principal bundle. We start by showing that the zero map is a strong principal
connection for the bundle.

######  Lemma 6.1.

We have the equality

(28)
$\displaystyle\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1})=\mathcal{O}_{q}(\mathrm{SU}_{n+1})\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})=\,\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\mathcal{O}_{q}(\mathrm{SU}_{n+1}).$

Hence the zero map on $\Omega^{(0,1)}_{q}(\mathrm{SU}_{n+1})$ is a principal
connection, and it is the unique principal connection for the bundle.

###### Proof.

It follows from [44, §2.3] (see also the discussion in [30, §2.3]) that

$\displaystyle\mathcal{O}_{q}(\mathrm{SU}_{n+1})\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})=$
$\displaystyle\,\mathcal{O}_{q}(\mathrm{SU}_{n+1})\Big{(}\mathcal{O}_{q}(\mathrm{SU}_{n+1})\square_{\mathcal{O}(\mathbb{T}^{n})}V_{\varnothing}^{(0,1)}\Big{)}$
$\displaystyle=$ $\displaystyle\,\mathcal{O}_{q}(\mathrm{SU}_{n+1})\otimes
V_{\varnothing}^{(0,1)},$

which gives us the first identity in (29).

To prove the second identity, we follow the approach of [4, Lemma 5.7]. Note
first that the Lusztig fodc is generated as a right
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$-module by elements of the form
$\overline{\partial}b$, for $b\in\mathcal{O}_{q}(\mathrm{F}_{n+1})$. Thus we
just need to show that, for any $a\in\mathcal{O}_{q}(\mathrm{SU}_{n+1})$,

(29) $\displaystyle
a\overline{\partial}b\in\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\mathcal{O}_{q}(\mathrm{SU}_{n+1}).$

Consider the set of algebra generators $z_{i}^{\varpi_{x}}$ of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ presented in (17). Since
$z_{i}^{\varpi_{x}}$ is a highest weight vector with respect to the left
action of $U_{q}(\mathfrak{sl}_{n+1})$ on
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$, it follows from (4) that it is contained
in the kernel of $\overline{\partial}$, the exterior derivative of our fodc on
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. Thus, for any
$b\in\mathcal{O}_{q}(\mathrm{F}_{n+1})$, we see that

(30) $\displaystyle
z_{i}^{\varpi_{x}}\overline{\partial}b=\overline{\partial}(z_{i}^{\varpi_{x}}b)-\overline{\partial}(z_{i}^{\varpi_{x}})b=\overline{\partial}(z_{i}^{\varpi_{x}}b).$

Consider next the element

$\displaystyle(S\otimes\mathrm{id})\circ\Delta(z^{\varpi_{x}}_{N_{x}})=\sum_{a=1}^{N_{x}}S(c^{\varpi_{x}}_{f_{N_{x}},v_{a}})\otimes
c^{\varpi_{x}}_{f_{a},v_{N_{x}}}=\sum_{a=1}^{N_{x}}S(c^{\varpi_{x}}_{f_{N_{x}},v_{a}})\otimes
z^{\varpi_{x}}_{a}\in{\mathcal{E}}_{-\varpi_{x}}\otimes{\mathcal{E}}_{\varpi_{x}},$

where the fact that $S(z^{\varpi_{x}}_{f_{N_{x}},v_{a}})$ is an element of
${\mathcal{E}}_{-\varpi_{x}}$ follows from the calculation

$\displaystyle K_{i}\triangleright
S(c^{\varpi_{x}}_{f_{N_{x}},v_{a}})=S(c^{\varpi_{x}}_{K_{i}f_{N_{x}},v_{a}})=q^{-(\alpha_{i},\varpi_{x})}S(c^{\varpi_{x}}_{f_{N_{x}},v_{a}}).$

Multiplying both sides of the identity gives us the new identity

(31)
$\displaystyle\varepsilon(z^{\varpi_{x}}_{N_{x}})=1=\sum_{a=1}^{N_{x}}S(z^{\varpi_{x}}_{f_{N_{x}},v_{a}})z^{\varpi_{x}}_{a}.$

Inserting this into (30) gives us the expression

$\displaystyle
z_{i}^{\varpi_{x}}\overline{\partial}b=\sum_{a=1}^{N_{x}}\overline{\partial}(z_{i}^{\varpi_{x}}bS(z^{\varpi_{x}}_{f_{N_{x}},v_{a}})z^{\varpi_{x}}_{a})=\sum_{a=1}^{N_{x}}\overline{\partial}(z_{i}^{\varpi_{x}}bS(z^{\varpi_{x}}_{f_{N_{x}},v_{a}}))z^{\varpi_{x}}_{a}.$

From the fact that our $\mathcal{P}$-grading is an algebra grading, we can
conclude that

$\displaystyle
z_{i}^{\varpi_{x}}bS(z^{\varpi_{x}}_{f_{N_{x}},v_{a}})\in{\mathcal{E}}_{0}=\mathcal{O}_{q}(\mathrm{F}_{n+1}).$

Thus we see that (29) is satisfied for $a=z^{\varpi_{x}}_{i}$. Consider next
the dual generator $\overline{z}^{\varpi_{x}}_{i}$, and note that

$\displaystyle\overline{z}^{\varpi_{x}}_{i}\overline{\partial}b=\overline{\partial}(\overline{z}^{\varpi_{x}}_{i}b)-\overline{\partial}(\overline{z}^{\varpi_{x}}_{i})b.$

An analogous application of the identity in (31) shows us that this element is
again contained in
$\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\mathcal{O}_{q}(\mathrm{SU}_{n+1})$,
which is to say, (29) is again satisfied. Thus the second identity in (28) is
satisfied. It is now clear that the zero map is a strong principal connection,
and moreover, that the zero map is the unique principal connection for the
bundle. ∎

The following corollary explicitly describes the connection associated to
$\Pi$ for a relative Hopf submodule of $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$.
It is a direct consequence of the various constructions involved, and so, we
omit it.

######  Proposition 6.2.

For any simple relative Hopf module
${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}_{0}$, considered as a sub-object of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$, the connection associated to $\Pi$ is
given explicitly by

$\displaystyle\overline{\partial}_{{\mathcal{F}}}:{\mathcal{F}}\to\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{F}},$
$\displaystyle f\mapsto
j\left(\sum_{\gamma\,\in\,\Delta^{+}}(E_{\gamma}\triangleright f)\otimes
e_{\gamma}\right)\\!.$

Since the zero map is obviously an $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module
map, the following corollary now follows directly from [7, Theorem 4.1].

######  Corollary 6.3.

For every relative Hopf module ${\mathcal{F}}\in{}^{A}_{B}\mathrm{Mod}_{0}$,
the associated connection
$\overline{\partial}_{{\mathcal{F}}}:{\mathcal{F}}\to\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{F}}$
is a bimodule connection.

We will now follow the representation theoretic argument given in [10] and
establish uniqueness and flatness for relative line module connections.

######  Lemma 6.4.

With respect to the Lusztig fodc, each line bundle ${\mathcal{E}}_{\gamma}$
admits a unique left $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-covariant connection
$\overline{\partial}_{{\mathcal{E}}_{\gamma}}$. Moreover, this connection is
flat.

###### Proof.

We know from Proposition 5.8 that

$\displaystyle\Phi\\!\left(\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{E}}_{\lambda}\right)\simeq\bigoplus_{\gamma\in\Delta^{+}}\Phi({\mathcal{E}}_{\gamma+\lambda}).$

Thus we see that the only morphism

$\displaystyle\Phi({\mathcal{E}}_{\lambda})\to\Phi\\!\left(\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{E}}_{\lambda}\right)\\!.$

is the zero morphism. Since the difference of two connections is a left
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$-module map, it now follows from Takeuchi’s
equivalence that $\overline{\partial}_{{\mathcal{E}}_{\lambda}}$ is the unique
left $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-covariant connection on
${\mathcal{E}}_{\lambda}$. (For more details see [10, Proposition 3.4].)

Proposition 5.8 also implies that the only morphism

$\Phi({\mathcal{E}}_{\lambda})\to\Phi\left(\Omega^{(0,2)}\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}{\mathcal{E}}_{\lambda}\right)$

is the zero map. Thus, since curvature is a left
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$-module map, we can similarly conclude that
$\overline{\partial}_{{\mathcal{E}}_{\lambda}}$ is flat. (For more details see
[10, Proposition 3.3].) ∎

This means that we have a complex

$\overline{\partial}_{{\mathcal{E}}_{\lambda}}:\Omega^{(0,\bullet)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}\mathcal{E}_{\lambda}\to\Omega^{(0,\bullet)}_{q}(\mathrm{F}_{n+1})\otimes_{\mathcal{O}_{q}(\mathrm{F}_{n+1})}\mathcal{E}_{\lambda}.$

which is a direct $q$-deformation of the Dolbeault anti-holomorphic complex of
the line bundles over the $A$-series full flag manifolds. We denote the
cohomology groups of the complex by
$H^{(0,\bullet)}_{\overline{\partial}}({\mathcal{E}}_{\lambda})$.

###  6.3. A Borel–Weil Theorem for $\mathcal{O}_{q}(\mathrm{F}_{n+1})$

In this subsection we establish the main result of this section, namely a
noncommutative Borel–Weil theorem for the full quantum flag manifold
$\mathcal{O}_{q}(\mathrm{F}_{n+1})$. This gives a noncommutative differential
geometric realisation of all the irreducible type-1 finite-dimensional
representations of $U_{q}(\mathfrak{sl}_{n+1})$.

######  Theorem 6.5.

For the full quantum flag manifold $\mathcal{O}_{q}(\mathrm{F}_{n+1})$, it
holds that

  1. 1.

$H^{0}(\mathcal{E}_{\lambda})$ is an irreducible
$U_{q}(\mathfrak{sl}_{n+1})$-module of highest weight $-w_{0}(\lambda)$, for
all $\lambda\in\mathcal{P}^{+}$,

  2. 2.

$H^{0}(\mathcal{E}_{\lambda})=0$, for all
$\lambda\in\mathcal{P}\,\backslash\,\mathcal{P}^{+}$.

###### Proof.

It follows from the construction of $\overline{\partial}$, as presented in
§2.2, that for any $e\in{\mathcal{E}}_{\lambda}$, we have
$\overline{\partial}(e)=0$ if and only if

(32) $\displaystyle E_{\gamma}\triangleright e=0,$ $\displaystyle\textrm{ for
all }\gamma\in\Delta^{+},$

which is to say, if and only if $e$ is a highest weight vector, of weight
$\lambda$.

If $\lambda\in\mathcal{P}/\mathcal{P}^{+}$, then it is clear that no such
element exists, meaning that $H^{0}(\mathcal{E}_{\lambda})=0$. For
$\lambda\in\mathcal{P}^{+}$, with respect to the Peter–Weyl decomposition of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$, the only highest weight elements of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ of weight $\lambda$ are of the form

$v\otimes v_{\lambda}\in V_{\lambda}^{\vee}\otimes V_{\lambda}\simeq
C(V_{\lambda}),$

where $v_{\lambda}$ is a highest weight vector of $V_{\lambda}$. Thus we see
that $H^{0}(\mathcal{E}_{\lambda})$ is an irreducible
$U_{q}(\mathfrak{sl}_{n+1})$-module of highest weight $-w_{0}(\lambda)$, for
all $\lambda\in\mathcal{P}^{+}$. ∎

Restricting to the case of $\lambda=0$, we get the following immediate
corollary.

######  Corollary 6.6.

The Lusztig fodc $\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})$ is _connected_ , which
is to say

$\mathrm{ker}\Big{(}\overline{\partial}:\mathcal{O}_{q}(\mathrm{F}_{n+1})\to\Omega^{(0,1)}_{q}(\mathrm{F}_{n+1})\Big{)}=0.$

We note that, by contrast, the fodc on $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is
not connected. However, this can be rectified by the addition of the anti-
holomorphic forms, corresponding to the negative root vectors. This will be
addressed in future work.

###  6.4. Quantum Homogeneous Coordinate Rings

Classically, for each $\lambda\in\mathcal{P}^{+}$, the line bundle
${\mathcal{E}}_{\lambda}$ is positive, or equivalently ample. We denote by
$S_{\lambda}[\mathrm{F}_{n+1}]$ the homogeneous coordinate ring of the
associated complex projective Kodaira embedding of $\mathrm{F}_{n+1}$. This
ring admits a direct generalisation to the quantum group setting: For
$\lambda=\sum_{i=1}^{n}\lambda_{i}\varpi_{i}$, the _quantum homogeneous
$\lambda$-coordinate ring_ is the subalgebra

$\displaystyle
S_{q,\lambda}[\mathrm{F}_{n+1}]:=\Big{\langle}z^{\lambda}U_{q}(\mathfrak{sl}_{n+1})\Big{\rangle}\subseteq\mathcal{O}_{q}(\mathrm{F}_{n+1}).$

These algebras have appeared many times in the literature, see for example
[43, 40, 23]. Moreover they constitute an important class of examples for
noncommutative projective algebraic geometry [37, 36], and for quantum cluster
algebras [15, 16].

We will now use the quantum Borel–Weil theorem established above to give a
novel noncommutative complex geometric realisation of the quantum homogeneous
coordinate rings, generalising the classical holomorphic section presentation
of the ring. The proof directly generalises the proof of the analogous
statement for the irreducible quantum quantum flag manifolds given in [4, 5],
and so, we omit it.

######  Proposition 6.7.

For the full quantum flag manifold $\mathcal{O}_{q}(\mathrm{F}_{n-1})$, and
any $\lambda\in\mathcal{P}^{+}$,

(33) $\displaystyle
S_{q,\lambda}[\mathrm{F}_{n+1}]=\bigoplus_{k\in\mathbb{Z}_{\geq
0}}H^{0}(\mathcal{E}_{k\lambda}),$

where each summand on the right-hand side is considered as a subspace of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$.

##  7\. Outside the Nice Setting

The reduced decompositions $\mathbf{j}$ and $\mathbf{j}^{\prime}$ (see A.6 for
the definition of $\mathbf{j}^{\prime}$) are distinguished, among the reduced
decompositions of the longest element of the Weyl group $S_{n+1}$, as those
that are _nice_ in the sense of Littleman. The notion of a nice decomposition
is defined for all Weyl groups, and is given in terms of crystal graphs for
finite-dimensional $U_{q}(\mathfrak{g})$-modules. See [24] for more details.

In this section we consider some low-dimensional examples of Lusztig’s root
vectors for non-nice decompositions, or more precisely, for their associated
commutator classes. Explicitly, we examine when the span of the root vectors
gives a quantum tangent space for the full quantum flag manifold. Since the
case of $U_{q}(\mathfrak{sl}_{2})$ is trivial, and for
$U_{q}(\mathfrak{sl}_{3})$ both decompositions are nice, we start with the
case of $U_{q}(\mathfrak{sl}_{4})$, treating its $6$ non-nice commutator
classes in detail. We then treat the case of $U_{q}(\mathfrak{sl}_{5})$,
examining those non-nice classes, from a total of $62$ such classes, that are
one or two braid moves from our choice of nice class.

###  7.1. The Opposite Involution, Reduced Decompositions, and DC

In this subsection we look at the action of the opposite involution on the
differential calculi associated to reduced decompositions of $w_{0}$ the
longest element of the Weyl group. In particular, we will show that the
dimension of the maximal prolongation of a fodc is invariant under the
opposite involution, as defined in Appendix A.6.

######  Lemma 7.1.

For any reduced decomposition $\mathbf{i}$ of $w_{0}$, the space of positive
root vectors $\mathfrak{n}_{\mathbf{i}}$ is a left, or a right, tangent space
if and only if $\mathfrak{n}_{\mathbf{i}^{\prime}}$ is a left, respectively
right, tangent space.

###### Proof.

This follows directly from Proposition A.4, and the fact that
$\overline{w_{0}}$ is a Hopf algebra automorphism of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. ∎

The following lemma now describes, in the case that we have tangent spaces,
the relationship between the corresponding ideals.

######  Lemma 7.2.

Assume that $\mathfrak{n}_{\mathbf{i}}$ is a left, or a right, tangent space,
and let us denote by $I\subseteq\mathcal{O}_{q}(\mathrm{SU}_{n+1})^{+}$ the
corresponding ideal. Then $\mathfrak{n}_{\mathbf{i}^{\prime}}$ is a left,
respectively right, tangent space with corresponding ideal
$\overline{w_{0}}^{\,*}(I)$.

###### Proof.

Let us assume that $y\in I$, which is to say, let us assume that

(34) $\displaystyle\big{\langle}y,X\big{\rangle}=0,$ $\displaystyle\textrm{
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \, for all
}X\in\mathfrak{n}_{\textbf{i}}.$

Equivalently, $y\in I$ if and only if

$\displaystyle\Big{\langle}\overline{w_{0}}^{*}(y),\overline{w_{0}}(X)\Big{\rangle}=0,$
$\displaystyle\textrm{ for all }X\in\mathfrak{n}_{\textbf{i}}.$

Thus, denoting by $I^{\prime}\subseteq\mathcal{O}_{q}(\mathrm{SU}_{n+1})^{+}$
the ideal corresponding to the tangent space
$\mathfrak{n}_{\mathbf{i}^{\prime}}=\overline{w_{0}}(\mathfrak{n}_{\mathbf{i}}),$
we see that $\overline{w_{0}}^{*}(y)\in I^{\prime}$, and so
$\overline{w_{0}}^{*}(I)\subseteq I^{\prime}$. In the other direction, we see
that $z\in I^{\prime}$ if and only if

$\displaystyle
0=\Big{\langle}z,\overline{w_{0}}(X)\Big{\rangle}=\Big{\langle}\overline{w_{0}}^{*}(z),X\Big{\rangle},$
$\displaystyle\textrm{ for all }X\in\mathfrak{n}_{\mathbf{i}}.$

Thus, since this is equivalent to having $\overline{w_{0}}^{*}(z)\in I$, we
see that we have established the opposite inclusion, and hence equality. ∎

######  Lemma 7.3.

Let $\mathbf{i}$ be a reduced decomposition of the longest element of the Weyl
group such that $\mathfrak{n}_{\mathbf{i}}$ is a right or a left tangent
space, and denote by $\Lambda_{\mathbf{i}}^{\bullet}$ the associated quantum
exterior algebra. Then, denoting by $\Lambda_{\mathbf{i}^{\prime}}^{\bullet}$
the quantum exterior algebra of the quantum tangent space
$\mathfrak{n}_{\mathbf{i}^{\prime}}=w_{0}(\mathfrak{n}_{\mathbf{i}})$, an
algebra isomorphism
$\Lambda^{\bullet}_{\mathbf{i}}\to\Lambda^{\bullet}_{\mathbf{i}’}$ is
determined by

$\displaystyle[a_{1}]\wedge\cdots\wedge[a_{k}]\leavevmode\nobreak\
\mapsto\leavevmode\nobreak\
[\overline{w_{0}}^{*}(a_{1})]\wedge\cdots\wedge[\overline{w_{0}}^{*}(a_{k})],$
$\displaystyle\textrm{ for }k\in\mathbb{Z}_{\geq 0}.$

###### Proof.

Note first that we have a linear isomorphism

$\displaystyle p:\Lambda^{1}_{\mathbf{i}}\to\Lambda^{1}_{\mathbf{i}’},$
$\displaystyle[a]\mapsto[\overline{w_{0}}^{*}(a)].$

We can uniquely extend this to an algebra isomorphism

$p:\mathcal{T}\big{(}\Lambda^{1}_{\mathbf{i}}\big{)}\to\mathcal{T}\big{(}\Lambda^{1}_{\mathbf{i}^{\prime}}\big{)}.$

For any element $y\in I$, we now see that

$\displaystyle
p\big{(}[y_{(1)}]\otimes[y_{(2)}]\big{)}=[\overline{w_{0}}^{*}(y_{(1)})]\otimes[\overline{w_{0}}^{*}(y_{(2)})]=[\overline{w_{0}}^{*}(y)_{(1)}]\otimes[\overline{w_{0}}^{*}(y)_{(2)}]\in(\overline{w_{0}}^{*}(I))^{(2)}.$

An analogous calculation in the opposite direction then confirms that $p^{-1}$
maps $(\overline{w_{0}}^{*}(I))^{(2)}$ to $I^{(2)}$. Thus $p$ restricts to an
algebra isomorphism between $\Lambda^{\bullet}_{\mathbf{i}}$ and
$\Lambda^{\bullet}_{\mathbf{i}’}$ as claimed. ∎

The following proposition now follows directly. In particular, it means that
one need only check the classical dimension requirement for $\mathbf{i}$ or
$\mathbf{i}^{\prime}$.

######  Proposition 7.4.

A reduced decomposition $\mathbf{i}$ gives a tangent space whose associated
quantum exterior algebra has classical dimension if and only if the same is
true for $\mathbf{i}^{\prime}$.

###  7.2. The Case of $U_{q}(\mathfrak{sl}_{4})$

For the quantised enveloping algebra $U_{q}(\mathfrak{sl}_{4})$, the Weyl
group is $S_{4}$, and its longest element has $8$ commutator classes of
reduced decompositions. We present them in diagrammatic form, each node being
a choice of representative from a commutator class and an edge between two
nodes indicating that the two decompositions differ by a braid move:

$321323$$321232$$312132$$123212$$232123$$231213$$212321$$123121$.

Clearly, operating on a reduced decomposition by the opposite automorphism
gives a well-defined map between commutator classes. We present this action in
the following diagram, where a blue arrow between two decompositions denotes
that the automorphism maps one decomposition to another.

$321323$$321232$$312132$$123212$$232123$$231213$$212321$$123121$

We can calculate the associated spaces of root vectors using the formulae
given in Appendix A. The set of $6$ roots vectors associated to each
decomposition always contains the simple root vectors $E_{1},E_{2}$, and
$E_{3}$. We present the other three non-simple root vectors in the table
below.

Decomposition | Non-Simple Root Vectors
---|---
$321323$ |  $[E_{3},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{2},E_{1}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{3},E_{2}]_{-},E_{1}]_{-}$
$321232$ |  $[E_{1},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{3},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{3},E_{2}]_{-},E_{1}]_{-}$
$232123$ |  $[E_{2},E_{1}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{2},E_{3}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{3},E_{2}]_{-},E_{1}]_{-}$
$213213$ |  $[E_{2},E_{1}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{2},E_{3}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{2},E_{3}]_{-},E_{1}]_{-}$
$132132$ |  $[E_{1},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{3},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{1},E_{2}]_{-},E_{3}]_{+}$
$123212$ |  $[E_{1},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{3},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{1},[E_{2},E_{3}]_{-}]_{-}$
$212321$ |  $[E_{2},E_{1}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{2},E_{3}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{1},E_{2}]_{-},E_{3}]_{-}$
$123121$ |  $[E_{1},E_{2}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [E_{2},E_{3}]_{-},\leavevmode\nobreak\ \leavevmode\nobreak\ [[E_{1},E_{2}]_{-},E_{3}]_{-}$

The formulae in Example 3.2 show us how the coproduct operates on the length
two commutators. A direct calculation shows that the coproduct of
$[[E_{2},E_{3}]_{-},E_{1}]_{-}$ is equal to

$\displaystyle[[E_{2},E_{3}]_{-},E_{1}]_{-}\otimes K_{1}K_{2}K_{3}+q^{-1}\nu
E_{1}\otimes[E_{2},E_{3}]_{-}K_{1}+q^{-1}\nu E_{3}\otimes[E_{2},E_{1}]_{-}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\leavevmode\nobreak\ \leavevmode\nobreak\ q^{-2}\nu^{2}E_{1}E_{3}\otimes
E_{2}K_{1}+1\otimes[[E_{2},E_{3}]_{-},E_{1}]_{-},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

and moreover, that the coproduct of $[[E_{1},E_{2}]_{-},E_{3}]_{+}$ is equal
to

$\displaystyle[[E_{1},E_{2}]_{-},E_{3}]_{+}\otimes
K_{1}K_{2}K_{3}+q^{-1}\nu[E_{2},E_{3}]_{+}\otimes
E_{1}K_{3}+\nu[E_{1},E_{2}]_{-}\otimes E_{3}K_{1}K_{2}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ +(q^{-1}-1)\nu
E_{2}\otimes
E_{1}E_{3}+1\otimes[[E_{1},E_{2}]_{-},E_{3}]_{+}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

Together with the action of the opposite automorphism, this allows us to
determine which spaces of positive root vectors give tangent spaces for the
full quntum flag manifold $\mathcal{O}_{q}(\mathrm{F}_{4})$. In this higher
rank situation, we find that $T\oplus\mathbb{C}1$ is no longer always a
subcoalgebra. For some cases it is a right coideal of
$\mathcal{O}_{q}(\mathrm{F}_{4})^{\circ}$, and for some cases it is a left
coideal. This means that the corresponding fodc is either left, or
respectively right, covariant. We now present this in the following diagram,
building on the graphical presentation of the reduced decomposition above:

Two-SidedLeftRightLeftRightLeftRightTwo-Sided.

Finally, we address the question of the dimension of the maximal prolongations
of the fodc associated to the different commutator classes of reduced
decompositions. The following result states that those that have classical
dimension are precisely those that give two-sided coideals. The proof is a
direct calculation, analogous to the proof of Lemma 3.9, and so, it is
omitted.

######  Proposition 7.5.

For the Drinfeld–Jimbo quantised enveloping algebra
$U_{q}(\mathfrak{sl}_{4})$, the commutator classes of $\mathbf{j}$ and
$\mathbf{j}^{\prime}$ are the only commutator classes of reduced
decompositions of $w_{0}$, the longest element of the Weyl group $S_{n+1}$,
whose associated tangent spaces have dc with classical total degree.

We can represent the above proposition diagrammatically as follows:

ClassNon-ClassNon-ClassNon-ClassNon-ClassNon-ClassNon-ClassClass

where _Class_ , and _Non-Class_ , indicate whether or not the maximal
prolongation of the fodc associated to the reduced decomposition has
classical, or non-classical, dimension respectively. Thus we again see that
the nice decompositions are confirmed as exceptional amongst the $8$
commutator classes.

###  7.3. The Case of $U_{q}(\mathfrak{sl}_{5})$

In this subsection we treat the rank four case of $U_{q}(\mathfrak{sl}_{5})$.
For this example the number of reduced decompositions rises to $62$. A full
table of the reduced decomposition classes can be found in [47, Figure 3].
Here we content ourselves with examining the sub-graph consisting of the
reduced decomposition classes generated by the decomposition $4321432434$ and
at most two braid moves:

A direct calculation shows, that for certain decompositions, the associated
positive root vectors form neither a left nor a right tangent space, a
phenomenon that does not occur for lower orders. We now present in
diagrammatic form the tangent space characterisation of the reduced
decomposition classes:

Just as for $U_{q}(\mathfrak{sl}_{3})$, we get a dual picture by operating by
the opposite automorphism. We will not treat the remaining $44$ cases.
Instead, in the next section we make a conjecture and pose some natural
questions.

###  7.4. Some Conjectures

We finish with some conjectures, identifying interesting future directions of
research. Motivated by the observations of §7.2 and §7.3, we conjecture that
the Lusztig fodc, and the _dual_ fodc associated to the reduced decomposition
$\mathbf{j}^{\prime}$, are distinguished by having tangent spaces $T$ for
which $T\oplus\mathbb{C}1$ is a coideal of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})^{\circ}$.

######  Conjecture 7.6.

The two reduced decompositions $\mathbf{j}$ and $\mathbf{j}^{\prime}$, of the
longest element of the Weyl group of $\mathfrak{sl}_{n+1}$, are the only
reduced decompositions for which the span of the positive root vectors is a
right and left tangent space.

From the discussions of §7.2 and §7.3, we see that the attribute of giving a
two-sided, left, or right tangent space, or not giving a tangent spaces at
all, separates equivalence classes of reduced decompositions into four
distinct non-empty families. It is natural to ask if these divisions admit a
natural combinatorical description in one of the many equivalent formulations
of reduced decomposition classes, such as rhombic tilings [12, 13] or
pseudoline arrangements [14]. This gives the following problem.

######  Problem 7.7.

For the Drinfeld–Jimbo quantum group $U_{q}(\mathfrak{sl}_{n+1})$, determine
those reduced decompositions of the longest elements of the Weyl group for
which the associated space of positive root vectors is a right or a left
quantum tangent space?

Motivated by the low-order maximal prolongation results presented in so far in
this subsection, and by Conjecture 7.6, we make the following second
conjecture.

######  Conjecture 7.8.

The two reduced decompositions $\mathbf{j}$ and $\mathbf{j}^{\prime}$, of the
longest element of the Weyl group of $\mathfrak{sl}_{n+1}$, are the only
reduced decompositions for which the associated space of positive root vectors
forms a tangent space whose associated fodc has a maximal prolongation of
classical dimension.

Another natural question to ask is whether the approach of this paper can be
extended to the $B,C$, and $D$-series, or even to the exceptionals. This gives
the second problem of the paper.

######  Problem 7.9.

Extend the approach of the present paper to the general Drinfeld–Jimbo setting
$U_{q}(\mathfrak{g})$, for $\mathfrak{g}$ a complex semisimple Lie algebra?

This problem can be expected to be quite challenging. Outside of the
$A$-series setting, while there exist various formulae for the number of
reduced decompositions

very little is even known about the number of commutator classes of reduced
decompositions of the longest elements of the Weyl groups. However, it offers
the exciting prospect of investigating these problems using ideas from
noncommutative geometry.

###  7.5. A General Class of Tangent Spaces for $U_{q}(\mathfrak{sl}_{3})$

In this final subsection, we will examine in more detail the case of
$U_{q}(\mathfrak{sl}_{3})$. Classically, $\mathfrak{sl}_{3}$ is of course the
first $A$-series Lie algebra with a non-simple root vector. While there are no
non-nice reduced decompositions for the rank $2$ case, it is still possible to
consider other, more general deformations of this commutator bracket
$[E_{2},E_{1}]$.

######  Proposition 7.10.

The coproduct of $[E_{2},E_{1}]_{\theta}$ is given by

$[E_{2},E_{1}]_{\theta}\otimes K_{1}K_{2}+(q^{-1}-\theta)E_{2}\otimes
E_{1}K_{2}+(1-q^{-1}\theta)E_{1}\otimes
E_{2}K_{1}+1\otimes[E_{2},E_{1}]_{\theta}.$

Thus, for all $\theta\in\mathbb{C}$, the subspace

$T_{\theta}:=\mathrm{span}_{\mathrm{C}}\Big{\\{}E_{1},\,E_{2},\,[E_{2},E_{1}]_{\theta}\Big{\\}}\subseteq\mathcal{O}_{q}(\mathrm{SU}_{3})^{\circ},$

is a quantum tangent space for $\mathcal{O}_{q}(\mathrm{F}_{3})$.

We denote the maximal prolongation of the corresponding left
$\mathcal{O}_{q}(\mathrm{SU}_{3})$-covariant fodc by
$\Omega^{(0,\bullet)}_{q,\theta}$. Just as for the Lusztig tangent spaces, we
can now read off the module structure of the associated tangent space
$\Lambda^{1}$.

######  Corollary 7.11.

All non-zero actions of non-diagonal generators $u_{lk}$ on the basis elements
$e_{ji}$ are given by

$\displaystyle e_{21}\,u_{32}=(q-\theta)e_{31},$ $\displaystyle
e_{32}\,u_{21}=(q^{-1}-\theta)e_{31},$

with the diagonal actions the same as for the Lusztig fodc.

Just as in Example 3.12 we can present this in diagrammatic form

where the basis elements are arranged in lower triangular form and we have
drawn an arrow from one basis element $e$ to another $e^{\prime}$ if there
exists a generator $u_{ij}$ such that $eu_{ij}$ is a scalar multiple of
$e^{\prime}$.

######  Proposition 7.12.

For any $\theta\neq q^{\pm 1}$, it holds that $e_{21}\wedge
e_{32}=-\theta\,e_{32}\wedge e_{21}$ with all other products of generators
equal to zero. In other words,

$\Omega^{(0,2)}_{q,\theta}=\mathcal{O}_{q}(\mathrm{SU}_{3})\otimes
e_{21}\wedge e_{32}=\mathcal{O}_{q}(\mathrm{SU}_{3})\otimes e_{32}\wedge
e_{21}$

and all higher forms are trivial. In particular, for $\theta\neq q^{\pm 1}$,
the dc have non-classical dimension.

###### Proof.

A generating set for the ideal
$I\subseteq\mathcal{O}_{q}(\mathrm{SU}_{3})^{+}$ of the fodc is given by the
union of $G_{1}$ and $G_{2}$ (as defined in Proposition 3.7) and the sets

$\displaystyle G_{3}\leavevmode\nobreak\ \leavevmode\nobreak\ :=$
$\displaystyle\,\\{u_{21}u_{21},u_{31}u_{31},u_{32}u_{32},u_{21}u_{31},u_{32}u_{31}\\},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ G_{3,\theta}:=$
$\displaystyle\,\\{u_{21}u_{32}-(q-\theta)u_{31},\,u_{32}u_{21}-(q^{-1}-\theta)u_{31}\\}.$

Just as for the Lusztig calculi, the generators contained in $G_{1}$ and
$G_{2}$ give the zero relation. From the elements of $G_{3}$ we get the
relations

(35) $\displaystyle e_{21}\wedge e_{21}=e_{32}\wedge e_{32}=e_{31}\wedge
e_{31}=0,$ $\displaystyle e_{21}\otimes e_{31}=-(q+q^{-1}-\theta)e_{31}\otimes
e_{21},$

while from the elements of $G_{3,\theta}$ we get the relations

(36) $\displaystyle e_{31}\wedge e_{32}=-(q+q^{-1}-\theta)e_{32}\wedge
e_{31},$ $\displaystyle e_{21}\wedge e_{32}=-\theta e_{32}\wedge e_{21}.$

Consider now the right action of the element $u_{32}$ on $e_{21}\otimes
e_{21}$:

$\displaystyle(e_{21}\otimes e_{21})u_{32}=$
$\displaystyle\,(q-\theta)(qe_{31}\otimes e_{21}+e_{21}\otimes e_{31}),$

meaning that we get the relation $e_{21}\wedge e_{31}=-qe_{31}\wedge e_{21}.$
Combining this with the second relation in in (35), we get that

$e_{21}\wedge e_{31}=e_{31}\wedge e_{21}=0.$

Similarly, acting on $e_{32}\otimes e_{32}$ by $u_{21}$ and combining with the
first relation in (36) allows us to conclude that $e_{31}\wedge e_{32}$ and
$e_{32}\wedge e_{31}$ are also zero. The claimed description of the degree two
and higher forms now follows immediately. ∎

######  Remark 7.13.

It is important to note that, in the proof of Proposition 7.12, the set of
relations produced from our choice of generating set are not closed under the
right action of $\mathcal{O}_{q}(\mathrm{SU}_{3})$. Hence, they do not give a
complete set of relations. We contrast this with the proof of Lemma 3.9, where
the image of the chosen generating set under $\omega$ is a right
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$-module of
$\Lambda^{(0,1)}_{q}(\mathrm{SU}_{n+1})$.

##  Appendix A The $A$-Series root vectors

In this appendix we give necessary preliminaries concerning the Drinfeld–Jimbo
quantum group $U_{q}(\mathfrak{sl}_{n+1})$, Lusztig’s braid group action, and
the root vectors associated to a reduced decomposition of the longest element
of the $A$-series Weyl group. We also explicitly derive, for sake of
completeness, the root vectors associated to the reduced decomposition
discussed in §3.1.

###  A.1. The $A_{n}$ Root System

In this subsection, so as to set notation, we recall some basic definitions
and results about the $A_{n}$-root system associated to the special linear Lie
algebra $\mathfrak{sl}_{n+1}$. Let $\\{\varepsilon_{i}\\}_{i=1}^{n+1}$ be the
standard basis of $\mathbb{R}^{n+1}$, and endow it with its canonical
Euclidean structure. The root system $A_{n}$ is the pair $(V,\Delta)$, where
$V$ is the subspace of $\mathbb{R}^{n+1}$ given by

$V:=\bigg{\\{}(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\,|\,\sum_{i=1}^{n+1}x_{i}=0\bigg{\\}},$

and the set of roots is the finite set

$\Delta:=\Big{\\{}\alpha_{ij}:=\varepsilon_{i}-\varepsilon_{j}\,|\,1\leq i\neq
j\leq n+1\Big{\\}}.$

We take the standard subset of positive roots

$\Delta^{+}:=\Big{\\{}\alpha_{ij}:=\varepsilon_{i}-\varepsilon_{j}\,|\,1\leq
i<j\leq n+1\Big{\\}},$

and we denote by $\mathcal{Q}^{+}$ the $\mathbb{Z}_{\geq 0}$-span of
$\Delta^{+}$. The associated set of simple roots is then given by

$\Pi:=\Big{\\{}\alpha_{i}:=\alpha_{i,i+1}=\varepsilon_{i}-\varepsilon_{i+1}\,|\,i=1,\dots,n\Big{\\}},$

and we denote the associated Cartan matrix by $(a_{ij})$.

The _standard partial order_ $\prec$ on $\Delta$ is defined by
$\beta\prec\gamma$ if $\gamma-\beta\in\mathcal{Q}^{+}$. We denote the
associated set of fundamental weights by $\\{\varpi_{1},\dots,\varpi_{n}\\}$,
and write ${\mathcal{P}}^{+}$ for the $\mathbb{Z}_{\geq 0}$-span of the
fundamental weights. We recall that the Weyl group of the root system is the
symmetric group $S_{n+1}$.

In addition to the partial order $\succ$, we will also be interested in
certain total orders known as convex orders [33]: A _convex order_ for
$\Delta^{+}$ is a total order such that for any $\beta,\gamma\in\Delta^{+}$
satisfying $\beta+\gamma\in\Delta^{+}$ and $\beta<\gamma$, we necessarily have
that $\beta<\beta+\gamma<\gamma$. As shown in [33], convex orderings
correspond to reduced decompositions of the longest element $w_{0}$ of the
Weyl group. For a reduced decomposition $w_{0}=s_{i_{1}}\cdots s_{i_{d}}$,
with $d:=|\Delta^{+}|$, we write

$\displaystyle\beta_{1}:=\alpha_{1},$ $\displaystyle\beta_{k}:=s_{i_{1}}\cdots
s_{i_{k-1}}(\alpha_{k}),$ $\displaystyle\textrm{ for }k=2,\dots,d.$

A convex order $<$ on $\Delta^{+}$ is induced by the fact that
$\Delta^{+}=\\{\beta_{1},\dots,\beta_{d}\\}$. Throughout the paper we deal
with $\mathbf{j}$ the reduced decomposition

$\displaystyle w_{0}=(s_{n}s_{n-1}\cdots s_{1})(s_{n}s_{n-1}\cdots
s_{2})\cdots(s_{n}s_{n-1})s_{n}.$

The associated convex order is given by

$\displaystyle\alpha_{ij}<\alpha_{kl},$ $\displaystyle\textrm{ whenever
}i>k,\textrm{ or whenever }i=k\textrm{ and }j>l.$

This order will be used in Theorem 3.10 and in §3.5.

######  Example A.1.

For the root system $A_{3}$, the convex order associated to the decomposition

$w_{0}=(s_{3}s_{2}s_{1})(s_{3}s_{2})s_{1}$

can be represented graphically as follows: Identify the positive roots with
the strictly upper triangular root vectors in $\mathfrak{sl}_{4}$. Then we
have the diagram

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where an arrow pointing from one node to another denotes that the origin root
is greater than the range root. The total order is then given by the
transitive closure of these inequalities.

###  A.2. The Drinfeld–Jimbo Quantised Enveloping Algebra
$U_{q}(\mathfrak{sl}_{n+1})$

Here we recall the definition of the Hopf algebra
$U_{q}(\mathfrak{sl}_{n+1})$, following the conventions of [21, §7]. For any
$q\in{\mathbb{R}}$, such that $q\neq-1,0,1$, the _Drinfeld–Jimbo quantised
enveloping_ $U_{q}(\mathfrak{sl}_{n+1})$ is the noncommutative associative
algebra generated by the elements $E_{i},F_{i},K_{i}$, and $K^{-1}_{i}$, for
$i=1,\ldots,n$, subject to the relations

$\displaystyle K_{i}E_{j}=q^{a_{ij}}E_{j}K_{i},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
K_{i}F_{j}=q^{-a_{ij}}F_{j}K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
K_{i}K_{j}=K_{j}K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ K_{i}K_{i}^{-1}=K_{i}^{-1}K_{i}=1,$
$\displaystyle
E_{i}F_{j}-F_{j}E_{i}=\delta_{ij}\frac{K_{i}-K^{-1}_{i}}{q-q^{-1}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $

where $\nu:=q-q^{-1}$, and the _quantum Serre relations_

$\displaystyle E_{i\pm 1}^{2}E_{i}-[2]_{q}E_{i\pm 1}E_{i}E_{i\pm
1}+E_{i}E^{2}_{i\pm 1}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ F_{i\pm 1}^{2}F_{i}-[2]_{q}F_{i\pm
1}F_{i}F_{i\pm 1}+F_{i}F^{2}_{i\pm 1}=0,$ $\displaystyle
E_{i}E_{i}=E_{j}E_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
F_{i}F_{j}=F_{j}F_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{ for
}|i-j|>1.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $

The formulae

$\displaystyle\Delta(K_{i})=K_{i}\otimes K_{i},$
$\displaystyle\Delta(E_{i})=E_{i}\otimes K_{i}+1\otimes E_{i},$
$\displaystyle\Delta(F_{i})=F_{i}\otimes 1+K_{i}^{-1}\otimes F_{i},$

and $\varepsilon(E_{i})=\varepsilon(F_{i})=0$, $\varepsilon(K_{i})=1$, define
a Hopf algebra structure on $U_{q}(\mathfrak{sl}_{n+1})$ satisfying

$\displaystyle S(E_{i})=-E_{i}K_{i}^{-1},\leavevmode\nobreak\
\leavevmode\nobreak\ S(F_{i})=-K_{i}F_{i},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
S(K_{i})=K_{i}^{-1}.$

We denote by $U_{q}(\mathfrak{h})$ the subalgebra of
$U_{q}(\mathfrak{sl}_{n+1})$ generated by the elements $K_{i},K^{-1}_{i}$, for
$i=1,\dots,n$. For any $\lambda\in\mathcal{P}^{+}$, we denote by $V_{\lambda}$
the associated finite-dimensional irreducible type-$1$ representation of
$U_{q}(\mathfrak{sl}_{n+1})$. For any $U_{q}(\mathfrak{sl}_{n+1})$-module $V$,
an element $v\in V$ is said to be a _weight vector_ if, for some
$\mathrm{wt}(v)\in\mathcal{P}^{+}$, we have
$K_{i}v=q^{(\alpha_{i},\mathrm{wt}(v))}v$, for all $i=1,\ldots,n$. We call
$\mathrm{wt}(v)$ the _weight_ of $v$.

###  A.3. The Quantum Coordinate Algebra $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$

Let $V$ be a finite-dimensional $U_{q}(\mathfrak{sl}_{n+1})$-module, $v\in V$,
and $f\in V^{*}$, the linear dual of $V$. Consider the linear functional
$c^{\textrm{\tiny$V$}}_{f,v}:U_{q}(\mathfrak{sl}_{n+1})\to{\mathbb{C}}$
defined by $c^{\textrm{\tiny$V$}}_{f,v}(X):=f\big{(}X(v)\big{)}$. The
coordinate ring of $V$ is the subspace

$\displaystyle
C(V):=\text{span}_{\mathbb{C}}\\!\left\\{c^{\textrm{\tiny$V$}}_{f,v}\,|\,v\in
V,\,f\in V^{*}\right\\}\subseteq U_{q}(\mathfrak{sl}_{n+1})^{*}.$

It is easily checked that $C(V)$ is contained in
$U_{q}(\mathfrak{sl}_{n+1})^{\circ}$, the Hopf dual of
$U_{q}(\mathfrak{sl}_{n+1})$, and moreover that a Hopf subalgebra of
$U_{q}(\mathfrak{sl}_{n+1})^{\circ}$ is given by

(37)
$\displaystyle\mathcal{O}_{q}(\mathrm{SU}_{n+1}):=\bigoplus_{\lambda\in\mathcal{P}^{+}}C(V_{\lambda}).$

We call $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ the quantum coordinate algebra of
$\mathrm{SU}_{n+1}$. Moreover, we call the decomposition of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ given in (37) the _Peter–Weyl
decomposition_ of $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$.

When $V=V_{\lambda}$, for $\lambda\in\mathcal{P}^{+}$, and $f\in V_{\lambda}$,
$v\in V_{\lambda}$, we find it convenient to $c^{\lambda}_{f,v}$ for the
associated coordinate functional. For any choice of weight basis
$\\{e_{i}\\}_{i=1}^{n}$ of $V_{\varpi_{1}}$, the associated _matrix
coefficients_ are

(38) $\displaystyle u_{ij}:=c^{\varpi_{1}}_{e^{i},e_{j}},$
$\displaystyle\textrm{ where }\\{e^{i}\\}_{i=1}^{n}\textrm{ is the dual basis
of }V^{*}_{\varpi_{1}}.$

Note that $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is generated as an algebra by
the elements $u_{ij}$. We fix, for once and for all, a weight basis satisfying

(39) $\displaystyle\langle u_{i+1,i},E_{i}\rangle=1,$ $\displaystyle\langle
u_{i,i+1},F_{i}\rangle=1,$ $\displaystyle\langle
u_{ii},K_{j}\rangle=q^{\delta_{j+1,i}-\delta_{ij}},$

and all other pairings are zero. It can be shown that the algebra
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is generated by the elements $u_{ij}$
subject to the relations

$\displaystyle u_{ij}u_{i^{\prime}j}=qu_{i^{\prime}j}u_{ij},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
u_{ij}u_{ij^{\prime}}=qu_{ij^{\prime}}u_{ij},$ $\displaystyle
u_{ij^{\prime}}u_{i^{\prime}j}=u_{i^{\prime}j}u_{ij^{\prime}},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
u_{ij}u_{i^{\prime}j^{\prime}}=u_{i^{\prime}j^{\prime}}u_{ij}+(q-q^{-1})u_{ij^{\prime}}u_{i^{\prime}j},$

for $i,i^{\prime},j,j^{\prime}=1,\dots,n+1$, such that
$i<i^{\prime},j<j^{\prime}$, along with the quantum determinant relation which
we omit. The Hopf algebra structure on $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ is
determined by $\Delta(u_{ij}):=\sum_{a=1}^{n+1}u_{ia}\otimes u_{aj}$ and
$\varepsilon(u_{ij}):=\delta_{ij}$.

###  A.4. Lusztig’s Root Vectors

In this subsection we recall Lusztig’s braid group action on
$U_{q}(\mathfrak{sl}_{n+1})$, and the space of positive root vectors
associated to a choice of a reduced decomposition of the longest element of
the Weyl group. See [26], or [21, §6.2], for a more detailed presentation.
Throughout, we will use twisted commutators, that is to say, for any algebra
$A$ and some constant $\lambda\in\mathbb{C}$, we denote $[a,b]_{c}:=ab-\lambda
ba,$ for $a,b\in A$.

To every $i=1,\ldots,n$, there corresponds an algebra automorphism $T_{i}$ of
$U_{q}(\mathfrak{sl}_{n+1})$ which acts on the generators, for $j=1,\dots,n$,
as

$\displaystyle T_{i}(K_{j})=K_{j}K_{i}^{-a_{ij}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
T_{i}(E_{i})=-F_{i}K_{i},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ T_{i}(F_{i})=-K_{i}^{-1}E_{i},$
$\displaystyle T_{k\pm 1}(E_{k})=-[E_{k\pm
1},E_{k}]_{q^{-1}},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ T_{i}(E_{k})=E_{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{ if }|i-k|>1,$ $\displaystyle T_{k\pm
1}(F_{k})=-[F_{k},F_{k\pm 1}]_{q},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ T_{i}(F_{k})=F_{k},\leavevmode\nobreak\
\,\,\leavevmode\nobreak\ \textrm{ if }|i-k|>1.$

The mapping $s_{i}\rightarrow T_{i}$, for $i=1,\dots,n$, determines a
homomorphism of the braid group $B_{n+1}$ into the group of algebra
automorphisms of $U_{q}(\mathfrak{sl}_{n+1})$. We associate, to any choice of
reduced decomposition $w_{0}=s_{i_{1}}\cdots s_{i_{d}}$ of the longest element
of the Weyl group, the following elements

$\displaystyle E_{\beta_{k}}:=T_{i_{1}}T_{i_{2}}\cdots
T_{i_{r-1}}(E_{i_{k}}),$ $\displaystyle\textrm{ for }r=1,\dots,d.$

We call any such element a _root vector_ of $U_{q}(\mathfrak{sl}_{n+1})$. The
set of all root vectors is linearly independent and we have an associated
quantum generalisation of the classical Poincaré–Birkoff–Witt theorem [21,
§6.2.3].

###  A.5. The Explicit Calculation

In this subsection we give an explicit description of the root vectors
associated to the longest word decomposition given in §3.1. We note first that
it follows from the discussions in [25, 1.8(d)] that

$\displaystyle T_{k,k\pm 1}(E_{k})=-E_{k\pm 1}.$

We find it convenient to introduce some notation. First we write

$\displaystyle
E_{ji}:=[\cdots[E_{j-1},E_{j-2}]_{q^{-1}},E_{j-3}]_{q^{-1}},\cdots,E_{i}]_{q^{-1}}.$

Then, for any $i\leq j$, we denote

$\displaystyle T_{ji}:=T_{j}\circ T_{j-1}\circ\cdots\circ T_{i}.$

The following more general formulae can now be derived from this lemma.

######  Lemma A.2.

For any $(i,j)\in\Delta^{+}$, it holds that

  1. 1.

$T_{n,i+1}(E_{i})=(-1)^{n-i}E_{n+1,i}$,

  2. 2.

$T_{n,i-1}(E_{ji})=(-1)^{j-i+1}E_{j-1,i-1}$.

###### Proof.

1\. Assume that, for some $n>b\geq i+1$, we have

$\displaystyle T_{b,i+1}(E_{i})=(-1)^{b-i}E_{b+1,i}.$

Then it follows that

$\displaystyle T_{b+1}\circ T_{b,i+1}(E_{i})=$
$\displaystyle\,(-1)^{b-i}T_{b+1}(E_{b+1,i})$ $\displaystyle=$
$\displaystyle\,(-1)^{b-i}T_{b+1}\\!\left([\cdots[E_{b},E_{b-1}]_{q^{-1}},E_{b-2}]_{q^{-1}},\cdots,E_{i}]_{q^{-1}}\right)$
$\displaystyle=$
$\displaystyle\,(-1)^{b-i}[\cdots[T_{b+1}(E_{b}),E_{b-1}]_{q^{-1}},E_{b-2}]_{q^{-1}},\cdots,E_{i}]_{q^{-1}}$
$\displaystyle=$
$\displaystyle\,(-1)^{b-i+1}[\cdots[[E_{b+1},E_{b}]_{q^{-1}},E_{b-1}]_{q^{-1}},E_{b-2}]_{q^{-1}},\cdots,E_{i}]_{q^{-1}}$
$\displaystyle=$ $\displaystyle\,(-1)^{(b+1)-i}E_{b+2,i}.$

Since the identity clearly holds for $T_{i+1}(E_{i})=-E_{i+2,i}$, the general
case now follows by an inductive argument.

2\. It follows from (A.5) that, for any $k=i,\dots,j-1$, we have

$\displaystyle T_{n,i-1}(E_{k})=$ $\displaystyle\,T_{n,k-1}\circ T_{k-2}\cdots
T_{i-1}(E_{ji})$ $\displaystyle=$ $\displaystyle\,T_{n,k+2}\circ T_{k}\circ
T_{k-1}(E_{k})$ $\displaystyle=$ $\displaystyle\,-T_{n,k+2}(E_{k-1})$
$\displaystyle=$ $\displaystyle\,-E_{k-1}.$

It now follows that

$\displaystyle T_{n,i-1}(E_{ji})=$
$\displaystyle\,T_{n,i-1}([\cdots[E_{j},E_{j-1}]_{q^{-1}},E_{j-2}]_{q^{-1}},\cdots,E_{i}]_{q^{-1}})$
$\displaystyle=$
$\displaystyle\,([\cdots[T_{n,i-1}(E_{j}),T_{n,i-1}(E_{j-1})]_{q^{-1}},T_{n,i-1}(E_{j-2})]_{q^{-1}},\cdots,T_{n,i-1}(E_{i})]_{q^{-1}})$
$\displaystyle=$
$\displaystyle\,(-1)^{j-i+1}[\cdots[E_{j-1},E_{j-2}]_{q^{-1}},E_{j-3}]_{q^{-1}},\cdots,E_{i-1}]_{q^{-1}})$
$\displaystyle=$ $\displaystyle\,(-1)^{j-i+1}E_{j-1,i-1},$

giving the claimed identity. ∎

######  Proposition A.3.

For the reduced decomposition $\mathbf{j}$, the associated root vectors are
given, up to sign, by the elements

$\left\\{E_{ji}\,|\,\textrm{ for }\alpha_{ij}\in\Delta^{+}\right\\}\\!.$

###### Proof.

Any root vector will, by definition, be of the form

$\displaystyle T_{n1}\circ T_{n2}\circ\cdots\circ T_{n,k+1}(E_{k}),$
$\displaystyle\textrm{ for some }1\leq k\leq n.$

From the first identity in Lemma A.2, we know that

$T_{n,k+1}(E_{k})=(-1)^{n-k}E_{n+1,k}.$

Moreover, the second identity in Lemma A.2 tells us that when we operate on
$E_{ji}$ by the operator $T_{n,i-1}$, we get the element $E_{j-1,i-1}$ up to
sign. Moreover, we see that each $E_{ji}$ can be realised in this way, giving
the claimed presentation of the root vectors. ∎

Finally, we note that each root vector is a weight vector with respect to the
adjoint action of $U_{q}(\mathfrak{sl}_{n+1})$ on itself. Explicitly, it holds
that

$\displaystyle|E_{ji}|=\alpha_{j-1}+\alpha_{j-2}+\cdots+\alpha_{i},$
$\displaystyle\textrm{ for }\alpha_{ij}\in\Delta^{+}.$

###  A.6. The Opposite Involution of $U_{q}(\mathfrak{sl}_{n+1})$

For the $A$-series Lie algebras, the element $-w_{0}$ permutes the simple
roots of $\mathfrak{sl}_{n+1}$, and this permutation corresponds to the non-
trivial symmetry of the associated Dynkin diagram. This in turn gives the Hopf
algebra involution

$\overline{w_{0}}:U_{q}(\mathfrak{sl}_{n+1})\to U_{q}(\mathfrak{sl}_{n+1})$

defined on generators as

$\displaystyle\overline{w_{0}}(E_{i}):=E_{i^{\prime}},$
$\displaystyle\overline{w_{0}}(K_{i})=K_{i^{\prime}},$
$\displaystyle\overline{w_{0}}(F_{i})=F_{i^{\prime}},$

where of course $i^{\prime}:=n+1-i$. We call $\overline{w_{0}}$ the _opposite
involution_ of $U_{q}(\mathfrak{sl}_{n+1})$. For more details on automorphisms
of Drinfeld–Jimbo quantum groups, see [21, §6.1.6].

We also have a natural associated inner automorphism of the Weyl group

$\displaystyle S_{n+1}\to S_{n+1},$ $\displaystyle s_{i}\mapsto
s_{i^{\prime}},$

and we call it the _opposite involution_ of $W$. This in turn gives a
permutation of the set of commutator classes of reduced decompositions, which
we denote by $\mathbf{i}\mapsto\mathbf{i}^{\prime}$, for any commutator class
$\mathbf{i}$.

######  Proposition A.4.

Let $\mathbf{i}$ be a reduced decomposition of $w_{0}$, and denote its
associated space of positive root vectors by $\mathfrak{n}^{+}_{\mathbf{i}}$.
Then it holds that

(40)
$\displaystyle\overline{w_{0}}(\mathfrak{n}^{+}_{\mathbf{i}})=\mathfrak{n}^{+}_{\mathbf{i}^{\prime}},$

for all reduced decompositions of the longest element of the Weyl group
$S_{n}$.

###### Proof.

It follows from the (A.5) that

$\displaystyle\overline{w_{0}}\circ
T_{i}(E_{k})=T_{i^{\prime}}\circ\overline{w_{0}}(E_{k}),$
$\displaystyle\textrm{ for all }i,k=1,\dots,n.$

It now follows that

$\displaystyle\overline{w_{0}}\circ T_{i}(E_{k_{1}}E_{k_{2}}\cdots
E_{k_{l}})=$
$\displaystyle\,\overline{w_{0}}(T_{i}(E_{k_{1}}))\overline{w_{0}}(T_{i}(E_{k_{2}}))\cdots\overline{w_{0}}(T_{i}(E_{k_{l}}))$
$\displaystyle=$
$\displaystyle\,T_{i^{\prime}}(\overline{w_{0}}(E_{k_{1}}))T_{i^{\prime}}(\overline{w_{0}}(E_{k_{2}})\cdots
T_{i^{\prime}}(\overline{w_{0}}(E_{k_{l}})$ $\displaystyle=$
$\displaystyle\,T_{i^{\prime}}\circ\overline{w_{0}}(E_{k_{1}}E_{k_{2}}\cdots
E_{k_{l}}).$

Thus we see that, for any $X\in U_{q}(\mathfrak{n}_{\mathbf{i}}^{+})$, we have
that

$\overline{w_{0}}\circ T_{i}(X)=T_{i^{\prime}}\circ\overline{w_{0}}(X).$

From this we now see that

$\displaystyle\overline{w_{0}}(T_{i_{1}}\circ\cdots T_{{i_{k}}}(E_{j}))=$
$\displaystyle T_{i^{\prime}_{1}}\circ\cdots\circ
T_{i^{\prime}_{k}}\circ\overline{w_{0}}(E_{j})=T_{i^{\prime}_{1}}\circ\cdots\circ
T_{i^{\prime}_{k}}(E_{j^{\prime}}).$

Thus we see that a root vector corresponding to the $w_{0}$ substring
$s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$ is mapped by $\overline{w_{0}}$ to the
root vector corresponding to the substring
$s_{i^{\prime}_{1}}s_{i^{\prime}_{2}}\cdots s_{i^{\prime}_{k}}$. This means
that $\overline{w_{0}}$ maps the positive quantum root space
$\mathfrak{n}^{+}_{\mathbf{i}}$ to the positive quantum root space
$\mathfrak{n}^{+}_{\mathbf{i}^{\prime}}$ as claimed. ∎

In other words, the above proposition says that the following diagram
commutes:

where the vertical arrows are given by the function that associates to a
reduced decomposition the space of the corresponding root vectors.

We finish with an explicit calculation of the automorphism of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ dual to $\overline{w_{0}}$. This is
undoubtedly known to the experts, but in the absence of a suitable reference,
we include a proof.

######  Proposition A.5.

The algebra map
$\overline{w_{0}}^{*}:\mathcal{O}_{q}(\mathrm{SU}_{n+1})\to\mathcal{O}_{q}(\mathrm{SU}_{n+1})$,
determined by

$\displaystyle\overline{w_{0}}^{*}(u_{ij})\mapsto-qS(u_{n+2-j,n+2-i}),$
$\displaystyle\textrm{ for }i,j=1,\dots,n+1,$

satisfies the identity

(41) $\displaystyle\langle\overline{w_{0}}(X),a\rangle=\langle
X,\overline{w_{0}}^{*}(a)\rangle,$ $\displaystyle\textrm{ for all }X\in
U_{q}(\mathfrak{sl}_{n+1}),\,a\in\mathcal{O}_{q}(\mathrm{SU}_{n+1}),$

which is to say, it is _dual_ to $\overline{w_{0}}$ with respect to the
bilinear pairing.

###### Proof.

A direct check of the defining relations of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$ confirms that $\overline{w_{0}}^{*}$
gives a well-defined Hopf algebra algebra automorphism of
$\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. Now since we have a dual pairing of Hopf
algebras, it is sufficient to check (41) for the generators of
$U_{q}(\mathfrak{sl}_{n+1})$ and $\mathcal{O}_{q}(\mathrm{SU}_{n+1})$. For the
generator $E_{i}$, we see that

$\displaystyle\big{\langle}\overline{w_{0}}(E_{i}),u_{n-i+2,n-i+1})\big{\rangle}=\big{\langle}E_{i^{\prime}},u_{n-i+2,n-i+1})\big{\rangle}=1,$

while on the other hand

$\displaystyle\big{\langle}E_{i},\overline{w_{0}}^{*}(u_{n-i+2,n-i+1})\big{\rangle}=-q\big{\langle}E_{i},S(u_{i+1,i})\big{\rangle}=-q\big{\langle}S(E_{i}),u_{i+1,i}\big{\rangle}=q\big{\langle}E_{i}K_{i},u_{i+1,i}\big{\rangle}=1.$

For all other matrix generators, both sides of (41) clearly give zero, which
is to say, the identity is trivially satisfied. The case of the generators
$F_{i}$ and $K_{i}$ is similarly confirmed. Thus we see that (41) holds. ∎

##  Appendix B Some Remarks on Filtered Algebras

We first present a simple system for constructing a filtration from a total
order on a set of algebra generators. Our motivation is to provide a
convenient framework in which to state the proof of Lemma 3.14, and we make no
claims of originality.

Consider an algebra $A$, together with a set of algebra generators $G$. This
gives a surjective algebra map
$\mathrm{proj}_{A}:\mathcal{T}(V)\twoheadrightarrow A$, where $V$ is the
linear span of the elements of $G$, and $\mathcal{T}(V)$ is its tensor
algebra. A set of _relations_ for $A$, with respect to $G$, is a choice of
subset $R\subseteq\mathrm{ker}(\mathrm{proj}_{A})$ that generates
$\mathrm{ker}(\mathrm{proj}_{A})$ as a two-sided ideal in $\mathcal{T}(V)$.

Let $X$ be a non-empty totally-ordered finite set with $n$ elements. Moreover,
let $V$ be an $n$-dimensional vector space with a basis $\\{e_{x}\\}_{x\in X}$
indexed by the elements of $X$. We identify $\mathbb{Z}_{\geq 0}^{n}$ and the
free abelian monoid generated by $X$, and we endow $\mathbb{Z}_{\geq 0}^{n}$
with the lexicographical order $<$ induced by the total order of $X$. A
$\mathbb{Z}^{n}_{\geq 0}$-algebra grading on $\mathcal{T}(V)$ is uniquely
determined by $|e_{x}|=x$. Consider a general element $f\in\mathcal{T}(V)$,
written as a sum $f=\sum_{d=1}^{m}f_{d}$, where each summand is homogeneous
with respect to the $\mathbb{Z}^{n}_{\geq 0}$-grading, and $|f_{i}|<|f_{m}|$,
for all $i=1,\dots,m-1$. We call $f_{m}$ the _leading term_ of $f$ and denote
$\mathrm{LT}(f):=f_{m}.$

An algebra filtration of the tensor algebra

$\mathscr{F}^{\mathcal{T}}=\bigcup_{\lambda\in\mathbb{Z}^{n}_{\geq
0}}\mathscr{F}_{\lambda}^{\mathcal{T}}\subseteq\mathcal{T}(V)$

can then be defined by

$\displaystyle\mathscr{F}_{\lambda}^{\mathcal{T}}:=\mathrm{span}_{\mathbb{C}}\left\\{e_{x_{1}}\otimes\cdots\otimes
e_{x_{k}}|\sum_{i=1}^{k}x_{i}\leq\lambda\right\\}.$

For any two-sided ideal $J\subseteq\mathcal{T}(V)$, we have the induced
filtration $\mathscr{F}$ on the quotient $A:=\mathcal{T}(V)/J$. Explicitly

$\displaystyle A=\bigcup_{\lambda\in\mathbb{Z}^{n}_{\geq
0}}\mathrm{proj}_{A}(\mathscr{F}_{\lambda}^{\mathcal{T}})=:\bigcup_{\lambda\in\mathbb{Z}^{n}_{\geq
0}}\mathscr{F}_{\lambda},$

where $\mathrm{proj}_{A}$ is the canonical projection from $\mathcal{T}(V)$ to
$A$. In Proposition B.1 below, we will produce a description of the associated
graded algebra

$\displaystyle\mathrm{gr}^{\mathscr{F}}=\bigoplus_{\lambda\in\mathbb{Z}^{n}_{\geq
0}}\mathrm{gr}^{\mathscr{F}}_{\lambda}:=\bigoplus_{\lambda\in\mathbb{Z}^{n}_{\geq
0}}\mathscr{F}_{\lambda}/\mathscr{F}_{<\lambda},$ $\displaystyle\textrm{ where
\leavevmode\nobreak\
}\mathscr{F}_{<\lambda}:=\bigcup_{\mu<\lambda}\mathscr{F}_{\mu}.$

Before this, we will introduce some notation and terminology. For any
$a\in\mathscr{F}_{\lambda}$, we denote the coset of $a$ in
$\mathrm{gr}^{\mathscr{F}}_{\lambda}$ by $[a]_{\lambda}$.

######  Proposition B.1.

Let $R\subseteq J$ be a set of relations for $A$. Then the associated graded
algebra $\mathrm{gr}^{\mathscr{F}}$ is generated by the elements of $V$ and a
relation set is given by

$\displaystyle\Big{\\{}\mathrm{LT}(r)\,|\,r\in
R\Big{\\}}\subseteq\mathcal{T}(V).$

###### Proof.

From the definition of the filtration, we see that the coset

$[e_{x}]_{x}\in\mathrm{gr}^{\mathscr{F}}_{x}=\mathscr{F}_{x}/\mathscr{F}_{<x}$

is a non-zero element, for every $x\in X$. Moreover, these cosets generate
$\mathrm{gr}^{\mathscr{F}}$ as an algebra. We denote by

$\mathrm{proj}_{\mathscr{F}}:\mathcal{T}(V)\to\mathrm{gr}^{\mathscr{F}}$

the associated surjective algebra map. Now for any element $y\in J$, we see
that

$\mathrm{proj}_{\mathscr{F}}(\mathrm{LT}(y))\in\mathrm{gr}^{\mathscr{F}}_{|\mathrm{LT}(y)|}.$

It now follows from the definition of
$\mathrm{gr}^{\mathscr{F}}_{|\mathrm{LT}(y)|}$ that

$\mathrm{proj}_{\mathscr{F}}(\mathrm{LT}(y))=[y]_{|\mathrm{LT}(y)|}=0,$

meaning that $\mathrm{LT}(y)\in\mathrm{ker}(\mathrm{proj}_{\mathscr{F}})$.

In the other direction, let $w$ be an element of the kernel of
$\mathrm{proj}_{\mathscr{F}}$. Write $w=\sum_{i=1}^{d}w_{i}$, where each
$w_{i}$ is homogeneous with respect to the $\mathbb{Z}^{n}_{\geq 0}$-grading
of $\mathcal{T}(V)$, each $w_{i}$ has a distinct degree, and
$\mathrm{LT}(w)=w_{n}$. Then we must have
$\mathrm{proj}_{\mathscr{F}}(w_{i})=0$, for all $i=1,\dots,n$. Consider now
the canonical surjective algebra map

$\displaystyle\mathrm{proj}_{A}:\mathcal{T}(V)\to A,$

We see that $\mathrm{proj}_{A}(w_{i})$ must be an element of the subspace
$\mathscr{F}_{<|w_{i}|}$. This means that we can choose an element

$\displaystyle z\in\mathscr{F}^{\mathcal{T}}_{<|w_{i}|},$
$\displaystyle\textrm{ such that \leavevmode\nobreak\
}\mathrm{proj}_{A}(z)=\mathrm{proj}_{A}(w_{i}).$

Thus $w_{i}-z$ is an element of $J$ satisfying $\mathrm{LT}(w_{i}-z)=w_{i}$,
and so, we have shown that every element of $\mathrm{ker}(\mathscr{F})$ is of
the form $\mathrm{LT}(y)$, for some $y\in J$.

Finally, let $R\subseteq J$ be a set of relations for $J$, then $J$ is spanned
by elements of the form $X\otimes r\otimes Y$, where $r\in R$, and $X$ and $Y$
are homogeneous elements of the tensor algebra. Now

$\mathrm{LT}(X\otimes r\otimes Y)=X\otimes\mathrm{LT}(y)\otimes Y.$

Thus we see that the leading terms of the relation set are a relation set for
the associated graded algebra. ∎

We finish with some remarks on Bongale’s theorem [2, Theorem 2] for filtered
Frobenius algebras (see §3.6 for the definition of Frobenius algebras). This
result was originally stated for $\mathbb{Z}$-filtrations, but as observed in
[22, Lemma 4.10], it holds more generally. We state it for the free abelian
monoid $\mathbb{Z}^{n}_{\geq 0}$ endowed with some total order $\geq$.
Consider a finite-dimensional algebra $A$, endowed with a filtration

$A=\bigcup_{\mu\in\mathbb{Z}^{n}_{\geq 0}}\mathscr{F}_{\mu}.$

Let us assume that the associated graded algebra $\mathrm{gr}^{\mathscr{F}}$
satisfies

$\displaystyle\mathrm{dim}\Big{(}\mathrm{gr}^{\mathscr{F}}_{0}\Big{)}=\mathrm{dim}\Big{(}\mathrm{gr}^{\mathscr{F}}_{\lambda}\Big{)}=1,$
$\displaystyle\mathrm{gr}^{\mathscr{F}}_{\mu}=0,\textrm{ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ for all }\mu>\lambda,$

for some $\lambda\in\mathbb{Z}^{n}_{\geq 0}$. _Bongale’s theorem_ states that
if $\mathrm{gr}^{\mathscr{F}}$ is a Frobenius algebra, then a Frobenius
structure for $A$ is given by

$\displaystyle B:A\otimes A\to\mathbb{C},$
$\displaystyle(a,b)\mapsto\iota\big{(}[ab]_{\lambda}\big{)},$

where $\iota:\mathrm{gr}^{\mathscr{F}}_{\lambda}\to\mathbb{C}$ is any choice
of linear isomorphism. Moreover, all Frobenius structures are of this form.

## References

  * [1] E. Beggs and S. Majid, Quantum Riemannian Geometry, vol. 355 of Grundlehren der mathematischen Wissenschaften, Springer International Publishing, 1 ed., 2019.
  * [2] P. R. Bongale, Filtered Frobenius algebras, Math. Z., 97 (1967), pp. 320–325.
  * [3] K. A. Brown and K. R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002.
  * [4] A. Carotenuto, F. Díaz García, and R. Ó Buachalla, A Borel–Weil theorem for the irreducible quantum flag manifolds, Int. Math. Res. Not. IMRN, (2023), pp. 12977–13006.
  * [5] A. Carotenuto, C. Mrozinski, and R. Ó Buachalla, A Borel–Weil theorem for the quantum Grassmannians, Doc. Math., 28 (2023), pp. 261–314.  Doc. Math.
  * [6] A. Carotenuto and R. Ó Buachalla, Principal pairs of quantum homogeneous spaces.  arXiv preprint math.QA/2111.11284.
  * [7] , Bimodule connections for relative line modules over the irreducible quantum flag manifolds, SIGMA Symmetry Integrability Geom. Methods Appl., 18 (2022), pp. Paper No. 070, 21.
  * [8] B. Das, R. Ó Buachalla, and P. Somberg, Dolbeault–Dirac Fredholm operators on quantum homogeneous spaces.  arXiv preprint math.QA/1910.14007.
  * [9] B. Das, R. Ó Buachalla, and P. Somberg, A Dolbeault–Dirac spectral triple for quantum projective space, Doc. Math., 25 (2020), pp. 1079–1157.
  * [10] F. Díaz García, A. Krutov, R. Ó Buachalla, P. Somberg, and K. R. Strung, Holomorphic relative Hopf modules over the irreducible quantum flag manifolds, Lett. Math. Phys., 111 (2021), pp. Paper No. 10, 24.
  * [11] F. Díaz García, R. Ó Buachalla, and E. Wagner, A Dolbeault–Dirac spectral triple for the $B_{2}$-irreducible quantum flag manifold, Comm. Math. Phys., 395 (2022), pp. 365–403.
  * [12] S. Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups, J. Combin. Theory Ser. A, 77 (1997), pp. 193–221.
  * [13] L. Escobar, O. Pechenik, B. E. Tenner, and A. Yong, Rhombic tilings and Bott-Samelson varieties, Proc. Amer. Math. Soc., 146 (2018), pp. 1921–1935.
  * [14] S. Felsner and P. Valtr, Coding and counting arrangements of pseudolines, Discrete Comput. Geom., 46 (2011), pp. 405–416.
  * [15] J. E. Grabowski and S. Launois, Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases, Int. Math. Res. Not. IMRN, (2011), pp. 2230–2262.
  * [16] , Graded quantum cluster algebras and an application to quantum Grassmannians, Proc. Lond. Math. Soc. (3), 109 (2014), pp. 697–732.
  * [17] I. Heckenberger and S. Kolb, Differential calculus on quantum homogeneous spaces, Lett. Math. Phys., 63 (2003), pp. 255–264.
  * [18] , The locally finite part of the dual coalgebra of quantized irreducible flag manifolds, Proc. London Math. Soc. (3), 89 (2004), pp. 457–484.
  * [19] , De Rham complex for quantized irreducible flag manifolds, J. Algebra, 305 (2006), pp. 704–741.
  * [20] , On the Bernstein–Gelfand–Gelfand resolution for Kac–Moody algebras and quantized enveloping algebras, Transform. Groups, 12 (2007), pp. 647–655.
  * [21] A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Texts and Monographs in Physics, Springer-Verlag, 1997.
  * [22] U. Krähmer and M. Tucker-Simmons, On the Dolbeault-Dirac operator of quantized symmetric spaces, Trans. London Math. Soc., 2 (2015), pp. 33–56.
  * [23] V. Lakshmibai and N. Reshetikhin, Quantum deformations of flag and Schubert schemes, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), pp. 121–126.
  * [24] P. Littelmann, Cones, crystals, and patterns, Transform. Groups, 3 (1998), pp. 145–179.
  * [25] G. Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc., 3 (1990), pp. 257–296.
  * [26] , Introduction to quantum groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010.  Reprint of the 1994 edition.
  * [27] S. Majid, Quantum and braided-Lie algebras, J. Geom. Phys., 13 (1994), pp. 307–356.
  * [28] , Noncommutative Riemannian and spin geometry of the standard $q$-sphere, Comm. Math. Phys., 256 (2005), pp. 255–285.
  * [29] M. Matassa, Kähler structures on quantum irreducible flag manifolds, J. Geom. Phys., 145 (2019), pp. 103477, 16.
  * [30] R. Ó Buachalla, Noncommutative complex structures on quantum homogeneous spaces, J. Geom. Phys., 99 (2016), pp. 154–173.
  * [31] , Noncommutative Kähler structures on quantum homogeneous spaces, Adv. Math., 322 (2017), pp. 892–939.
  * [32] R. Ó Buachalla, J. Šťoviček, and A.-C. van Roosmalen, A Kodaira vanishing theorem for noncommutative Kähler structures.  arXiv preprint math.QA/1801.08125.
  * [33] P. Papi, A characterization of a special ordering in a root system, Proc. Amer. Math. Soc., 120 (1994), pp. 661–665.
  * [34] P. Podleś, Differential calculus on quantum spheres, Lett. Math. Phys., 18 (1989), pp. 107–119.
  * [35] A. Polishchuk and L. Positselski, Quadratic algebras, vol. 37 of University Lecture Series, American Mathematical Society, Providence, RI, 2005\.
  * [36] L. Rigal and P. Zadunaisky, Quantum toric degeneration of quantum flag and Schubert varieties, Transform. Groups, 26 (2021), pp. 1113–1143.
  * [37] A. L. Rosenberg, Noncommutative schemes, Compositio Math., 112 (1998), pp. 93–125.
  * [38] P. Schauenburg, Hopf modules and Yetter–Drinfeld modules, J. Algebra, 169 (1994), pp. 874–890.
  * [39] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, vol. 72, 1990, pp. 167–195.  Hopf algebras.
  * [40] Y. S. Soibelman, On the quantum flag manifold, Funktsional. Anal. i Prilozhen., 26 (1992), pp. 90–92.
  * [41] J. V. Stokman, The quantum orbit method for generalized flag manifolds, Math. Res. Lett., 10 (2003), pp. 469–481.
  * [42] M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
  * [43] E. Taft and J. Towber, Quantum deformation of flag schemes and Grassmann schemes. I. A $q$-deformation of the shape-algebra for ${\rm GL}(n)$, J. Algebra, 142 (1991), pp. 1–36.
  * [44] M. Takeuchi, Relative Hopf modules—equivalences and freeness criteria, J. Algebra, 60 (1979), pp. 452–471.
  * [45] C. Voigt and R. Yuncken, Equivariant Fredholm modules for the full quantum flag manifold of ${\rm SU}_{q}(3)$, Doc. Math., 20 (2015), pp. 433–490.
  * [46] S. L. Woronowicz, Twisted ${\rm SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci., 23 (1987), pp. 117–181.
  * [47] G. M. Ziegler, Higher Bruhat orders and cyclic hyperplane arrangements, Topology, 32 (1993), pp. 259–279.