbb{C}})\otimes V$ and
$\beta\in{\mathfrak{X}}^{*}_{A}({\mathbb{C}})\otimes W^{*}$. The pairing acts
on the first tensor factors while in the second factor the elements are
multiplied as above: $\langle u\pi,v\xi\rangle=\langle
u,v\rangle\pi\xi\in\mathop{\mathrm{Hom}}\nolimits(W,V)$, where
$u\in{\mathfrak{X}}_{A}({\mathbb{C}})$,
$v\in{\mathfrak{X}}^{*}({\mathbb{C}})$, $\pi\in V$ and $\xi\in W^{*}$. In
particular,

$\bigg{\langle}\sum\limits_{i_{1},\dots,i_{k}}u_{i_{1}\dots
i_{k}}e_{i_{1}\dots i_{k}},\sum\limits_{j_{1},\dots,j_{k}}v_{j_{1}\dots
j_{k}}e^{j_{1}\dots
j_{k}}\bigg{\rangle}=\sum\limits_{i_{1},\dots,i_{k},j_{1}\dots,j_{k}}\langle
u_{i_{1}\dots i_{k}},v_{j_{1}\dots j_{k}}\rangle
E_{i_{1}}{}^{j_{1}}\otimes\cdots\otimes E_{i_{k}}{}^{j_{k}}.$

By using this notation we can write the pairing of degree $1$ elements in
matrix form:

$\displaystyle\langle X,X^{*}\rangle=\langle
x^{i}e_{i},x_{j}e^{j}\rangle=\delta^{i}_{j}E_{i}{}^{j}=1.$

Consider the operators
$S_{(k)}=\sum_{i_{1},\dots,i_{k},j_{1},\dots,j_{k}}S^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}E_{i_{1}}{}^{j_{1}}\otimes\cdots\otimes
E_{i_{k}}{}^{j_{k}}\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$. For $k=1$ the operator $S_{(1)}$ is the $n\times n$ unit matrix.
The formula (5.4) in matrix form reads

$\displaystyle\langle X\otimes\cdots\otimes X,X^{*}\otimes\cdots\otimes
X^{*}\rangle=S_{(k)}$ (5.6)

(here and below dots mean that we have $k$ tensor factors). The conditions
(5.5) is written as

$\displaystyle A^{(a,a+1)}S_{(k)}=S_{(k)}A^{(a,a+1)}=0,$ (5.7)

$a=1,\dots,k-1$. In terms of $P=1-2A$ we can rewrite (5.7) in the form

$\displaystyle P^{(a,a+1)}S_{(k)}=S_{(k)}P^{(a,a+1)}=S_{(k)}.$

Analogously, one can define the algebra $\Xi^{*}_{A}({\mathbb{C}})$ generated
by the elements $\psi^{1},\dots,\psi^{n}$ over ${\mathbb{C}}$ with the
commutation relations
$\psi^{i}\psi^{j}=\sum_{k,l}A^{ij}_{kl}\psi^{k}\psi^{l}$, i.e.,

$\displaystyle(1-A)(\Psi^{*}\otimes\Psi^{*})=0,$

where $\Psi^{*}=\sum\limits_{i=1}^{n}\psi^{i}e_{i}$. The pairing
$\langle\cdot,\cdot\rangle\colon{\Xi}_{A}({\mathbb{C}})\times{\Xi}^{*}_{A}({\mathbb{C}})\to{\mathbb{C}}$
is defined by the formula

$\displaystyle\langle\psi^{i_{1}}\cdots\psi^{i_{l}},\psi_{j_{1}}\cdots\psi_{j_{k}}\rangle=\delta_{kl}A^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}},$ (5.8)

where $A^{i_{1}\dots i_{k}}_{j_{1}\dots j_{k}}\in{\mathbb{C}}$,
$A^{i}_{j}=\delta^{i}_{j}$, $A^{\varnothing}_{\varnothing}=1$. The formula
(5.8) for $k=l$ can be written as

$\displaystyle\langle\Psi^{*}\otimes\cdots\otimes\Psi^{*},\Psi\otimes\cdots\otimes\Psi\rangle=A_{(k)},$
(5.9)

where $A_{(k)}=\sum\limits_{i_{1},\dots,i_{k},j_{1},\dots,j_{k}}A^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}E_{i_{1}}{}^{j_{1}}\otimes\cdots\otimes
E_{i_{k}}{}^{j_{k}}$, $A_{(1)}=1$. Again, we should require

$\displaystyle S^{(a,a+1)}A_{(k)}=A_{(k)}S^{(a,a+1)}=0,$

where $a=1,\dots,k-1$, $S=1-A$, or, equivalently,

$\displaystyle P^{(a,a+1)}A_{(k)}=A_{(k)}P^{(a,a+1)}=-A_{(k)}.$

######  Lemma 5.3.

Let $V$ and $W$ be vector spaces. For any operators
$T\in\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k},V\big{)}$ and
$\widetilde{T}\in\mathop{\mathrm{Hom}}\nolimits\big{(}W,({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ we have

$\displaystyle\big{\langle}T(X\otimes\cdots\otimes
X),(X^{*}\otimes\cdots\otimes
X^{*})\widetilde{T}\big{\rangle}=TS_{(k)}\widetilde{T},$ (5.10)
$\displaystyle\big{\langle}T(\Psi^{*}\otimes\cdots\otimes\Psi^{*}),(\Psi\otimes\cdots\otimes\Psi)\widetilde{T}\big{\rangle}=TA_{(k)}\widetilde{T}$
(5.11)

$($we understand $T(X\otimes\cdots\otimes X)$, $(X^{*}\otimes\cdots\otimes
X^{*})\widetilde{T}$, $T(\Psi^{*}\otimes\cdots\otimes\Psi^{*})$ and
$(\Psi\otimes\cdots\otimes\Psi)\widetilde{T}$ as elements of
${\mathfrak{X}}_{A}({\mathbb{C}})\otimes V$,
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})\otimes W^{*}$,
$\Xi^{*}_{A}({\mathbb{C}})\otimes V$ and $\Xi_{A}({\mathbb{C}})\otimes W^{*}$
respectively$)$.

###### Proof.

By substituting $T(X\otimes\cdots\otimes
X)=\sum\limits_{i_{1},\dots,i_{k}}x^{i_{1}}\cdots x^{i_{k}}Te_{i_{1}\dots
i_{k}}$ and $(X^{*}\otimes\cdots\otimes
X^{*})\widetilde{T}=\sum\limits_{j_{1},\dots,j_{k}}x_{j_{1}}\cdots
x_{j_{k}}e^{j_{1}\dots j_{k}}\widetilde{T}$ we derive

$\displaystyle\big{\langle}T(X\otimes\cdots\otimes
X),(X^{*}\otimes\cdots\otimes X^{*})\widetilde{T}\big{\rangle}$
$\displaystyle\qquad{}=\sum_{i_{1},\dots,i_{k},j_{1},\dots,j_{k}}\langle
x^{i_{1}}\otimes\cdots\otimes x^{i_{k}},x_{j_{1}}\otimes\cdots\otimes
x_{j_{k}}\rangle Te_{i_{1}\dots i_{k}}e^{j_{1}\dots j_{k}}\widetilde{T}$
$\displaystyle\qquad{}=\sum_{i_{1},\dots,i_{k},j_{1},\dots,j_{k}}S^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}T(E_{i_{1}}{}^{j_{1}}\otimes\cdots\otimes
E_{i_{k}}{}^{j_{k}})\widetilde{T}=TS_{(k)}\widetilde{T}.$

The formula (5.11) are proved in a similar way (it can be considered as the
formula (5.10) for the algebras
$\Xi_{A}({\mathbb{C}})={{\mathfrak{X}}}^{*}_{1-A}({\mathbb{C}})$ and
${\Xi}^{*}_{A}({\mathbb{C}})={\mathfrak{X}}_{1-A}({\mathbb{C}})$). ∎

######  Corollary 5.4.

Let $T\in\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k},V\big{)}$ and
$\widetilde{T}\in\mathop{\mathrm{Hom}}\nolimits\big{(}W,({\mathbb{C}}^{n})^{\otimes
k}\big{)}$.

  * $(a)$

If $T(X\otimes\cdots\otimes X)=0$ then $TS_{(k)}=0$.

  * $(b)$

If $(X^{*}\otimes\cdots\otimes X^{*})\widetilde{T}=0$ then
$S_{(k)}\widetilde{T}=0$.

  * $(c)$

If $(\Psi\otimes\cdots\otimes\Psi)\widetilde{T}=0$ then
$A_{(k)}\widetilde{T}=0$.

  * $(d)$

If $T(\Psi^{*}\otimes\cdots\otimes\Psi^{*})=0$ then $TA_{(k)}=0$.

In particular, these statements are valid for
$T,\widetilde{T}\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ and for $T\in\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}$,
$\widetilde{T}\in({\mathbb{C}}^{n})^{\otimes k}$.

###### Proof.

The statement $(a)$ are obtained from (5.10) in the case
$W=({\mathbb{C}}^{n})^{\otimes k}$,
$\widetilde{T}=1\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$. For $V=({\mathbb{C}}^{n})^{\otimes k}$,
$T=1\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ we obtain $(b)$. The statements $(c)$ and $(d)$ are obtained from
(5.11) in the same way. In the cases $V=W=({\mathbb{C}}^{n})^{\otimes k}$ and
$V=W={\mathbb{C}}$ we obtain
$\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k},V\big{)}=\mathop{\mathrm{Hom}}\nolimits\big{(}W,({\mathbb{C}}^{n})^{\otimes
k}\big{)}=\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k})\big{)}$ and
$\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k},V\big{)}=\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}$,
$\mathop{\mathrm{Hom}}\nolimits\big{(}W,({\mathbb{C}}^{n})^{\otimes
k}\big{)}=({\mathbb{C}}^{n})^{\otimes k}$ respectively. ∎

###  5.3 Pairing operators

Let
$A\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$
be an idempotent and $S=1-A$. We formulate some conditions on the operators
(5.6) and (5.9) that guarantee non-degeneracy of the corresponding pairings.

######  Definition 5.5.

Operators
$S_{(k)},\,A_{(k)}\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ are called pairing operators (for the idempotent $A$) if they
satisfy the following conditions:

$\displaystyle A^{(a,a+1)}S_{(k)}=S_{(k)}A^{(a,a+1)}=0,$ (5.12)
$\displaystyle(X^{*}\otimes\cdots\otimes
X^{*})S_{(k)}=(X^{*}\otimes\cdots\otimes X^{*}),$ (5.13) $\displaystyle
S_{(k)}(X\otimes\cdots\otimes X)=(X\otimes\cdots\otimes X),$ (5.14)
$\displaystyle S^{(a,a+1)}A_{(k)}=A_{(k)}S^{(a,a+1)}=0,$ (5.15)
$\displaystyle(\Psi\otimes\cdots\otimes\Psi)A_{(k)}=(\Psi\otimes\cdots\otimes\Psi),$
(5.16) $\displaystyle
A_{(k)}(\Psi^{*}\otimes\cdots\otimes\Psi^{*})=(\Psi^{*}\otimes\cdots\otimes\Psi^{*}).$
(5.17)

We call them the $k$-th $S$-operator and the $k$-th $A$-operator.

The conditions (5.14) and (5.16) for $k=1$ implies that $S_{(1)}=A_{(1)}=1$.
For $k=2$ the equations (5.12)–(5.14) and (5.15)–(5.17) have the solutions
$S_{(k)}=S$ and $S_{(k)}=A$ respectively. Let us prove the uniqueness of
solutions of these equations for any $k$ (we do not prove their existence for
$k>2$ in a general case).

######  Proposition 5.6.

The pairing operators are unique. In particular, $S_{(2)}=S$ and $A_{(2)}=A$.

###### Proof.

Let $S^{\prime}_{(k)}$ and $\bar{S}_{(k)}$ be $k$-th $S$-operators for the
same idempotent $A$. By applying the part $(a)$ of Corollary 5.4 for
$S_{(k)}=S^{\prime}_{(k)}$, $T=1-\bar{S}_{(k)}$ and the part $(b)$ of
Corollary 5.4 for $S_{(k)}=\bar{S}_{(k)}$, $T=1-S^{\prime}_{(k)}$ we obtain
$S^{\prime}_{(k)}=\bar{S}_{(k)}S^{\prime}_{(k)}$ and
$\bar{S}_{(k)}=\bar{S}_{(k)}S^{\prime}_{(k)}$. Hence we obtain
$S^{\prime}_{(k)}=\bar{S}_{(k)}$. The uniqueness of the $A$-operators follows
similarly from the parts $(c)$ and $(d)$ of Corollary 5.4. ∎

######  Proposition 5.7.

Pairing operators $S_{(k)}$ and $A_{(k)}$ are idempotents; they are orthogonal
for $k\geqslant 2$:

$\displaystyle S_{(k)}^{2}=S_{(k)},\qquad A_{(k)}^{2}=A_{(k)},\qquad
S_{(k)}A_{(k)}=A_{(k)}S_{(k)}=0.$ (5.18)
$\displaystyle(X^{*}\otimes\cdots\otimes X^{*})A_{(k)}=0,\qquad
A_{(k)}(X\otimes\cdots\otimes X)=0,$ (5.19)
$\displaystyle(\Psi\otimes\cdots\otimes\Psi)S_{(k)}=0,\qquad
S_{(k)}(\Psi^{*}\otimes\cdots\otimes\Psi^{*})=0.$ (5.20)

###### Proof.

Due to Corollary 5.4 the formula (5.14) gives $S_{(k)}^{2}=S_{(k)}$, while the
formula (5.16) implies $A_{(k)}^{2}=A_{(k)}$. Further, from (5.12) and (5.15)
we derive $S_{(k)}A_{(k)}=S_{(k)}P^{(a,a+1)}A_{(k)}=-S_{(k)}A_{(k)}$ and
$A_{(k)}S_{(k)}=A_{(k)}P^{(a,a+1)}S_{(k)}=-A_{(k)}S_{(k)}$. The formulae
(5.19) and (5.20) follows from the orthogonality and (5.13), (5.14), (5.16),
(5.17). For example, the relation (5.14) implies
$A_{(k)}(X\otimes\cdots\otimes X)=A_{(k)}S_{(k)}(X\otimes\cdots\otimes X)=0$.
∎

Below we suppose that $S_{(k)}$ and $A_{(k)}$ are pairing operators for an
idempotent $A$. Note that in general the sum $S_{(k)}+A_{(k)}$ does not
coincide with the identity operator.

The following formulae generalise the equalities (5.18). They are proved in
the same way.

######  Proposition 5.8.

If $\ell\leqslant k$ then

$\displaystyle
S_{(\ell)}^{(a,\dots,a+\ell-1)}S_{(k)}=S_{(k)}S_{(\ell)}^{(a,\dots,a+\ell-1)}=S_{(k)},\quad
A_{(\ell)}^{(a,\dots,a+\ell-1)}A_{(k)}=A_{(k)}A_{(\ell)}^{(a,\dots,a+\ell-1)}=A_{(k)},$
$\displaystyle
A_{(\ell)}^{(a,\dots,a+\ell-1)}S_{(k)}=S_{(k)}A_{(\ell)}^{(a,\dots,a+\ell-1)}=0,\qquad
S_{(\ell)}^{(a,\dots,a+\ell-1)}A_{(k)}=A_{(k)}S_{(\ell)}^{(a,\dots,a+\ell-1)}=0,$

where $a=1,\dots,k-\ell+1$ and we used the notation
$T^{(a,\dots,a+\ell-1)}=\mathop{\mathrm{id}}\nolimits^{\otimes(a-1)}\otimes
T\otimes\mathop{\mathrm{id}}\nolimits^{\otimes(k-\ell-a+1)}$ for
$T\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes\ell}\big{)}$.

Since $\Xi_{A}({\mathbb{C}})={{\mathfrak{X}}}^{*}_{S}({\mathbb{C}})$ and
${\Xi}^{*}_{A}({\mathbb{C}})={\mathfrak{X}}_{S}({\mathbb{C}})$ the conditions
(5.12), (5.13), (5.14) interchange with the conditions (5.15), (5.16), (5.17)
under the substitutions

$\displaystyle A\leftrightarrow S,\qquad P\leftrightarrow-P,\qquad
A_{(k)}\leftrightarrow S_{(k)},\qquad X\leftrightarrow\Psi^{*},\qquad
X^{*}\leftrightarrow\Psi.$ (5.21)

Thus if $S_{(k)}$ and $A_{(k)}$ are pairing operators for $A$ then $A_{(k)}$
and $S_{(k)}$ are pairing operators for $S=1-A$.

The following property of the pairing operators shows their role for the
quadratic algebras. Recall that we have identifications
${\mathfrak{X}}_{A}({\mathbb{C}})_{1}=({\mathbb{C}}^{n})^{*}$ and
$\Xi_{A}({\mathbb{C}})_{1}={\mathbb{C}}^{n}$. Let us identify the higher
graded components ${\mathfrak{X}}_{A}({\mathbb{C}})_{k}$ and
$\Xi_{A}({\mathbb{C}})_{k}$ with subspaces of
$\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}$ and
$\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}$ by using the idempotents
$S_{(k)}$ and $A_{(k)}$ respectively.

######  Proposition 5.9.

We have the following isomorphisms of vector spaces:

$\displaystyle\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S_{(k)}\cong{\mathfrak{X}}_{A}({\mathbb{C}})_{k},\qquad\xi\mapsto\xi(X\otimes\cdots\otimes
X),$ $\displaystyle S_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}\cong{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k},\qquad\
\pi\mapsto(X^{*}\otimes\cdots\otimes X^{*})\pi,$ $\displaystyle
A_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}\cong\Xi_{A}({\mathbb{C}})_{k},\qquad\
\pi\mapsto(\Psi\otimes\cdots\otimes\Psi)\pi,$
$\displaystyle\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}A_{(k)}\cong\Xi^{*}_{A}({\mathbb{C}})_{k},\qquad\xi\mapsto\xi(\Psi^{*}\otimes\cdots\otimes\Psi^{*}).$
(5.22)

The pairings
$\langle\cdot,\cdot\rangle\colon{\mathfrak{X}}_{A}({\mathbb{C}})_{k}\times{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$
and
$\langle\cdot,\cdot\rangle\colon\Xi^{*}_{A}({\mathbb{C}})_{k}\times\Xi_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$
correspond to the multiplication $(\xi,\pi)\mapsto\xi\pi\in{\mathbb{C}}$.

###### Proof.

All these maps are linear. Consider the map (5.22). Due to (5.14) the image of
the covector $\xi=e^{i_{1}\dots i_{k}}S_{(k)}$ is $e^{i_{1}\dots
i_{k}}S_{(k)}(X\otimes\cdots\otimes X)=e^{i_{1}\dots
i_{k}}(X\otimes\cdots\otimes X)=x^{i_{1}}\cdots x^{i_{k}}$. Hence this map is
surjective. Let us check injectivity. If $\xi(X\otimes\cdots\otimes X)=0$ then
by using Corollary 5.4 for $T=\xi$ we obtain $\xi S_{(k)}=0$. Since
$\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}S_{(k)}$ we have
$\xi=\xi S_{(k)}=0$. The bijectivity of other maps are proved in the same way.
The last statement follows from Lemma 5.3, for example,
$\big{\langle}\xi(X\otimes\cdots\otimes X),(X^{*}\otimes\cdots\otimes
X^{*})\pi\big{\rangle}=\xi S_{(k)}\pi=\xi\pi$. ∎

Let $d_{k}=\mathop{\mathrm{rk}}\nolimits S_{(k)}$ and
$r_{k}=\mathop{\mathrm{rk}}\nolimits A_{(k)}$. Then Proposition 5.9 implies

$\displaystyle\dim{\mathfrak{X}}_{A}({\mathbb{C}})_{k}=\dim{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}=d_{k},\qquad\dim\Xi_{A}({\mathbb{C}})_{k}=\dim\Xi^{*}_{A}({\mathbb{C}})_{k}=r_{k}.$
(5.23)

######  Proposition 5.10.

Let $S_{(k)}$ and $A_{(k)}$ be the pairing operators for the idempotent $A$.
Then the pairings
$\langle\cdot,\cdot\rangle\colon{\mathfrak{X}}_{A}({\mathbb{C}})\times{\mathfrak{X}}^{*}_{A}({\mathbb{C}})\to{\mathbb{C}}$
and
$\langle\cdot,\cdot\rangle\colon\Xi^{*}_{A}({\mathbb{C}})\times\Xi_{A}({\mathbb{C}})\to{\mathbb{C}}$
defined by the formulae (5.6) and (5.9) are non-degenerate.

###### Proof.

Since the pairing
$\langle\cdot,\cdot\rangle\colon{\mathfrak{X}}_{A}({\mathbb{C}})\times{\mathfrak{X}}^{*}_{A}({\mathbb{C}})\to{\mathbb{C}}$
respects the grading its non-degeneracy is equivalent to non-degeneracy of the
restricted pairings
$\langle\cdot,\cdot\rangle\colon{\mathfrak{X}}_{A}({\mathbb{C}})_{k}\times{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$.
Due to Proposition 5.9 it is enough to prove that the pairings
$\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}S_{(k)}\times
S_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}\to{\mathbb{C}}$ given by
$(\xi,\pi)\mapsto\xi\pi$ are non-degenerate, but this fact follows from the
part (1) of Proposition 5.1. The non-degeneracy of the pairing
$\langle\cdot,\cdot\rangle\colon\Xi^{*}_{A}({\mathbb{C}})\times\Xi_{A}({\mathbb{C}})\to{\mathbb{C}}$
is obtained in the same way. ∎

The space $({\mathbb{C}}^{n})^{\otimes k}$ is decomposed into the direct sum
of $S_{(k)}({\mathbb{C}}^{n})^{\otimes k}$ and
$\big{(}1-S_{(k)}\big{)}({\mathbb{C}}^{n})^{\otimes k}$. Let
$v_{\alpha}=v_{\alpha}^{S_{(k)}}\in({\mathbb{C}}^{n})^{\otimes k}$ be
eigenvectors: $S_{(k)}v_{\alpha}=v_{\alpha}$ for $\alpha=1,\dots,d_{k}$ and
$S_{(k)}v_{\alpha}=0$ for $\alpha=d_{k}+1,\dots,n^{k}$. Then $(v_{\alpha})$ is
a basis of $({\mathbb{C}}^{n})^{\otimes k}$ such that
$(v_{\alpha})_{\alpha\leqslant d_{k}}$ and $(v_{\alpha})_{\alpha>d_{k}}$ are
bases of the subspaces $S_{(k)}({\mathbb{C}}^{n})^{\otimes k}$ and
$\big{(}1-S_{(k)}\big{)}({\mathbb{C}}^{n})^{\otimes k}$ respectively. Let
$(v^{\alpha}=v^{\alpha}_{S_{(k)}})$ be dual basis of
$\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}$, i.e.,
$v^{\alpha}v_{\beta}=\delta^{\alpha}_{\beta}$. Since
$v^{\alpha}S_{(k)}v_{\beta}=\delta^{\alpha}_{\beta}$ for $\beta\geqslant
d_{k}$ and $v^{\alpha}S_{(k)}v_{\beta}=0$ for $\beta>d_{k}$ we obtain
$v^{\alpha}S_{(k)}=v_{\alpha}$, $\alpha=1,\dots,d_{k}$, $v^{\alpha}S_{(k)}=0$,
$\alpha=d_{k}+1,\dots,n^{k}$. In coordinates we have
$v_{\alpha}=\sum_{i_{1},\dots,i_{k}}v_{\alpha}^{i_{1}\dots i_{k}}e_{i_{1}\dots
i_{k}}$, $v^{\alpha}=\sum_{i_{1},\dots,i_{k}}v^{\alpha}_{i_{1}\dots
i_{k}}e^{i_{1}\dots i_{k}}$ for some $v_{\alpha}^{i_{1}\dots
i_{k}},v^{\alpha}_{i_{1}\dots i_{k}}\in{\mathbb{C}}$ such that
$\sum_{i_{1},\dots,i_{k}}v_{\alpha}^{i_{1}\dots i_{k}}v^{\beta}_{i_{1}\dots
i_{k}}=\delta^{\beta}_{\alpha}$ and
$\sum_{\alpha=1}^{n^{k}}v_{\alpha}^{i_{1}\dots i_{k}}v^{\alpha}_{j_{1}\dots
j_{k}}=\delta^{i_{1}}_{j_{1}}\cdots\delta^{i_{k}}_{j_{k}}$.

Analogously, let
$w_{\alpha}=v_{\alpha}^{A_{(k)}}=\sum_{i_{1},\dots,i_{k}}w_{\alpha}^{i_{1}\dots
i_{k}}e_{i_{1}\dots i_{k}}$ and
$w^{\alpha}=v^{\alpha}_{A_{(k)}}=\sum_{i_{1},\dots,i_{k}}w^{\alpha}_{i_{1}\dots
i_{k}}e^{i_{1}\dots i_{k}}$ form dual bases of $({\mathbb{C}}^{n})^{\otimes
k}$ and $\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}$ such that
$A_{(k)}w_{\alpha}=w_{\alpha}$ for $\alpha=1,\dots,r_{k}$ and
$A_{(k)}w_{\alpha}=0$ for $\alpha=r_{k}+1,\dots,n^{k}$. Then
$w^{\alpha}A_{(k)}=w^{\alpha}$, $\alpha=1,\dots,r_{k}$ and
$w^{\alpha}A_{(k)}=0$, $\alpha=r_{k}+1,\dots,n^{k}$.

######  Proposition 5.11.

The elements

$\displaystyle x_{(k)}^{\alpha}=v^{\alpha}(X\otimes\cdots\otimes
X)=\sum_{i_{1},\dots,i_{k}}v^{\alpha}_{i_{1}\dots i_{k}}x^{i_{1}}\cdots
x^{i_{k}},\qquad\alpha=1,\dots,d_{k},$ (5.24) $\displaystyle
x^{(k)}_{\alpha}=(X^{*}\otimes\cdots\otimes
X^{*})v_{\alpha}=\sum_{i_{1},\dots,i_{k}}v_{\alpha}^{i_{1}\dots
i_{k}}x_{i_{1}}\cdots x_{i_{k}},\qquad\alpha=1,\dots,d_{k},$ (5.25)
$\displaystyle\psi^{(k)}_{\alpha}=(\Psi\otimes\cdots\otimes\Psi)w_{\alpha}=\sum_{i_{1},\dots,i_{k}}w_{\alpha}^{i_{1}\dots
i_{k}}\psi_{i_{1}}\cdots\psi_{i_{k}},\qquad\alpha=1,\dots,r_{k},$ (5.26)
$\displaystyle\psi_{(k)}^{\alpha}=w^{\alpha}(\Psi^{*}\otimes\cdots\otimes\Psi^{*})=\sum_{i_{1},\dots,i_{k}}w^{\alpha}_{i_{1}\dots
i_{k}}\psi^{i_{1}}\cdots\psi^{i_{k}},\qquad\alpha=1,\dots,r_{k},$ (5.27)

form the dual bases of the $k$-th graded components
${\mathfrak{X}}_{A}({\mathbb{C}})_{k}$,
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}$, $\Xi_{A}({\mathbb{C}})_{k}$,
$\Xi^{*}_{A}({\mathbb{C}})_{k}$.

###### Proof.

This is consequence of Proposition 5.9 and the fact that
$(v^{\alpha})_{\alpha=1}^{d_{k}}$, $(v_{\alpha})_{\alpha=1}^{d_{k}}$,
$(w_{\alpha})_{\alpha=1}^{r_{k}}$ and $(w^{\alpha})_{\alpha=1}^{r_{k}}$ are
dual bases of $\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}S_{(k)}$,
$S_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}$,
$A_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}$ and
$\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}A_{(k)}$ respectively. ∎

In contrast to the uniqueness the existence of the pairing operators is not
guaranteed for an arbitrary idempotent $A$. In many interesting cases the
pairing operators can be found explicitly. In general situation we can claim
the existence of $S_{(1)}$, $A_{(1)}$, $S_{(2)}$ and $A_{(2)}$ only. In
Section 6.3 we consider cases when the third pairing operators do not exist.
Now we give necessary and sufficient conditions for existence of a pairing
operator.

######  Theorem 5.12.

Let
$A\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$
be an idempotent and $k\geqslant 2$. Consider the subspaces

$\displaystyle V_{k}=\big{\\{}\pi\in({\mathbb{C}}^{n})^{\otimes k}\mid
A^{(a,a+1)}\pi=0\big{\\}},\qquad\overline{V}_{k}=\big{\\{}\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}\mid\xi A^{(a,a+1)}=0\big{\\}},$ $\displaystyle
W_{k}=\big{\\{}\pi\in({\mathbb{C}}^{n})^{\otimes k}\mid
S^{(a,a+1)}\pi=0\big{\\}},\qquad\overline{W}_{k}=\big{\\{}\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}\mid\xi S^{(a,a+1)}=0\big{\\}}.$

We have isomorphisms ${\mathfrak{X}}_{A}({\mathbb{C}})_{k}\cong V_{k}^{*}$,
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\cong\overline{V}_{k}^{*}$,
$\Xi_{A}({\mathbb{C}})_{k}\cong\overline{W}_{k}^{*}$,
$\Xi^{*}_{A}({\mathbb{C}})_{k}\cong W_{k}^{*}$ given by the pairings

$\displaystyle\big{\langle}\xi(X\otimes\cdots\otimes
X),\pi\big{\rangle}=\xi\pi,\qquad\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*},\quad\pi\in V_{k},$ (5.28)
$\displaystyle\big{\langle}(X^{*}\otimes\cdots\otimes
X^{*})\pi,\xi\big{\rangle}=\xi\pi,\qquad\pi\in({\mathbb{C}}^{n})^{\otimes
k},\quad\xi\in\overline{V}_{k},$ (5.29)
$\displaystyle\big{\langle}(\Psi\otimes\cdots\otimes\Psi)\pi,\xi\big{\rangle}=\xi\pi,\qquad\pi\in({\mathbb{C}}^{n})^{\otimes
k},\quad\xi\in\overline{W}_{k},$ (5.30)
$\displaystyle\big{\langle}\xi(\Psi^{*}\otimes\cdots\otimes\Psi^{*}),\pi\big{\rangle}=\xi\pi,\qquad\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*},\quad\pi\in W_{k}.$ (5.31)

  * •

The $k$-th $S$-operator $S_{(k)}$ for the idempotent $A$ exists iff the spaces
$V_{k}$ and $\overline{V}_{k}$ are dual via the natural pairing, that is $\dim
V_{k}=\dim\overline{V}_{k}$ and there are bases $(v_{\alpha})$ and
$(v^{\alpha})$ of $V_{k}$ and $\overline{V}_{k}$ such that
$v^{\alpha}v_{\beta}=\delta^{\alpha}_{\beta}$. This pairing operator has the
form $S_{(k)}=\sum_{\alpha}v_{\alpha}v^{\alpha}$.

  * •

The $k$-th $A$-operator $A_{(k)}$ for the idempotent $A$ exists iff the spaces
$W_{k}$ and $\overline{W}_{k}$ are dual via the natural pairing, that is $\dim
W_{k}=\dim\overline{W}_{k}$ and there are bases $(w_{\alpha})$ and
$(w^{\alpha})$ of $W_{k}$ and $\overline{W}_{k}$ such that
$w^{\alpha}w_{\beta}=\delta^{\alpha}_{\beta}$. This pairing operator is
$A_{(k)}=\sum_{\alpha}w_{\alpha}w^{\alpha}$.

###### Proof.

Due to the symmetry (5.21) it is enough to prove the statements concerning
$V_{k}$ and $\overline{V}_{k}$.

By definition the algebra ${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$ is the
quotient of the tensor algebra
$T{\mathbb{C}}^{n}=\bigoplus_{k\in{\mathbb{N}}_{0}}({\mathbb{C}}^{n})^{\otimes
k}$ by its two sided ideal $I$ generated by the elements
$\sum_{s,t=1}^{n}e_{\rm st}A^{\rm st}_{ij}$. In the $k$-th graded component we
have ${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}=({\mathbb{C}}^{n})^{\otimes
k}/I_{k}$, where $I_{k}$ is a subspace of $({\mathbb{C}}^{n})^{\otimes k}$
spanned by the vectors $\sum_{s,t=1}^{n}e_{i_{1}\dots
i_{a-1}}\otimes\big{(}e_{\rm st}A^{\rm st}_{i_{a}i_{a+1}}\big{)}\otimes
e_{i_{a+2}\dots i_{k}}=A^{(a,a+1)}e_{i_{1}\dots i_{k}}$, where
$a=1,\dots,k-1$, $i_{1},\dots,i_{k}=1,\dots,n$. That is
$I_{k}=\sum_{a=1}^{k-1}A^{(a,a+1)}({\mathbb{C}}^{n})^{\otimes k}$. This
definition implies that the element $x_{i}$ is the class $[e_{i}]$, hence
$[e_{i_{1}\dots i_{k}}]=[e_{i_{1}}\otimes\cdots\otimes
e_{i_{k}}]=x_{i_{1}}\cdots x_{i_{k}}=(X^{*}\otimes\cdots\otimes
X^{*})e_{i_{1}\dots i_{k}}$, so the canonical projection
$p_{k}\colon({\mathbb{C}}^{n})^{\otimes
k}\to{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}$ has the form
$p_{k}\colon\pi\mapsto(X^{*}\otimes\cdots\otimes X^{*})\pi$. The subspace
orthogonal to $\mathop{\mathrm{Ker}}\nolimits p_{k}=I_{k}$ is
$I_{k}^{\bot}=\big{\\{}\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}\mid\xi I_{k}=0\big{\\}}=\overline{V}_{k}$.

Note that for any subspace $U_{0}$ of a vector space $U$ we have an
isomorphism $U_{0}^{\bot}\cong(U/U_{0})^{*}$ via the pairing
$\big{\langle}[u],\lambda\big{\rangle}=\lambda(u)$, where $\lambda\in
U_{0}^{\bot}$ and $[u]\in U/U_{0}$ is the class of a vector $u\in U$. Hence we
obtain the isomorphism
$\overline{V}_{k}=I_{k}^{\bot}\cong\big{(}({\mathbb{C}}^{n})^{\otimes
k}/I_{k}\big{)}^{*}=\big{(}{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\big{)}^{*}$
given by the pairing (5.29). In the same way we obtain the isomorphism
$V_{k}\cong\big{(}{\mathfrak{X}}_{A}({\mathbb{C}})_{k}\big{)}^{*}$.

If $S_{(k)}$ exists then there are dual bases
$(v_{\alpha})_{\alpha=1}^{d_{k}}$ and $(v^{\alpha})_{\alpha=1}^{d_{k}}$ of the
spaces $S_{(k)}({\mathbb{C}}^{n})^{\otimes k}$ and
$\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}^{*}S_{(k)}$. Since
$S_{(k)}A^{(a,a+1)}=0$ we have $\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S_{(k)}\subset\overline{V}_{k}$. The equality (5.23) and the
isomorphism
$\overline{V}_{k}^{*}\cong{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}$ imply
$\dim\overline{V}_{k}=\dim{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}=d_{k}=\dim\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S_{(k)}$, hence $\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S_{(k)}=\overline{V}_{k}$. Similarly we obtain $\dim V_{k}=d_{k}$
and $S_{(k)}({\mathbb{C}}^{n})^{\otimes k}=V_{k}$. Thus
$(v_{\alpha})_{\alpha=1}^{d_{k}}$ and $(v^{\alpha})_{\alpha=1}^{d_{k}}$ are
dual bases of $V_{k}$ and $\overline{V}_{k}$ respectively.

Conversely, let $d=\dim V_{k}=\dim\overline{V}_{k}$ and let
$(v_{\alpha})_{\alpha=1}^{d}$ and $(v^{\alpha})_{\alpha=1}^{d}$ be dual bases
of $V_{k}$ and $\overline{V}_{k}$. Since the vectors $v_{\alpha}$ are not
orthogonal to $\overline{V}_{k}$ they do not belong to $I_{k}$, that is
$V_{k}\cap I_{k}=0$. Moreover, $\dim
I_{k}=n^{k}-\dim\overline{V}_{k}=n^{k}-\dim V_{k}$ and hence
$({\mathbb{C}}^{n})^{\otimes k}=V_{k}\oplus I_{k}$. This implies that the
restriction of the projection $p_{k}$ to the subspace $V_{k}$ is an
isomorphism and hence the elements
$x_{\alpha}^{(k)}:=(X^{*}\otimes\cdots\otimes X^{*})v_{\alpha}$ form a basis
of ${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}$. In particular, $x_{i_{1}}\cdots
x_{i_{k}}=\sum_{\alpha=1}^{d}c_{i_{1}\dots i_{k}}^{\alpha}x^{(k)}_{\alpha}$
for some $c_{i_{1}\dots i_{k}}^{\alpha}\in{\mathbb{C}}$, so that
$(X^{*}\otimes\cdots\otimes
X^{*})=\sum_{\alpha=1}^{d}x^{(k)}_{\alpha}\xi^{\alpha}$, where
$\xi^{\alpha}=\sum_{i_{1},\dots,i_{k}=1}^{n}c_{i_{1}\dots
i_{k}}^{\alpha}e^{i_{1}\dots i_{k}}\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}$. By multiplying this by $A^{(a,a+1)}$ from the right and taking
into account (5.3) we obtain
$\sum_{\alpha=1}^{d}x^{(k)}_{\alpha}\xi^{\alpha}A^{(a,a+1)}=0$. Since the
elements $x^{(k)}_{\alpha}$ are linearly independent, we have
$\xi^{\alpha}A^{(a,a+1)}=0$ for all $a=1,\dots,k-1$, so that
$\xi^{\alpha}\in\overline{V}_{k}$. Multiplication of the same relation by
$v_{\beta}$ from the right gives
$\xi^{\alpha}v_{\beta}=\delta^{\alpha}_{\beta}$, hence
$\xi^{\alpha}=v^{\alpha}$. Let
$S_{(k)}=\sum_{\alpha=1}^{d}v_{\alpha}v^{\alpha}$, then we have
$(X^{*}\otimes\cdots\otimes
X^{*})S_{(k)}=\sum_{\alpha,\beta=1}^{d}x^{(k)}_{\alpha}v^{\alpha}v_{\beta}v^{\beta}=\sum_{\alpha=1}^{d}x^{(k)}_{\alpha}v^{\alpha}=(X^{*}\otimes\cdots\otimes
X^{*})$. Analogously we obtain $S_{(k)}(X\otimes\cdots\otimes
X)=(X\otimes\cdots\otimes X)$. Since $S_{(k)}$ satisfies also
$A^{(a,a+1)}S_{(k)}=S_{(k)}A^{(a,a+1)}=0$ it is the $k$-th $S$-operator for
the idempotent $A$. ∎

######  Remark 5.13.

Theorem 5.12 means in fact that the non-degeneracies of the pairings
$\overline{V}_{k}\times V_{k}\to{\mathbb{C}}$ and $\overline{W}_{k}\times
W_{k}\to{\mathbb{C}}$ given by $\langle\xi,\pi\rangle=\xi\pi$ imply that they
induce non-degenerate pairings
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\times{\mathfrak{X}}_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$
and
$\Xi_{A}({\mathbb{C}})_{k}\times\Xi^{*}_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$
respectively. Conversely, if there exists a non-degenerate pairing
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\times{\mathfrak{X}}_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$
or
$\Xi_{A}({\mathbb{C}})_{k}\times\Xi^{*}_{A}({\mathbb{C}})_{k}\to{\mathbb{C}}$
then the induced pairing $\overline{V}_{k}\times V_{k}\to{\mathbb{C}}$ or
$\overline{W}_{k}\times W_{k}\to{\mathbb{C}}$ is non-degenerate but it may
differ from the corresponding pairing defined by
$\langle\xi,\pi\rangle=\xi\pi$. In other words, the conditions
$\dim{\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}=\dim{\mathfrak{X}}_{A}({\mathbb{C}})_{k}$
and $\dim\Xi^{*}_{A}({\mathbb{C}})_{k}=\dim\Xi_{A}({\mathbb{C}})_{k}$ do not
guarantee the existence of $S_{(k)}$ and $A_{(k)}$ respectively (e.g., see
Section 6.3).

######  Remark 5.14.

If some pairing $S$\- or $A$-operators do not exist then one can consider the
dual space
$\big{(}{\mathfrak{X}}_{A}({\mathbb{C}})\big{)}^{*}=\bigoplus_{k=0}^{\infty}V_{k}$
or
$\big{(}\Xi_{A}({\mathbb{C}})\big{)}^{*}=\bigoplus_{k=0}^{\infty}\overline{W}_{k}$
(without a structure of algebra) instead of the algebra
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$ or $\Xi^{*}_{A}({\mathbb{C}})$
respectively. The algebra structures on the spaces
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$ and $\Xi^{*}_{A}({\mathbb{C}})$ are
auxiliary. They are not in agreement with the algebra structures of
${\mathfrak{X}}_{A}({\mathbb{C}})$ and $\Xi_{A}({\mathbb{C}})$, but they are
used to define these spaces in a more convenient way.

###  5.4 Minor operators

Let
$S_{(k)},A_{(k)}\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ and
$\widetilde{S}_{(k)},\widetilde{A}_{(k)}\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k}\big{)}$ be pairing operators for idempotents
$A\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$
and
$\widetilde{A}\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$
respectively. Let $X$, $X^{*}$, $\Psi$, $\Psi^{*}$ denote the same as previous
subsection. The corresponding column- and row-vectors for $\widetilde{A}$ we
denote by $\widetilde{X}$, $\widetilde{X}^{*}$, $\widetilde{\Psi}^{*}$,
$\widetilde{\Psi}$.

By virtue of Proposition 5.9 any graded linear operator
${\mathfrak{X}}_{A}({\mathbb{C}})\to{\mathfrak{X}}_{\widetilde{A}}({\mathfrak{R}})$
is given by the formula

$\displaystyle\xi(X\otimes\cdots\otimes X)\mapsto\xi
T_{k}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)},\qquad\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}$ (5.32)

for some operators
$T_{k}\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},({\mathbb{C}}^{n})^{\otimes k}\big{)}$ such that
$S_{(k)}T_{k}\widetilde{S}_{(k)}=T_{k}\widetilde{S}_{(k)}$. In the same time a
graded linear operator
${\mathfrak{X}}^{*}_{\widetilde{A}}({\mathbb{C}})\to{\mathfrak{X}}^{*}_{A}({\mathfrak{R}})$
has the form

$\displaystyle(\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*})\pi\mapsto(X^{*}\otimes\cdots\otimes
X^{*})T_{k}\pi,\qquad\pi\in({\mathbb{C}}^{m})^{\otimes k}$ (5.33)

for some
$T_{k}\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},({\mathbb{C}}^{n})^{\otimes k}\big{)}$ such that
$S_{(k)}T_{k}\widetilde{S}_{(k)}=S_{(k)}T_{k}$.

Analogously, a graded linear operator
$\Xi_{\widetilde{A}}({\mathbb{C}})\to\Xi_{A}({\mathfrak{R}})$ can be written
as

$\displaystyle(\widetilde{\Psi}\otimes\cdots\otimes\widetilde{\Psi})\pi\mapsto(\Psi\otimes\cdots\otimes\Psi)R_{k}\pi,\qquad\pi\in({\mathbb{C}}^{m})^{\otimes
k}$ (5.34)

for
$R_{k}\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},({\mathbb{C}}^{n})^{\otimes k}\big{)}$ such that
$A_{(k)}R_{k}\widetilde{A}_{(k)}=A_{(k)}R_{k}$, while a graded linear operator
$\Xi^{*}_{A}({\mathbb{C}})\to\Xi^{*}_{\widetilde{A}}({\mathfrak{R}})$ has the
form

$\displaystyle\xi(\Psi^{*}\otimes\cdots\otimes\Psi^{*})\mapsto\xi
R_{k}\big{(}\widetilde{\Psi}^{*}\otimes\cdots\otimes\widetilde{\Psi}^{*}\big{)},\qquad\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}$ (5.35)

for some operators
$R_{k}\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},({\mathbb{C}}^{n})^{\otimes k}\big{)}$ such that
$A_{(k)}R_{k}\widetilde{A}_{(k)}=R_{k}\widetilde{A}_{(k)}$.

Note that $T_{k}$ and $R_{k}$ can be replaced by
$S_{(k)}T_{k}\widetilde{S}_{(k)}$ and $A_{(k)}R_{k}\widetilde{A}_{(k)}$
respectively and this does not change the maps (5.32), (5.33), (5.34), (5.35).
Hence we can always suppose $S_{(k)}T_{k}\widetilde{S}_{(k)}=T_{k}$ and
$A_{(k)}R_{k}\widetilde{A}_{(k)}=R_{k}$.

Denote
${\mathfrak{X}}^{*}_{A}({\mathfrak{R}})={\mathfrak{R}}\otimes{\mathfrak{X}}^{*}_{A}({\mathbb{C}})$
and
$\Xi^{*}_{A}({\mathfrak{R}})={\mathfrak{R}}\otimes\Xi^{*}_{A}({\mathbb{C}})$.
The pairings are invariant in the following sense.

######  Proposition 5.15.

Let
$t\colon{\mathfrak{X}}_{A}({\mathbb{C}})\to{\mathfrak{X}}_{\widetilde{A}}({\mathfrak{R}})$
and
$t^{*}\colon{\mathfrak{X}}^{*}_{\widetilde{A}}({\mathbb{C}})\to{\mathfrak{X}}^{*}_{A}({\mathfrak{R}})$
be the operators (5.32) and (5.33) defined by the same
$T_{k}\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},({\mathbb{C}}^{n})^{\otimes k}\big{)}$ satisfying
$S_{(k)}T_{k}\widetilde{S}_{(k)}=T_{k}$. Let
$r\colon\Xi_{\widetilde{A}}({\mathbb{C}})\to\Xi_{A}({\mathfrak{R}})$ and
$r^{*}\colon\Xi^{*}_{A}({\mathbb{C}})\to\Xi^{*}_{\widetilde{A}}({\mathfrak{R}})$
be the operators (5.34) and (5.35) defined by the same
$R_{k}\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},({\mathbb{C}}^{n})^{\otimes k}\big{)}$ satisfying
$A_{(k)}R_{k}\widetilde{A}_{(k)}=R_{k}$. Then for any
$u\in{\mathfrak{X}}_{A}({\mathbb{C}})$,
$v\in{\mathfrak{X}}^{*}_{\widetilde{A}}({\mathbb{C}})$,
$\nu\in\Xi^{*}_{A}({\mathbb{C}})$, $\mu\in\Xi_{\widetilde{A}}({\mathbb{C}})$
we have

$\displaystyle\langle u,t^{*}(v)\rangle=\langle
t(u),v\rangle,\qquad\langle\nu,r(\mu)\rangle=\langle r^{*}(\nu),\mu\rangle,$
(5.36)

where the pairings
${\mathfrak{X}}_{A}({\mathbb{C}})\times{\mathfrak{X}}^{*}_{A}({\mathbb{C}})\to{\mathbb{C}}$,
$\Xi^{*}_{A}({\mathbb{C}})\times\Xi_{A}({\mathbb{C}})\to{\mathbb{C}}$ and
${\mathfrak{X}}_{\widetilde{A}}({\mathbb{C}})\times{\mathfrak{X}}^{*}_{\widetilde{A}}({\mathbb{C}})\to{\mathbb{C}}$,
$\Xi^{*}_{\widetilde{A}}({\mathbb{C}})\times\Xi_{\widetilde{A}}({\mathbb{C}})\to{\mathbb{C}}$
are defined by the pairing operators $S_{(k)}$, $A_{(k)}$ and
$\widetilde{S}_{(k)}$, $\widetilde{A}_{(k)}$ respectively.

###### Proof.

Let $u=\xi(X\otimes\cdots\otimes X)$ and
$v=\big{(}\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*}\big{)}\pi$.
Then by using Lemma 5.3 we have

$\displaystyle\langle u,t^{*}(v)\rangle$
$\displaystyle=\big{\langle}\xi(X\otimes\cdots\otimes
X),(X^{*}\otimes\cdots\otimes X^{*})T_{k}\pi\big{\rangle}=\xi
S_{(k)}T_{k}\pi=\xi T_{k}\widetilde{S}_{(k)}\pi$
$\displaystyle=\big{\langle}\xi
T_{k}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)},\big{(}\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*}\big{)}\pi\big{\rangle}=\langle
t(u),v\rangle.$

The second formula (5.36) is proved similarly. ∎

Recall that in Section 2.5 we introduced the homomorphisms
$f_{M}\colon{\mathfrak{X}}_{A}({\mathbb{C}})\to{\mathfrak{X}}_{\widetilde{A}}({\mathbb{C}})$
and $f^{M}\colon\Xi_{\widetilde{A}}({\mathbb{C}})\to\Xi_{A}({\mathbb{C}})$ for
a $\big{(}A,\widetilde{A}\big{)}$-Manin matrix
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$.
Their definition on generators can be written in the matrix form as

$\displaystyle f_{M}(X)=M\widetilde{X},\qquad f^{M}(\widetilde{\Psi})=\Psi M.$

Since homomorphisms preserve multiplications we obtain

$\displaystyle f_{M}(X\otimes\cdots\otimes X)=M^{(1)}\cdots
M^{(k)}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)},$ (5.37)
$\displaystyle
f^{M}(\widetilde{\Psi}\otimes\cdots\otimes\widetilde{\Psi})=(\Psi\otimes\cdots\otimes\Psi)M^{(1)}\cdots
M^{(k)}.$ (5.38)

For general elements of ${\mathfrak{X}}_{A}({\mathbb{C}})$ and
$\Xi_{\widetilde{A}}({\mathbb{C}})$ the values of the maps $f_{M}$ and $f^{M}$
are obtained via multiplication of these formulae by $\xi$ and $\pi$
respectively, so these maps have the form (5.32) and (5.34) for
$T_{k}=R_{k}=M^{(1)}\cdots M^{(k)}$. In this way we obtain the following
generalisation of (2.13) and (2.16).

######  Proposition 5.16.

Any $\big{(}A,\widetilde{A}\big{)}$-Manin matrix
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
satisfy the relations

$\displaystyle M^{(1)}\cdots M^{(k)}\widetilde{S}_{(k)}=S_{(k)}M^{(1)}\cdots
M^{(k)}\widetilde{S}_{(k)},$ (5.39) $\displaystyle A_{(k)}M^{(1)}\cdots
M^{(k)}=A_{(k)}M^{(1)}\cdots M^{(k)}\widetilde{A}_{(k)}.$ (5.40)

###### Proof.

Note that the left hand sides of (5.37) and (5.38) are invariant under
multiplication by $S_{(k)}$ from the left and by $\widetilde{A}_{(k)}$ from
the right respectively. As consequence, we obtain

$\displaystyle S_{(k)}M^{(1)}\cdots
M^{(k)}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)}=M^{(1)}\cdots
M^{(k)}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)},$
$\displaystyle(\Psi\otimes\cdots\otimes\Psi)M^{(1)}\cdots
M^{(k)}\widetilde{A}_{(k)}=(\Psi\otimes\cdots\otimes\Psi)M^{(1)}\cdots
M^{(k)}.$

Then, (5.39) and (5.40) are derived by application of Corollary 5.4 for
$V={\mathfrak{R}}\otimes({\mathbb{C}}^{n})^{\otimes k}$,
$T=S_{(k)}M^{(1)}\cdots M^{(k)}-M^{(1)}\cdots
M^{(k)}\in\mathop{\mathrm{Hom}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k},V\big{)}$ and for $W={\mathfrak{R}}^{*}\otimes({\mathbb{C}}^{m})^{\otimes
k}$, $\widetilde{T}=M^{(1)}\cdots M^{(k)}\widetilde{A}_{(k)}-M^{(1)}\cdots
M^{(k)}\in\mathop{\mathrm{Hom}}\nolimits\big{(}W,({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ respectively. ∎

For an arbitrary
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
define the linear operators
$t_{M}\colon{\mathfrak{X}}_{A}({\mathbb{C}})\to{\mathfrak{X}}_{\widetilde{A}}({\mathfrak{R}})$
and
$t^{*}_{M}\colon{\mathfrak{X}}^{*}_{\widetilde{A}}({\mathbb{C}})\to{\mathfrak{X}}^{*}_{A}({\mathfrak{R}})$
by the operators $T_{k}=S_{(k)}M^{(1)}\cdots M^{(k)}\widetilde{S}_{(k)}$, that
is

$\displaystyle t_{M}(X\otimes\cdots\otimes X)=S_{(k)}M^{(1)}\cdots
M^{(k)}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)},$ (5.41)
$\displaystyle
t^{*}_{M}\big{(}\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*}\big{)}=(X^{*}\otimes\cdots\otimes
X^{*})M^{(1)}\cdots M^{(k)}\widetilde{S}_{(k)}.$ (5.42)

Analogously, for an arbitrary matrix
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
define the linear operators
$r_{M}\colon\Xi_{\widetilde{A}}({\mathbb{C}})\to\Xi_{A}({\mathfrak{R}})$ and
$r^{*}_{M}\colon\Xi^{*}_{A}({\mathbb{C}})\to\Xi^{*}_{\widetilde{A}}({\mathfrak{R}})$
by the operators $R_{k}=A_{(k)}M^{(1)}\cdots M^{(k)}\widetilde{A}_{(k)}$, that
is

$\displaystyle
r_{M}\big{(}\widetilde{\Psi}\otimes\cdots\otimes\widetilde{\Psi}\big{)}=(\Psi\otimes\cdots\otimes\Psi)M^{(1)}\cdots
M^{(k)}\widetilde{A}_{(k)},$ (5.43) $\displaystyle
r_{M}^{*}(\Psi^{*}\otimes\cdots\otimes\Psi^{*})=A_{(k)}M^{(1)}\cdots
M^{(k)}\big{(}\widetilde{\Psi}^{*}\otimes\cdots\otimes\widetilde{\Psi}^{*}\big{)}.$
(5.44)

######  Remark 5.17.

If $M$ is an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix then the maps (5.41)
and (5.43) are homomorphisms: $t_{M}=f_{M}$ and $r_{M}=f_{M}$ (however, the
maps $t^{*}_{M}$ and $r_{M}^{*}$ are not homomorphisms). Conversely, if
$t_{M}$ or $r_{M}$ is a homomorphism then $M$ is an
$\big{(}A,\widetilde{A}\big{)}$-Manin matrix. The maps $t^{*}_{M}$ and
$r_{M}^{*}$ are homomorphisms iff $M$ is a
$\big{(}S,\widetilde{S}\big{)}$-Manin matrix, where $S=1-A$,
$\widetilde{S}=1-\widetilde{A}$.

For a matrix
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
we introduce minor operator corresponding to a pair of operators
$T\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ and
$\widetilde{T}\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k}\big{)}$ by the formula

$\displaystyle\mathop{\mathrm{Min}}\nolimits^{T}_{\widetilde{T}}M$
$\displaystyle:=TM^{(1)}\cdots M^{(k)}\widetilde{T}.$ (5.45)

From (5.41)–(5.44) we obtain

$\displaystyle\mathop{\mathrm{Min}}\nolimits^{S_{(k)}}_{\widetilde{S}_{(k)}}M$
$\displaystyle=\big{\langle}t_{M}(X\otimes\cdots\otimes
X),\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*}\big{\rangle}=\big{\langle}X\otimes\cdots\otimes
X,t^{*}_{M}\big{(}\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*}\big{)}\big{\rangle},$
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}_{\widetilde{A}_{(k)}}M$
$\displaystyle=\big{\langle}\Psi^{*}\otimes\cdots\otimes\Psi^{*},r_{M}(\widetilde{\Psi}\otimes\cdots\otimes\widetilde{\Psi})\big{\rangle}=\big{\langle}r^{*}_{M}(\Psi^{*}\otimes\cdots\otimes\Psi^{*}),\widetilde{\Psi}\otimes\cdots\otimes\widetilde{\Psi}\big{\rangle}.$

If $M$ is an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix then due to
Proposition 5.16 these operators take the form

$\displaystyle\mathop{\mathrm{Min}}\nolimits^{S_{(k)}}_{\widetilde{S}_{(k)}}M$
$\displaystyle=S_{(k)}M^{(1)}\cdots M^{(k)}\widetilde{S}_{(k)}=M^{(1)}\cdots
M^{(k)}\widetilde{S}_{(k)},$ (5.46)
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}_{\widetilde{A}_{(k)}}M$
$\displaystyle=A_{(k)}M^{(1)}\cdots
M^{(k)}\widetilde{A}_{(k)}=A_{(k)}M^{(1)}\cdots M^{(k)}.$ (5.47)

In these case these minor operators are defined by one operator only:
$\widetilde{S}_{(k)}$ and $A_{(k)}$ respectively, so we can denote them as

$\displaystyle\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M$
$\displaystyle:=M^{(1)}\cdots
M^{(k)}\widetilde{S}_{(k)}=\big{\langle}f_{M}(X\otimes\cdots\otimes
X),\widetilde{X}^{*}\otimes\cdots\otimes\widetilde{X}^{*}\big{\rangle},$
(5.48) $\displaystyle\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M$
$\displaystyle:=A_{(k)}M^{(1)}\cdots
M^{(k)}=\big{\langle}\Psi^{*}\otimes\cdots\otimes\Psi^{*},f^{M}\big{(}\widetilde{\Psi}\otimes\cdots\otimes\widetilde{\Psi}\big{)}\big{\rangle}.$
(5.49)

######  Definition 5.18.

Let
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
be an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix. Then the minor operators
(5.48) and (5.49) are called $S$-minor and $A$-minor operators respectively.
We also call them minor operators for $\big{(}A,\widetilde{A}\big{)}$-Manin
matrix. Their entries

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}$ $\displaystyle=e^{i_{1}\dots
i_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}e_{j_{1}\dots
j_{k}}=\big{\langle}f_{M}\big{(}x^{i_{1}}\cdots
x^{i_{k}}\big{)},\widetilde{x}_{j_{1}}\cdots\widetilde{x}_{j_{k}}\big{\rangle},$
(5.50)
$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}$ $\displaystyle=e^{i_{1}\dots
i_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}e_{j_{1}\dots
j_{k}}=\big{\langle}\psi^{i_{1}}\cdots\psi^{i_{k}},f^{M}\big{(}\widetilde{\psi}_{j_{1}}\cdots\widetilde{\psi}_{j_{k}}\big{)}\big{\rangle}$
(5.51)

are called $S$-minors and $A$-minors of the order $k$ or simply minors for
$\big{(}A,\widetilde{A}\big{)}$-Manin matrix (here $\widetilde{x}_{i}$ and
$\widetilde{\psi}_{i}$ are the generators of
${\mathfrak{X}}^{*}_{\widetilde{A}}({\mathbb{C}})$ and
$\Xi_{\widetilde{A}}({\mathbb{C}})$ respectively).

In terms of $y^{i}=f_{M}(x^{i})=\sum_{j=1}^{m}M^{i}_{j}\widetilde{x}^{j}$ and
$\phi_{j}=f^{M}\big{(}\widetilde{\psi}_{j}\big{)}=\sum_{i=1}^{n}\psi_{i}M^{i}_{j}$
the minors can be written as

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}=\big{\langle}y^{i_{1}}\cdots
y^{i_{k}},\widetilde{x}_{j_{1}}\cdots\widetilde{x}_{j_{k}}\big{\rangle},$
(5.52)
$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots
j_{k}}=\big{\langle}\psi^{i_{1}}\cdots\psi^{i_{k}},\phi_{j_{1}}\cdots\phi_{j_{k}}\big{\rangle}.$
(5.53)

They are coefficients in the decompositions

$\displaystyle y^{i_{1}}\cdots
y^{i_{k}}=\sum_{j_{1},\dots,j_{k}=1}^{m}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}\;\widetilde{x}^{j_{1}}\cdots\widetilde{x}^{j_{k}},$
(5.54) $\displaystyle\phi_{j_{1}}\cdots\phi_{j_{k}}=\sum_{i_{1}\dots
i_{k}=1}^{n}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}\;\psi_{i_{1}}\cdots\psi_{i_{k}}.$ (5.55)

In the matrix form these formulae are written as

$\displaystyle(Y\otimes\cdots\otimes
Y)=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)},$
$\displaystyle(\Phi\otimes\cdots\otimes\Phi)=(\Psi\otimes\cdots\otimes\Psi)\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)},$

where $Y=\sum_{i=1}^{n}y^{i}e_{i}$ and $\Phi=\sum_{j=1}^{m}\phi_{j}e^{j}$.

The formulae (5.54) and (5.55) are not decompositions by bases. However, due
to Proposition 5.9 they have the form of the decompositions considered in the
part (2) of Proposition 5.1. For example, for the formula (5.54) we need to
set $V=\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}^{*}$ and
$W=({\mathbb{C}}^{m})^{\otimes k}$, while the operator $S_{(k)}$ acting on
$\xi\in\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}^{*}$ from the right plays
the role of the idempotent $A$. Thus the minors of an
$\big{(}A,\widetilde{A}\big{)}$-Manin matrix $M$ are the coefficients of the
decompositions (5.54) and (5.55) satisfying the conditions

$\displaystyle\sum_{l_{1},\dots,l_{k}=1}^{m}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{l_{1}\dots l_{k}}\widetilde{S}^{l_{1}\dots l_{k}}_{j_{1}\dots
j_{k}}=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}},$
$\displaystyle\sum_{l_{1},\dots,l_{k}=1}^{n}A^{i_{1}\dots i_{k}}_{l_{1}\dots
l_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}^{l_{1}\dots
l_{k}}_{j_{1}\dots
j_{k}}=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{i_{1}\dots
i_{k}}_{j_{1}\dots j_{k}}.$

In operator form these symmetries can be written as

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{S}_{(k)}=\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M,\qquad
A_{(k)}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}=\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M.$

The expression for the $S$\- and $A$-minors of an
$\big{(}A,\widetilde{A}\big{)}$-Manin matrix $M$ depends on the pairing
operators $\widetilde{S}_{(k)}$ and $A_{(k)}$ only, hence they are defined if
these pairing operators exist even if $S_{(k)}$ and $\widetilde{A}_{(k)}$ do
not exist. The condition that $M$ is a $\big{(}A,\widetilde{A}\big{)}$-Manin
matrix implies the symmetry of the minor $S$\- and $A$-operators with respect
to the upper and lower indices:

$\displaystyle
S^{(a,a+1)}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}=\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M,\qquad\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\widetilde{A}^{(a,a+1)}=\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M.$

If $S_{(k)}$ and $\widetilde{A}_{(k)}$ do exist these symmetries can be
written in the form

$\displaystyle
S_{(k)}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}=\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M,\qquad\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\widetilde{A}_{(k)}=\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M.$

###  5.5 Properties of the minor operators

The determinant of usual complex matrices is a homomorphism:
$\mathop{\mathrm{det}}\nolimits(MN)=\mathop{\mathrm{det}}\nolimits(M)\mathop{\mathrm{det}}\nolimits(N)$.
The generalisation of this property to the case of $k\times k$ minors is (a
generalisation of the) Cauchy–Binet formula:

$\displaystyle\mathop{\mathrm{det}}\nolimits\big{(}(MN)_{IJ}\big{)}=\sum_{L=(l_{1}<\dots<l_{k})}\mathop{\mathrm{det}}\nolimits(M_{IL})\mathop{\mathrm{det}}\nolimits(N_{LJ}).$

The right hand side corresponds to the product of the $A$-minor operators.
This property is generalised to Manin matrices.

######  Proposition 5.19.

Let $S_{(k)}$, $A_{(k)}$, $\widetilde{S}_{(k)}$, $\widetilde{A}_{(k)}$ and
$\widehat{S}_{(k)}$, $\widehat{A}_{(k)}$ be the pairing operators for
idempotents
$A\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$,
$\widetilde{A}\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{m}\otimes{\mathbb{C}}^{m})$
and
$\widehat{A}\in\mathop{\mathrm{End}}\nolimits\big{(}{\mathbb{C}}^{l}\otimes{\mathbb{C}}^{l}\big{)}$
respectively. Let $M$ and $N$ be $n\times m$ and $m\times l$ matrices over an
algebra ${\mathfrak{R}}$. Suppose that the entries of the first one commute
with the entries of the second one: $\big{[}M^{i}_{j},N^{k}_{l}\big{]}=0$.

  * •

If $N$ is an $(\widetilde{A},\widehat{A})$-Manin matrix then

$\displaystyle\mathop{\mathrm{Min}}\nolimits^{S_{(k)}}_{\widehat{S}_{(k)}}(MN)=\big{(}\mathop{\mathrm{Min}}\nolimits^{S_{(k)}}_{\widetilde{S}_{(k)}}M\big{)}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widehat{S}_{(k)}}N\big{)}.$
(5.56)

  * •

If $M$ is an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix then

$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}_{\widehat{A}_{(k)}}(MN)=\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\big{(}\mathop{\mathrm{Min}}\nolimits^{\widetilde{A}_{(k)}}_{\widehat{A}_{(k)}}N\big{)}.$
(5.57)

  * •

If $M$ and $N$ are Manin matrices for $\big{(}A,\widetilde{A}\big{)}$ and
$\big{(}\widetilde{A},\widehat{A}\big{)}$ respectively then $MN$ is a Manin
matrix for $\big{(}A,\widehat{A}\big{)}$ and the formulae (5.56), (5.57) take
the form

$\displaystyle\mathop{\mathrm{Min}}\nolimits_{\widehat{S}_{(k)}}(MN)=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widehat{S}_{(k)}}N\big{)},$
(5.58)
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}(MN)=\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\big{(}\mathop{\mathrm{Min}}\nolimits^{\widetilde{A}_{(k)}}N\big{)}.$
(5.59)

###### Proof.

The first and second statements follow from Proposition 5.16. For instance,
the formula (5.57) is derived in the following way:

$\displaystyle A_{(k)}(MN)^{(1)}\cdots(MN)^{(k)}\widehat{A}_{(k)}$
$\displaystyle=A_{(k)}M^{(1)}\cdots M^{(k)}N^{(1)}\cdots
N^{(k)}\widehat{A}_{(k)}$ $\displaystyle=A_{(k)}M^{(1)}\cdots
M^{(k)}\widetilde{A}_{(k)}N^{(1)}\cdots N^{(k)}\widehat{A}_{(k)},$

where we used the commutativity in the form $N^{(i)}M^{(j)}=M^{(j)}N^{(i)}$,
$i<j$. The last statement is implied by Proposition 2.27 and the formulae
(5.46), (5.47). ∎

Now we give formulae for permutations of rows and columns. For any $n$ and
$\sigma\in{\rm GL}(n,{\mathbb{C}})$ denote the conjugation by the element
$\sigma^{\otimes k}=\sigma\otimes\cdots\otimes\sigma$ as

$\displaystyle\iota_{\sigma}T:=\sigma^{\otimes k}T\big{(}\sigma^{\otimes
k}\big{)}^{-1},$ (5.60)

where $T\in\mathop{\mathrm{End}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$ and $\big{(}\sigma^{\otimes
k}\big{)}^{-1}=\sigma^{-1}\otimes\cdots\otimes\sigma^{-1}$. Note that
$\iota_{\sigma}S_{(k)}$ and $\iota_{\sigma}A_{(k)}$ are pairing operators for
the idempotent
$\iota_{\sigma}A=(\sigma\otimes\sigma)A\big{(}\sigma^{-1}\otimes\sigma^{-1}\big{)}$.

######  Proposition 5.20.

For any matrix
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
and operators $\sigma\in{\rm GL}(n,{\mathbb{C}})$, $\tau\in{\rm
GL}(m,{\mathbb{C}})$ we have

$\displaystyle\mathop{\mathrm{Min}}\nolimits^{\iota_{\sigma}S_{(k)}}_{\iota_{\tau}\widetilde{S}_{(k)}}\big{(}\sigma
M\tau^{-1}\big{)}=\sigma^{\otimes
k}\big{(}\mathop{\mathrm{Min}}\nolimits^{S_{(k)}}_{\widetilde{S}_{(k)}}M\big{)}\big{(}\tau^{\otimes
k}\big{)}^{-1},$ (5.61)
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{\iota_{\sigma}A_{(k)}}_{\iota_{\tau}\widetilde{A}_{(k)}}\big{(}\sigma
M\tau^{-1}\big{)}=\sigma^{\otimes
k}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}_{\widetilde{A}_{(k)}}M\big{)}\big{(}\tau^{\otimes
k}\big{)}^{-1}.$ (5.62)

If $M$ is an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix then

$\displaystyle\mathop{\mathrm{Min}}\nolimits_{\iota_{\tau}\widetilde{S}_{(k)}}\big{(}\sigma
M\tau^{-1}\big{)}=\sigma^{\otimes
k}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\big{(}\tau^{\otimes
k}\big{)}^{-1},$ (5.63)
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{\iota_{\sigma}A_{(k)}}\big{(}\sigma
M\tau^{-1}\big{)}=\sigma^{\otimes
k}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\big{(}\tau^{\otimes
k}\big{)}^{-1}.$ (5.64)

###### Proof.

The formulae (5.61), (5.62) follow directly from the definition (5.45). If $M$
is an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix, then Proposition 2.22
implies that $\sigma M\tau^{-1}$ is a
$\big{(}\iota_{\sigma}A,\iota_{\tau}\widetilde{A}\big{)}$-Manin matrix. Thus
we obtain (5.63), (5.64). ∎

In particular, Proposition 5.20 gives the minors of ${}^{\sigma}M=\sigma M$
and ${}_{\tau}M=M\tau^{-1}$ for a matrix
$M\in{\mathfrak{R}}\otimes\mathop{\mathrm{Hom}}\nolimits({\mathbb{C}}^{m},{\mathbb{C}}^{n})$
and permutations $\sigma\in{\mathbf{S}}_{n}$, $\tau\in{\mathbf{S}}_{m}$.

Let us consider the minor operators
$\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M$ and
$\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M$ for an
$\big{(}A,\widetilde{A}\big{)}$-Manin matrix $M$ as $n^{k}\times m^{k}$
matrices over ${\mathfrak{R}}$ with the entries (5.52) and (5.53). They are
also Manin matrices for some pairs of idempotents.

######  Proposition 5.21.

Let $M$ be an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix. Let $S_{(k)}$,
$A_{(k)}$ and $\widetilde{S}_{(k)}$, $\widetilde{A}_{(k)}$ are the pairing
operators for $A$ and $\widetilde{A}$. For any $k,\ell\geqslant 1$ we have

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\otimes\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(\ell)}}M\big{)}\widetilde{S}_{(k+\ell)}=M^{(1)}\cdots
M^{(k+\ell)}\widetilde{S}_{(k+\ell)}=\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k+\ell)}}M,$
(5.65) $\displaystyle
A_{(k+\ell)}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\otimes\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(\ell)}}M\big{)}=A_{(k+\ell)}M^{(1)}\cdots
M^{(k+\ell)}=\mathop{\mathrm{Min}}\nolimits^{A_{(k+\ell)}}M.$ (5.66)

In particular, $\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M$ and
$\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M$ are
$\big{(}1-S_{(2k)},1-\widetilde{S}_{(2k)}\big{)}$\- and
$\big{(}A_{(2k)},\widetilde{A}_{(2k)}\big{)}$-Manin matrices respectively.

###### Proof.

The formulae (5.65) and (5.66) follow from Proposition 5.8. To prove the
second statement one needs to put $\ell=k$ in these formulae and to apply
Proposition 5.16 for $2k$. ∎

Let us finally write the minor operators in terms of bases. Denote
$y^{\alpha}_{(k)}=v^{\alpha}(Y\otimes\cdots\otimes Y)$,
$\widetilde{x}^{\alpha}_{(k)}=\widetilde{v}^{\alpha}\big{(}\widetilde{X}\otimes\cdots\otimes\widetilde{X}\big{)}$,
$\phi_{\alpha}^{(k)}=(\Phi\otimes\cdots\otimes\Phi)\widetilde{w}_{\alpha}$,
where $\widetilde{v}^{\alpha}=v^{\alpha}_{\widetilde{S}_{(k)}}$ and
$\widetilde{w}_{\alpha}=v_{\alpha}^{\widetilde{A}_{(k)}}$ are eigenvectors of
the idempotents $\widetilde{S}_{(k)}$ and $(\widetilde{A}_{(k)})^{\top}$
respectively. Let $\widetilde{d}_{k}$ and $\widetilde{r}_{k}$ be the ranks of
$\widetilde{S}_{(k)}$ and $\widetilde{A}_{(k)}$. Consider matrix entries of
the minor operators:

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{\alpha}_{\beta}:=v^{\alpha}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{v}_{\beta}=v^{\alpha}M^{(1)}\cdots
M^{(k)}\widetilde{v}_{\beta},$ (5.67)
$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}_{\delta}^{\gamma}:=w^{\gamma}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\widetilde{w}_{\delta}=w^{\gamma}M^{(1)}\cdots
M^{(k)}\widetilde{w}_{\delta},$ (5.68)

where $\alpha\leqslant d_{k}$, $\beta\leqslant\widetilde{d}_{k}$,
$\gamma\leqslant r_{k}$ and $\delta\leqslant\widetilde{r}_{k}$. Then the
formulae (5.54), (5.55) are rewritten as

$\displaystyle
y^{\alpha}_{(k)}=\sum_{\gamma=1}^{\widetilde{d}_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{\alpha}_{\gamma}\;\widetilde{x}^{\gamma}_{(k)},\qquad\phi_{\beta}^{(k)}=\sum_{\gamma=1}^{r_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}_{\beta}^{\gamma}\;\psi_{\gamma}^{(k)},$
(5.69)

where $\alpha=1,\dots,d_{k}$, $\beta=1,\dots,\widetilde{r}_{k}$. These are
decompositions by bases. They generalise the formulae given in Section 5.1.

Note that $y^{\alpha}_{(k)}=f_{M}\big{(}x^{\alpha}_{(k)}\big{)}$ and
$\phi^{(k)}_{\alpha}=f^{M}\big{(}\widetilde{\psi}^{(k)}_{\alpha}\big{)}$. Thus
the formulae (5.69) describe the homomorphisms
$f_{M}\colon{\mathfrak{X}}_{A}({\mathbb{C}})\to{\mathfrak{X}}_{\widetilde{A}}({\mathfrak{R}})$
and $f^{M}\colon\Xi_{\widetilde{A}}({\mathbb{C}})\to\Xi_{A}({\mathfrak{R}})$
in terms of bases (5.24)–(5.27).

The formulae (5.58) and (5.59) are written in terms of bases in the form

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widehat{S}_{(k)}}(MN)\big{)}^{\alpha}_{\gamma}=\sum_{\beta=1}^{\widetilde{d}_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{\alpha}_{\beta}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widehat{S}_{(k)}}N\big{)}^{\beta}_{\gamma},$
(5.70)
$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}(MN)\big{)}^{\alpha}_{\gamma}=\sum_{\beta=1}^{\widetilde{r}_{k}}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}^{\alpha}_{\beta}\big{(}\mathop{\mathrm{Min}}\nolimits^{\widetilde{A}_{(k)}}N\big{)}^{\beta}_{\gamma}.$
(5.71)

###  5.6 Minors for left-equivalent idempotents

In contrast with the quadratic algebras ${\mathfrak{X}}_{A}({\mathbb{C}})$ and
$\Xi_{A}({\mathbb{C}})$ their dual algebras
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$ and $\Xi^{*}_{A}({\mathbb{C}})$ do not
coincide with the corresponding algebras for a left-equivalent idempotent
$A^{\prime}$ in general, but they do coincide for a right-equivalent
idempotent $A^{\prime}$. Indeed, by applying Proposition 2.12 to for
$A^{\top}$ and $(A^{\prime})^{\top}$ we see that the following 4 conditions
are equivalent:
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})={\mathfrak{X}}^{*}_{A^{\prime}}({\mathbb{C}})$,
$\Xi^{*}_{A}({\mathbb{C}})=\Xi^{*}_{A^{\prime}}({\mathbb{C}})$, $A$ is right-
equivalent to $A^{\prime}$, $S=1-A$ is left-equivalent to
$S^{\prime}=1-A^{\prime}$.

######  Proposition 5.22.

Let $S_{(k)}$, $A_{(k)}$ and $S^{\prime}_{(k)}$, $A^{\prime}_{(k)}$ be the
pairing operators for idempotents $A$ and $A^{\prime}$. If $A$ is left-
equivalent to $A^{\prime}$ then $A_{(k)}$ is left-equivalent to
$A^{\prime}_{(k)}$, while $S_{(k)}$ is right-equivalent to $S^{\prime}_{(k)}$
for each $k\geqslant 0$. If $A$ is right-equivalent to $A^{\prime}$ then
$A_{(k)}$ is right-equivalent to $A^{\prime}_{(k)}$, while $S_{(k)}$ is left-
equivalent to $S^{\prime}_{(k)}$ for each $k\geqslant 0$.

###### Proof.

The left equivalence of $A$ and $A^{\prime}$ implies
$\Xi_{A}({\mathbb{C}})=\Xi_{A^{\prime}}({\mathbb{C}})$ and hence we obtain the
equality
$(\Psi\otimes\cdots\otimes\Psi)A_{(k)}=(\Psi\otimes\cdots\otimes\Psi)=(\Psi\otimes\cdots\otimes\Psi)A^{\prime}_{(k)}$.
By applying Corollary 5.4 we obtain
$A^{\prime}_{(k)}\big{(}1-A_{(k)}\big{)}=0$ and
$A_{(k)}\big{(}1-A^{\prime}_{(k)}\big{)}=0$. Then, due to Lemma 2.11 the
idempotents $A_{(k)}$ and $A^{\prime}_{(k)}$ are left-equivalent. Analogously,
we obtain $\big{(}1-S_{(k)}\big{)}S^{\prime}_{(k)}=0$,
$\big{(}1-S^{\prime}_{(k)}\big{)}S_{(k)}=0$. By Lemma 2.11 this implies the
left equivalence of $1-S_{(k)}$ and $1-S^{\prime}_{(k)}$, which means in turn
the right equivalence of $S_{(k)}$ and $S^{\prime}_{(k)}$ by Proposition 2.8.
∎

Let
$A,A^{\prime}\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n})$
be left-equivalent idempotents and let $S_{(k)}$, $A_{(k)}$ and
$S^{\prime}_{(k)}$, $A^{\prime}_{(k)}$ be the corresponding pairing operators.
From Proposition 5.22 we obtain

$\displaystyle A^{\prime}_{(k)}=G_{[k]}A_{(k)},\qquad
S^{\prime}_{(k)}=S_{(k)}G_{(k)}$

for some
$G_{[k]},G_{(k)}\in\mathop{\mathrm{Aut}}\nolimits\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}$. Then, by means of Proposition 5.9 the identification
${\mathfrak{X}}_{A}({\mathbb{C}})_{k}={\mathfrak{X}}_{A^{\prime}}({\mathbb{C}})_{k}$
induces an isomorphism $\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S_{(k)}\cong\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S^{\prime}_{(k)}$. Explicitly it has the form

$\displaystyle\xi\mapsto\xi^{\prime}=\xi G_{(k)}=\xi
S^{\prime}_{(k)},\qquad\xi\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S_{(k)}.$

Indeed, we have $\xi^{\prime}\in\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}S^{\prime}_{(k)}$ and $\xi^{\prime}(X\otimes\cdots\otimes X)=\xi
S^{\prime}_{(k)}(X\otimes\cdots\otimes X)=\xi(X\otimes\cdots\otimes X)$. The
inverse map is $\xi^{\prime}\mapsto\xi^{\prime}G_{(k)}^{-1}=\xi S_{(k)}$.
Analogously, the identification
$\Xi_{A}({\mathbb{C}})_{k}=\Xi_{A^{\prime}}({\mathbb{C}})_{k}$ gives
$A_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}\cong
A^{\prime}_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}$,

$\displaystyle\pi\mapsto\pi^{\prime}=G_{[k]}\pi=A^{\prime}_{(k)}\pi,\qquad\pi\in
A_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}.$

Proposition 2.10 implies the equalities of subspaces:
$S_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}=S^{\prime}_{(k)}\big{(}({\mathbb{C}}^{n})^{\otimes k}\big{)}$ and
$\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}A_{(k)}=\big{(}({\mathbb{C}}^{n})^{\otimes
k}\big{)}^{*}A^{\prime}_{(k)}$. Proposition 5.9 in turn gives the isomorphisms
of vector spaces
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})_{k}\cong{\mathfrak{X}}^{*}_{A^{\prime}}({\mathbb{C}})_{k}$
and $\Xi^{*}_{A}({\mathbb{C}})_{k}\cong\Xi^{*}_{A^{\prime}}({\mathbb{C}})_{k}$
(these are not homomorphisms of algebras). They are given by the formulae
$x^{A}_{i_{1}}\cdots x^{A}_{i_{k}}\mapsto x^{A^{\prime}}_{i_{1}}\cdots
x^{A^{\prime}}_{i_{k}}$ and
$\psi_{A}^{i_{1}}\cdots\psi_{A}^{i_{k}}\mapsto\psi_{A^{\prime}}^{i_{1}}\cdots\psi_{A^{\prime}}^{i_{k}}$,
where $x^{A}_{i}$, $x^{A^{\prime}}_{i}$, $\psi^{i}_{A}$ and
$\psi^{i}_{A^{\prime}}$ are the generators of the algebras
${\mathfrak{X}}^{*}_{A}({\mathbb{C}})$,
${\mathfrak{X}}^{*}_{A^{\prime}}({\mathbb{C}})$, $\Xi^{*}_{A}({\mathbb{C}})$
and $\Xi^{*}_{A^{\prime}}({\mathbb{C}})$ respectively.

Let $(v_{\alpha})_{\alpha=1}^{d_{k}}$, $(v^{\alpha})_{\alpha=1}^{d_{k}}$ and
$(w_{\alpha})_{\alpha=1}^{r_{k}}$, $(w^{\alpha})_{\alpha=1}^{r_{k}}$ be dual
bases of $S_{(k)}\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}$,
$\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}^{*}S_{(k)}$,
$A_{(k)}\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}$ and
$\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}^{*}A_{(k)}$ respectively. Then

$\displaystyle
v^{\prime}_{\alpha}=v_{\alpha},\qquad(v^{\prime})^{\alpha}=v^{\alpha}G_{(k)}=v^{\alpha}S^{\prime}_{(k)},\qquad\alpha=1,\dots,d_{k},$
$\displaystyle(w^{\prime})^{\alpha}=w^{\alpha},\qquad
w^{\prime}_{\alpha}=G_{[k]}w_{\alpha}=A^{\prime}_{(k)}w_{\alpha},\qquad\alpha=1,\dots,r_{k},$
(5.72)

are dual bases of $S^{\prime}_{(k)}\big{(}({\mathbb{C}}^{m})^{\otimes
k}\big{)}$, $\big{(}({\mathbb{C}}^{m})^{\otimes
k}\big{)}^{*}S^{\prime}_{(k)}$, $\big{(}({\mathbb{C}}^{m})^{\otimes
k}\big{)}^{*}A^{\prime}_{(k)}$ and
$A^{\prime}_{(k)}\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}$.

By substituting $(v^{\prime})^{\alpha}$ and $w^{\prime}_{\alpha}$ to the
formulae (5.24) and (5.26) we obtain the same bases of
${\mathfrak{X}}_{A^{\prime}}({\mathbb{C}})_{k}={\mathfrak{X}}_{A}({\mathbb{C}})_{k}$
and $\Xi_{A^{\prime}}({\mathbb{C}})_{k}=\Xi_{A}({\mathbb{C}})_{k}$:

$\displaystyle(v^{\prime})^{\alpha}(X^{*}\otimes\cdots\otimes
X^{*})=v^{\alpha}S^{\prime}_{(k)}(X^{*}\otimes\cdots
X^{*})=v^{\alpha}(X^{*}\otimes\cdots X^{*})=x_{(k)}^{\alpha},$
$\displaystyle(\Psi\otimes\cdots\otimes\Psi)w^{\prime}_{\alpha}=(\Psi\otimes\cdots\otimes\Psi)A^{\prime}_{(k)}w_{\alpha}=(\Psi\otimes\cdots\otimes\Psi)w_{\alpha}=\psi_{\alpha}^{(k)}.$
(5.73)

Note that the bases of ${\mathfrak{X}}^{*}_{A^{\prime}}({\mathbb{C}})_{k}$ and
${\mathfrak{X}}_{A}({\mathbb{C}})_{k}$ defined by the formula (5.25) for
$v^{\prime}_{\alpha}=v_{\alpha}$ are not identified since they are elements of
different algebras. The same is valid for the bases (5.27) of
$\Xi_{A^{\prime}}({\mathbb{C}})_{k}$ and $\Xi_{A}({\mathbb{C}})_{k}$.

Let $\widetilde{S}_{(k)}$, $\widetilde{A}_{(k)}$ and
$\widetilde{S}^{\prime}_{(k)}$, $\widetilde{A}^{\prime}_{(k)}$ be the pairing
operators for two left-equivalent idempotents
$\widetilde{A},\widetilde{A}^{\prime}\in\mathop{\mathrm{End}}\nolimits({\mathbb{C}}^{m}\otimes{\mathbb{C}}^{m})$.
We have $\widetilde{S}^{\prime}_{(k)}=\widetilde{S}_{(k)}\widetilde{G}_{(k)}$
and $\widetilde{A}^{\prime}_{(k)}=\widetilde{G}_{[k]}\widetilde{A}_{[k]}$ for
some matrices
$\widetilde{G}_{(k)},\widetilde{G}_{[k]}\in\mathop{\mathrm{Aut}}\nolimits\big{(}({\mathbb{C}}^{m})^{\otimes
k}\big{)}$. Let $M$ be an $\big{(}A,\widetilde{A}\big{)}$-Manin matrix. Due to
Proposition 2.21 this means that $M$ is an
$\big{(}A^{\prime},\widetilde{A}^{\prime}\big{)}$-Manin matrix. We can
consider different minor operators for $M$, but they are related by a
multiplication of a complex matrix in the following way.

######  Proposition 5.23.

For any $\big{(}A,\widetilde{A}\big{)}$-Manin matrix we have

$\displaystyle\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}^{\prime}_{(k)}}M=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{G}_{(k)}=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{S}^{\prime}_{(k)},$
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A^{\prime}_{(k)}}M=G_{[k]}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}=A^{\prime}_{(k)}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}.$
(5.74)

###### Proof.

The formulae are obtained by the definitions of the minors:

$\displaystyle\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}^{\prime}_{(k)}}M=M^{(1)}\cdots
M^{(k)}\widetilde{S}^{\prime}_{(k)}=M^{(1)}\cdots
M^{(k)}\widetilde{S}_{(k)}\widetilde{G}_{(k)}=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{G}_{(k)},$
$\displaystyle\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}^{\prime}_{(k)}}M=M^{(1)}\cdots
M^{(k)}\widetilde{S}^{\prime}_{(k)}\widetilde{S}^{\prime}_{(k)}=\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{S}^{\prime}_{(k)},$
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A^{\prime}_{(k)}}M=A^{\prime}_{(k)}M^{(1)}\cdots
M^{(k)}=G_{[k]}A_{(k)}M^{(1)}\cdots
M^{(k)}=G_{[k]}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)},$
$\displaystyle\mathop{\mathrm{Min}}\nolimits^{A^{\prime}_{(k)}}M=A^{\prime}_{(k)}A^{\prime}_{(k)}M^{(1)}\cdots
M^{(k)}=A^{\prime}_{(k)}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}.$
∎

Note that an $n^{k}\times m^{k}$ matrix is a
$\big{(}A_{(2k)},\widetilde{A}_{(2k)}\big{)}$-Manin matrix iff it is an
$\big{(}A^{\prime}_{(2k)},\widetilde{A}^{\prime}_{(2k)}\big{)}$-Manin matrix.
It is a $\big{(}1-S_{(k)},1-\widetilde{S}_{(k)}\big{)}$-Manin matrix iff it is
a $\big{(}1-S^{\prime}_{(2k)},1-\widetilde{S}^{\prime}_{(2k)}\big{)}$-Manin
matrix. Due to Proposition 5.21 the both matrices
$\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M$ and
$\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}^{\prime}_{(k)}}M$ are
$\big{(}1-S_{(k)},1-\widetilde{S}_{(k)}\big{)}$-Manin matrices as well as
$\big{(}1-S^{\prime}_{(2k)},1-\widetilde{S}^{\prime}_{(2k)}\big{)}$-Manin
matrices. They are related by the change of basis in the space
$({\mathbb{C}}^{m})^{\otimes k}$ corresponding to the matrix
$\widetilde{G}_{(k)}^{-1}$ (see Section 2.4). In the same way the matrices
$\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M$ and
$\mathop{\mathrm{Min}}\nolimits^{A^{\prime}_{(k)}}M$ are
$\big{(}A_{(k)},\widetilde{A}_{(k)}\big{)}$-Manin matrices as well as
$\big{(}A^{\prime}_{(2k)},\widetilde{A}^{\prime}_{(2k)}\big{)}$-Manin
matrices, and they are related by the change of basis in the space
$({\mathbb{C}}^{n})^{\otimes k}$ corresponding to the matrix $G_{[k]}$.

Let $(\widetilde{v}_{\alpha})_{\alpha=1}^{\widetilde{d}_{k}}$,
$(\widetilde{v}^{\alpha})_{\alpha=1}^{\widetilde{d}_{k}}$ and
$(\widetilde{w}_{\alpha})_{\alpha=1}^{\widetilde{r}_{k}}$,
$(\widetilde{w}^{\alpha})_{\alpha=1}^{\widetilde{r}_{k}}$ be dual bases of
$\widetilde{S}_{(k)}\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}$,
$\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}^{*}\widetilde{S}_{(k)}$,
$\widetilde{A}_{(k)}\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}$ and
$\big{(}({\mathbb{C}}^{m})^{\otimes k}\big{)}^{*}\widetilde{A}_{(k)}$. Let

$\displaystyle\widetilde{v}^{\prime}_{\alpha}=\widetilde{v}_{\alpha},\qquad(\widetilde{v}^{\prime})^{\alpha}=\widetilde{v}^{\alpha}\widetilde{G}_{(k)}=\widetilde{v}^{\alpha}\widetilde{S}^{\prime}_{(k)},\qquad\alpha=1,\dots,\widetilde{d}_{k},$
$\displaystyle(\widetilde{w}^{\prime})^{\alpha}=\widetilde{w}^{\alpha},\qquad\widetilde{w}^{\prime}_{\alpha}=\widetilde{G}_{[k]}\widetilde{w}_{\alpha}=\widetilde{A}^{\prime}_{(k)}\widetilde{w}^{\alpha},\qquad\alpha=1,\dots,\widetilde{r}_{k}.$

######  Proposition 5.24.

Consider the entries of minor operators (5.67) and (5.68) in the bases defined
above:

$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}^{\alpha}_{\beta}=v^{\alpha}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}_{(k)}}M\big{)}\widetilde{v}_{\beta},\qquad\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}^{\prime}_{(k)}}M\big{)}^{\alpha}_{\beta}=(v^{\prime})^{\alpha}\big{(}\mathop{\mathrm{Min}}\nolimits_{\widetilde{S}^{\prime}_{(k)}}M\big{)}\widetilde{v}^{\prime}_{\beta},$
$\displaystyle\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}^{\gamma}_{\delta}=w^{\gamma}\big{(}\mathop{\mathrm{Min}}\nolimits^{A_{(k)}}M\big{)}\widetilde{w}_{\delta},\qquad\big{(}\mathop{\mathrm{Min}}\nolimits^{A^{\prime}_{(k)}}M\big{)}^{\gamma}_{\delta}=(w^{\prime})^{\gamma}\big{(}\mathop{\mathrm{Min}}\nolimits^{A^{\prime}_{(k)}}M\big{)}\widetilde{w}^{\prime}_{\delta}.$

These entries coincide: