# Noncommutative invariants
of dihedral groups

Vesselin Drensky, Boyan Kostadinov  Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $A_{2}({\mathfrak{M}})$ and $L_{2}({\mathfrak{A}}^{2})$ be, respectively,
the 2-generated free metabelian associative and Lie algebra over the field of
complex numbers. In the associative case we find a finite set of generators of
the algebra $A_{2}({\mathfrak{M}})^{D_{2n}}$ of the dihedral group of order
$2n$, $n\geq 3$. In the Lie case we find a minimal system of generators as a
${\mathbb{C}}[x,y]^{D_{2n}}$-module of the $D_{2n}$-invariants
$L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ in the commutator ideal
$L_{2}^{\prime}({\mathfrak{A}}^{2})$ of $L_{2}({\mathfrak{A}}^{2})$. In both
cases we compute the Hilbert (or Poincaré) series of the algebras
$A_{2}({\mathfrak{M}})^{D_{2n}}$ and $L_{2}({\mathfrak{A}}^{2})^{D_{2n}}$.

###### Key words and phrases:

Algebras with polynomial identity, free metabelian associative algebra; free
metabelian Lie algebra; dihedral groups; noncommutative invariant theory.

###### 2020 Mathematics Subject Classification:

16R10; 17B01; 05A15; 15A72; 16R40; 16W22; 17B30; 20D10.

The research of the second named author was supported by the National
Programme of Ministry of Education and Science “Young Scientists and Post-
Doctoral Researchers - 2” and partially supported by Grant KP-06-N-32/1 of the
Bulgarian National Science Fund.

##  1\. Introduction

In the sequel we assume that $K$ is a field of characteristic 0. We shall
consider the case $K=\mathbb{C}$ only although our main results hold for any
sufficiently large field $K$. In classical commutative invariant theory the
general linear group $GL_{d}({\mathbb{C}})$ acts on the $d$-dimensional
complex vector space $V_{d}$ with basis $\\{v_{1},\ldots,v_{d}\\}$ and this
action induces an action on the algebra of polynomial functions
${\mathbb{C}}[X_{d}]={\mathbb{C}}[x_{1},\ldots,x_{d}]$. For a subgroup $G$ of
$GL_{d}({\mathbb{C}})$ the algebra of $G$-invariants ${\mathbb{C}}[X_{d}]^{G}$
consists of all polynomials which are fixed under the action of $G$. In one of
the main branches of noncommutative invariant theory one replaces the algebra
${\mathbb{C}}[X_{d}]$ with an algebra which shares some of the important
properties of ${\mathbb{C}}[X_{d}]$. Among the candidates for such an algebra
are the free associative algebra ${\mathbb{C}}\langle
X_{d}\rangle={\mathbb{C}}\langle x_{1},\ldots,x_{d}\rangle$, the free Lie
algebra $L_{d}$ or the $d$-generated relatively free algebra of a variety of
(associative, Lie, Jordan or other nonassociative) algebras, $d\geq 2$. In
this case it is more convenient to assume that $GL_{d}({\mathbb{C}})$ acts
directly on the vector space ${\mathbb{C}}X_{d}$ with basis $X_{d}$ instead of
on $V_{d}$. Going back to classical invariant theory this means that
$GL_{d}({\mathbb{C}})$ acts on the symmetric algebra $S(X_{d})$ of
${\mathbb{C}}X_{d}$ instead on the algebra of polynomial functions
${\mathbb{C}}[X_{d}]$. We refer to the surveys by Formanek [10] and the first
named author of the present paper [7] for more details.

###  1.1. Varieties of associative algebras.

Let $\mathfrak{V}$ be a variety of unitary associative algebras over a field
$K$ of characteristic 0 and let $A_{d}({\mathfrak{V}})$ be the relatively free
algebra of $\mathfrak{V}$ freely generated by $X_{d}$, $d\geq 2$. The general
linear group $GL_{d}(K)$ acts canonically on the vector space $KX_{d}$ and
this action is extended diagonally on the whole algebra
$A_{d}({\mathfrak{V}})$:

$g(f(x_{1},\ldots,x_{d}))=f(g(x_{1}),\ldots,g(x_{d})),\,f\in
A_{d}({\mathfrak{V}}),g\in GL_{d}(K).$

For a subgroup $G$ of $GL_{d}(K)$ the algebra of $G$-invariants is

$A_{d}({\mathfrak{V}})^{G}=\\{f(X_{d})\in A_{d}({\mathfrak{V}})\mid
g(f(X_{d}))=f(X_{d})\text{ for all }g\in G\\}.$

There are many equivalent conditions on the variety $\mathfrak{V}$ to the fact
that the algebra $A_{d}({\mathfrak{V}})^{G}$ is finitely generated for all
finite subgroups $G$ of $GL_{d}(K)$. See [14, Section 4.7] or [5, Theorem 2.1]
for the complete statement. In particular, due to the results by Kharchenko
[13], Lvov [18] and Anan’in [1] the following theorem holds.

######  Theorem 1.1.

Let $\mathfrak{V}$ be a variety of unitary associative algebras. The following
conditions on $\mathfrak{V}$ are equivalent. If some of them is satisfied for
some $d_{0}\geq 2$, then all of them hold for all $d\geq 2$:

(i) The algebra $A_{d}({\mathfrak{V}})^{G}$ is finitely generated for every
finite subgroup $G$ of $GL_{d}(K)$.

(ii) The variety $\mathfrak{V}$ satisfies the polynomial identity

$[x_{1},x_{2},\ldots,x_{2}]x_{3}^{n}[x_{4},x_{5},\ldots,x_{5}]=0$

for sufficiently long commutators and $n$ large enough.

Here and further $[z_{1},z_{2}]=z_{1}z_{2}-z_{2}z_{1}=z_{1}\,\text{ad}(z_{2})$
is the commutator of $z_{1}$ and $z_{2}$ and the longer commutators are left
normed:

$[z_{1},\ldots,z_{n-1},z_{n}]=[[z_{1},\ldots,z_{n-1}],z_{n}],\,n\geq 3.$

In particular Theorem 1.1 holds for the metabelian variety $\mathfrak{M}$
defined by the polynomial identity

$[x_{1},x_{2}][x_{3},x_{4}]=0.$

The metabelian variety plays a key role in the theory of PI-algebras. By a
theorem of Kemer [12, Corollary 5] if an associative algebra satisfies the
Engel identity

$[x_{2},x_{1},\ldots,x_{1}]=x_{2}\,\text{ad}^{n}(x_{1})=0$

then it is Lie nilpotent, i.e. $[x_{1},\ldots,x_{m}]=0$ for some $m$. As a
consequence there is a dichotomy for varieties of associative algebras. The
variety $\mathfrak{V}$ is either Lie nilpotent or contains $\mathfrak{M}$ as a
subvariety.

In the sequel we shall denote the free generators of $A_{2}({\mathfrak{M}})$
by $x$ and $y$. The first main result of our paper gives a system of
generators of the algebra $A_{2}({\mathfrak{M}})^{D_{2n}}$ of the invariants
of the dihedral group $D_{2n}$ of order $2n$ under the natural action of
$D_{2n}$ as the group of symmetries of the regular $n$-gon, $n\geq 3$. We also
compute the Hilbert series of the algebra $A_{2}({\mathfrak{M}})^{D_{2n}}$.
Our proof uses essentially the well known description of the algebra of
invariants ${\mathbb{C}}[x,y]^{D_{2n}}$.

###  1.2. Varieties of Lie algebras.

The metabelian variety of Lie algebras ${\mathfrak{A}}^{2}$ is defined by the
polynomial identity

$[[x_{1},x_{2}],[x_{3},x_{4}]]=0.$

Zelmanov [22] proved that a Lie algebra satisfying the Engel identity is
nilpotent. This implies that as in the case of associative algebras there is a
dichotomy. A variety is either nilpotent or contains ${\mathfrak{A}}^{2}$ as a
subvariety. Concerning invariant theory of relatively free algebras, by a
result of the first named author of this paper [6] if $G$ is a nontrivial
finite subgroup of $GL_{d}(K)$, $d\geq 2$, then the algebra of $G$-invariants
$L_{d}({\mathfrak{V}})^{G}$ in the relatively free algebra
$L_{d}({\mathfrak{V}})$ of the variety of Lie algebras $\mathfrak{V}$ is not
finitely generated if and only if ${\mathfrak{A}}^{2}\subseteq\mathfrak{V}$.
This result was generalized by Bryant and Papistas [3] for any finite subgroup
of the automorphism group of $L_{d}({\mathfrak{V}})$. A positive result in
this direction is obtained by Drensky, Fındık and Öğüşlü [9, Corollary 5].
They showed that if $G<GL_{d}(K)$ is a finite group then
$L_{d}^{\prime}({\mathfrak{A}}^{2})^{G}$ is a finitely generated
$K[X_{d}]^{G}$-module, where the variables $x_{i}\in X_{d}$ act on
$L_{d}^{\prime}({\mathfrak{A}}^{2})$ by the adjoint action
$x_{i}\to\text{ad}(x_{i})$.

Since the free metabelian Lie algebra $L_{d}({\mathfrak{A}}^{2})$ is naturally
embedded into the free metabelian associative algebra $A_{d}({\mathfrak{M}})$,
when $K=\mathbb{C}$ the results for the algebra of invariants
$A_{2}({\mathfrak{M}})^{D_{2n}}$ allow easily to obtain a minimal set of
generators of $L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ as a
${\mathbb{C}}[x,y]^{D_{2n}}$-module. As in the associative case, we compute
the Hilbert series of the algebra $L_{2}({\mathfrak{A}}^{2})^{D_{2n}}$.

##  2\. Preliminaries

From now on we shall work on the field of complex numbers.

###  2.1. Invariant theory of dihedral groups

The facts on invariant theory of dihedral groups which we shall need in the
sequel are well known. We include the proofs for completeness of the
exposition. Recall the Endlichkeitssatz of Emmy Noether and the theorem of
Chevalley-Shephard-Todd.

######  Theorem 2.1.

[19] For any finite subgroups $G$ of $GL_{d}({\mathbb{C}})$ the algebra of
invariants ${\mathbb{C}}[X_{d}]^{G}$ is finitely generated and has a
homogeneous system of generators of degree $\leq|G|$.

Recall that an element $g\in GL_{d}({\mathbb{C}})$ of finite order is a
(pseudo)reflection if it has an eigenvalue 1 of multiplicity $d-1$ and another
eigenvalue of multiplicity 1 which is a root of unity. The group
$G<GL_{d}({\mathbb{C}})$ is called a reflection group if it is generated by
reflections. For our purposes we need a stronger than in the original papers
[4, 20] form of the theorem of Chevalley-Shephard-Todd.

######  Theorem 2.2.

[21, Theorem 4.2.5] The following properties of the finite subgroup $G$ of
$GL_{d}({\mathbb{C}})$ are equivalent:

  1. (1)

$G$ is a finite reflection group;

  2. (2)

${\mathbb{C}}[X_{d}]$ is a free graded module over ${\mathbb{C}}[X_{d}]^{G}$
with a finite basis;

  3. (3)

${\mathbb{C}}[X_{d}]^{G}$ is generated by $d$ algebraically independent
homogeneous elements.

######  Corollary 2.3.

[21, Corollary 4.2.12] Let $G$ be a reflection group and let
${\mathbb{C}}[X_{d}]^{G}=\mathbb{C}[f_{1},\ldots,f_{n}]$, where $f_{i}$ is
homogeneous of degree $d_{i}$. Then up to order the integers $d_{i}$ are
uniquely determined by $G$. The order $|G|$ of $G$ is equal to
$\prod\limits_{i=1}^{n}d_{i}$ and the number of reflections in $G$ is equal to
$\sum\limits_{i=1}^{n}(d_{i}-1)$.

######  Proposition 2.4.

[11, Proposition 3.6] If $G$ is a reflection group then the free
${\mathbb{C}}[X_{d}]^{G}$-module ${\mathbb{C}}[X_{d}]$ is freely generated by
$|G|$ homogeneous elements.

The dihedral group $D_{2n}$, $n\geq 3$, acts on the two dimensional real
vector space $xOy$ as the group of symmetries of the regular $n$-gon. It is
generated by a rotation by angle $\displaystyle\frac{2\pi}{n}$ around the
origin and a reflection with respect to the axis $Ox$. According to Klein
([15, Kapitel II] in the German original or [16, Chapter II] in the English
translation) already Riemann used the following change of the coordinate
system:

$u=x+iy,v=x-iy.$

Then $D_{2n}$ is generated by the rotation

$\rho:u\longrightarrow\xi u,\,v\longrightarrow\bar{\xi}u,\text{ where
}\xi=e^{\frac{2\pi i}{n}},$

and the reflection

$\tau:u\longleftrightarrow v.$

Till the end of the paper we shall work with the generators of $D_{2n}$ with
respect to the coordinates $(u,v)$ instead of with the coordinates $(x,y)$.

######  Proposition 2.5.

The algebra of invariants ${\mathbb{C}}[u,v]^{D_{2n}}$ is generated by $uv$
and $u^{n}+v^{n}$.

###### Proof.

We shall give an elementary proof which uses only the Endlichkeitssatz of Emmy
Noether. Obviously the invariants of $\tau$ in ${\mathbb{C}}[u,v]$ are linear
combinations of

$u^{p+q}v^{p}+u^{q}v^{p+q}=(uv)^{p}(u^{q}+v^{q}).$

Since

$\varrho((uv)^{p}(u^{q}+v^{q}))=(uv)^{p}(\xi^{p}u^{q}+\bar{\xi}^{q}v^{q})$

we derive that the invariants of $\varrho$ occur for $q\equiv 0\,(\text{mod
}n)$. Hence the vector space ${\mathbb{C}}[u,v]^{D_{2n}}$ has a basis

$\\{(uv)^{p}(u^{qn}+v^{qn})\mid p,q\geq 0\\}$

and the algebra ${\mathbb{C}}[u,v]^{D_{2n}}$ is generated by

$\\{uv,u^{qn}+v^{qn}\mid q\geq 1\\}.$

Since $|D_{2n}|=2n$, the Endlichkeitssatz of Emmy Noether gives that
${\mathbb{C}}[u,v]^{D_{2n}}$ is generated by

$uv,u^{n}+v^{n},u^{2n}+v^{2n}.$

This completes the proof because

$u^{2n}+v^{2n}=(u^{n}+v^{n})^{2}-2(uv)^{n}$

and $u^{2n}+v^{2n}$ belongs to the subalgebra of ${\mathbb{C}}[u,v]$ generated
by $uv$ and $u^{n}+v^{n}$. ∎

Recall that the Hilbert (or Poincaré) series of a graded vector space
$W=\sum\limits_{i\geq 0}W^{(i)}$ with finite dimensional homogeneous
components $W^{(i)}$ is the formal power series

$H(W,t)=\sum_{i\geq 0}\dim(W^{(i)})t^{i}.$

######  Proposition 2.6.

The polynomial algebra $\mathbb{C}[u,v]$ is a free
$\mathbb{C}[u,v]^{D_{2n}}$-module with free generators

$W=(1,u,u^{2},\ldots,u^{n},v,v^{2},\ldots,v^{n-1}).$

###### Proof.

The fact that $\mathbb{C}[u,v]$ is a free $\mathbb{C}[u,v]^{D_{2n}}$-module
follows immediately from Theorem 2.2. By Proposition 2.5
$\mathbb{C}[u,v]^{D_{2n}}$ is a polynomial algebra generated by two elements
of degree 2 and $n$, respectively. Hence the Hilbert series of
$\mathbb{C}[u,v]$ and $\mathbb{C}[u,v]^{D_{2n}}$ are

$H(\mathbb{C}[u,v],t)=\frac{1}{(1-t)^{2}}\text{ and
}H(\mathbb{C}[u,v]^{D_{2n}},t)=\frac{1}{(1-t^{2})(1-t^{n})}.$

By Proposition 2.4 the $\mathbb{C}[u,v]^{D_{2n}}$-module $\mathbb{C}[u,v]$ is
generated by $2n$ homogeneous polynomials $z_{1},\ldots,z_{2n}$ and

$\mathbb{C}[u,v]=z_{1}\mathbb{C}[u,v]^{D_{2n}}\oplus\cdots\oplus
z_{2n}\mathbb{C}[u,v]^{D_{2n}}.$

Hence

$H(\mathbb{C}[u,v],t)=\sum_{j=1}^{2n}t^{\deg(z_{j})}H(\mathbb{C}[u,v]^{D_{2n}},t),$
$\sum_{j=1}^{2n}t^{\deg(z_{j})}=\frac{H(\mathbb{C}[u,v],t)}{H(\mathbb{C}[u,v]^{D_{2n}},t)}=(1+t)(1+t+\cdots+t^{n-1})$

and $\mathbb{C}[u,v]$ is generated as a $\mathbb{C}[u,v]^{D_{2n}}$-module by
one element of degree 0 and of degree $n$ and 2 elements of each degree
$1,2,\ldots,n-1$. Let

$Z=(z_{0},z_{1},z_{1}^{\prime},\ldots,z_{n-1},z_{n-1}^{\prime},z_{n})$

be such a system of free generators,

$\deg(z_{0})=0,\,\deg(z_{i})=\deg(z_{i}^{\prime})=i,\,i=0,1,\ldots,n-1,\,\deg(z_{n})=n.$

The next step of our proof is to prove that all elements of degree $\leq n$ in
$\mathbb{C}[u,v]$ can be expressed as linear combinations of the system $W$
with coefficients from $\mathbb{C}[u,v]^{D_{2n}}$. The monomials of degree
$\leq n$ in ${\mathbb{C}}[u,v]$ are of the form $u^{a}v^{b}$, $a+b\leq n$.

  * •

If $a=b$, then $u^{a}v^{a}=1\cdot(uv)^{a}$.

  * •

If $a>b$, then $u^{a}v^{b}=u^{a-b}\cdot(uv)^{b}$, $a-b\leq n$.

  * •

If $a<b$, then $u^{a}v^{b}=v^{b-a}\cdot(uv)^{b}$ if $b-a<n$. Since $a+b\leq
n$, if $b-a=n$, then $(a,b)=(0,n)$ and
$u^{a}v^{b}=v^{n}=1\cdot(u^{n}+v^{n})-u^{n}\cdot 1$.

Hence all elements of degree $\leq n$ belong to the
${\mathbb{C}}[u,v]^{D_{2n}}$-module generated by $W$. In particular, this
holds to the system $Z$ of $2n$ free generators of the
${\mathbb{C}}[u,v]^{D_{2n}}$-module ${\mathbb{C}}[u,v]$. Hence there exists a
$2n\times 2n$ matrix $T$ with entries in $\mathbb{C}[u,v]^{D_{2n}}$ such that
$Z^{t}=TW^{t}$, where $Z^{t}$ and $W^{t}$ are, respectively, the transposed of
$Z$ and $W$. On the other hand, since $Z$ is a system of free generators of
the ${\mathbb{C}}[u,v]^{D_{2n}}$-module ${\mathbb{C}}[u,v]$, there exists a
$2n\times 2n$ matrix $S$ such that $W^{t}=SZ^{t}$. This implies that
$Z^{t}=T(SZ^{t})$ and that the matrix $TS$ is the identity matrix. Hence the
matrix $T$ is invertible, i.e. $W$ is a free generating set of the free
${\mathbb{C}}[u,v]^{D_{2n}}$-module ${\mathbb{C}}[u,v]$. ∎

The next corollary follows immediately from Proposition 2.6 because

${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]\cong{\mathbb{C}}[u_{1},v_{1}]\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}].$

######  Corollary 2.7.

Let $D_{2n}$ act on ${\mathbb{C}}[u_{1},v_{1}]$ and
${\mathbb{C}}[u_{2},v_{2}]$ in the same way as on ${\mathbb{C}}[u,v]$. Then
${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$ is a free
${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$-module
freely generated by

$u_{1}^{a}u_{2}^{c},u_{1}^{a}v_{2}^{d},v_{1}^{b}u_{2}^{c},v_{1}^{b}v_{2}^{d},\,0\leq
a,c\leq n,1\leq b,d\leq n-1.$

###  2.2. Free metabelian associative algebra

The description of the free metabelian associative algebra
$A_{d}({\mathfrak{M}})$ is well known, see e.g. [17, Proposition 3] or [8,
Theorem 5.2.1].

######  Proposition 2.8.

The free metabelian algebra $A_{d}({\mathfrak{M}})$ has a basis consisting of
all

$x_{1}^{a_{1}}\cdots x_{d}^{a_{d}},\,x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}[x_{i_{1}},\dots,x_{i_{n}}],\,a_{1},\ldots,a_{d}\geq
0,i_{1}>i_{2}\leq\dots\leq i_{n}.$

It satisfies the identities

$x_{\varphi(1)}\cdots
x_{\varphi(m)}[x_{m+1},x_{m+2},x_{\psi(m+3)},\ldots,x_{\psi(m+n)}]$
$=x_{1}\cdots x_{m}[x_{m+1},x_{m+2},x_{m+3},\ldots,x_{m+n}],\,m\geq 0,n\geq
3,$

where $\varphi$ and $\psi$ are permutations of $\\{1,\ldots,m\\}$ and
$\\{m+3,\ldots,m+n\\}$, respectively.

######  Corollary 2.9.

(i) In the special case of $d=2$ the algebra $A_{2}({\mathfrak{M}})$ has a
basis with respect to the set $\\{u,v\\}$

$u^{a}v^{b},\,u^{a}v^{b}[v,u]u^{c}v^{d},\,a,b,c,d\geq 0.$

(ii) The commutator ideal $A_{2}^{\prime}({\mathfrak{M}})$ is a free
${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$-module generated by $[v,u]$, where
${\mathbb{C}}[u_{1},v_{1}]$ and ${\mathbb{C}}[u_{2},v_{2}]$ act on
$A_{2}^{\prime}({\mathfrak{M}})$ by multiplication by $u$ and $v$ from the
left and the right, respectively.

###### Proof.

As in Proposition 2.8, the metabelian identity implies that

$z_{2}z_{1}z_{3}^{\varepsilon}[z_{4},z_{5}]=z_{1}z_{2}z_{3}^{\varepsilon}[z_{4},z_{5}],\,[z_{1},z_{2}]z_{3}^{\varepsilon}z_{5}z_{4}=[z_{1},z_{2}]z_{3}^{\varepsilon}z_{4}z_{5},\,\varepsilon=0,1.$

Hence we can rewrite the basis of the algebra $A_{2}({\mathfrak{M}})$ with
respect to the set $\\{u,v\\}$ in the desired form and this proves (i). The
same equations imply that $A_{2}^{\prime}({\mathfrak{M}})$ is a
${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$-module. The basis of
$A_{2}^{\prime}({\mathfrak{M}})$ given in (i) gives the freeness of the
module. ∎

###  2.3. Free metabelian Lie algebra

As in the associative case, the structure of free metabelian Lie algebra
$L_{d}({\mathfrak{A}}^{2})$ is also well known, see e.g. [2, Section 4.7.1].
It has a basis

$x_{1},\ldots,x_{d},\,[x_{i_{1}},\dots,x_{i_{n}}],\,i_{1}>i_{2}\leq\dots\leq
i_{n}.$

Clearly, $L_{d}({\mathfrak{A}}^{2})$ may be considered as the Lie subalgebra
of $A_{d}({\mathfrak{M}})$ generated by $x_{1},\ldots,x_{d}$ with respect to
the commutator operation. In the special case $d=2$ the basis of
$L_{2}({\mathfrak{A}}^{2})$ consists of

$u,v,\,[v,u]\text{ad}^{a}(u)\text{ad}^{b}(v),\,a,b\geq 0,$

and $L_{d}^{\prime}({\mathfrak{A}}^{2})$ is a free ${\mathbb{C}}[u,v]$-module
generated by $[v,u]$ with respect to the action

$f(u,v):w\longrightarrow wf(\text{ad}(u),\text{ad}(v)),\quad w\in
L_{d}^{\prime}({\mathfrak{A}}^{2}),f(u,v)\in{\mathbb{C}}[u,v].$

##  3\. Main results

###  3.1. Associative case

Applying Corollary 2.9 (ii) we define an isomorphism of
${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$-modules

$\nu:A_{2}^{\prime}({\mathfrak{M}})\longrightarrow{\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]\text{
by }\nu(u^{a}v^{b}[v,u]u^{c}v^{d})=u_{1}^{a}v_{1}^{b}u_{2}^{c}v_{2}^{d}.$

Clearly, $\nu$ is also an isomorphism of
${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$-modules.
This allows to restate Corollary 2.7 in the following way.

######  Proposition 3.1.

Let $D_{2n}$ act on ${\mathbb{C}}[u_{1},v_{1}]$ and
${\mathbb{C}}[u_{2},v_{2}]$ in the same way as on ${\mathbb{C}}[u,v]$. Then
$A_{2}^{\prime}({\mathfrak{M}})$ is a free
${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$-module
freely generated by

$u^{a}[v,u]u^{c},u^{a}[v,u]v^{d},v^{b}[v,u]u^{c},v^{b}[v,u]v^{d},\,0\leq
a,c\leq n,1\leq b,d\leq n-1.$

The next proposition is the key step in the description of the algebra of
invariants $A_{2}({\mathfrak{M}})^{D_{2n}}$.

######  Proposition 3.2.

(i) A polynomial $f(u,v)\in A_{2}^{\prime}({\mathfrak{M}})$ belongs to
$A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$ if and only if it is fixed under the
action of the rotation $\varrho$ and the reflection $\tau$ which generate the
dihedral group $D_{2n}$, i.e.

$f(\xi u,\bar{\xi}v)=f(u,v),f(v,u)=f(u,v),\text{ where }\xi=e^{\frac{2\pi
i}{n}}.$

The polynomial
$h(u_{1},v_{1},u_{2},v_{2})=\nu(f(u,v))\in{\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$
belongs to ${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]^{D_{2n}}$ if and only if

$\varrho(h(u_{1},v_{1},u_{2},v_{2}))=h(\xi u_{1},\bar{\xi}v_{1},\xi
u_{2},\bar{\xi}v_{2})=h(u_{1},v_{1},u_{2},v_{2}),$
$\tau(h(u_{1},v_{1},u_{2},v_{2}))=h(v_{1},u_{1},v_{2},u_{2})=-h(u_{1},v_{1},u_{2},v_{2}).$

(ii) The
${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$-module
$A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$ is free with a set of free
generators

$u^{a}[v,u]u^{n-a}-v^{a}[v,u]v^{n-a},\,a=0,1,\ldots,n,\,u^{n}[v,u]u^{n}-v^{n}[v,u]v^{n},$
$u^{a}[v,u]v^{a}-v^{a}[v,u]u^{a},\,a=1,\ldots,n-1.$

###### Proof.

(i) A polynomial $f(u,v)\in A_{2}^{\prime}({\mathfrak{M}})$ belongs to
$A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$ if and only if it is fixed under the
action of the generators of $D_{2n}$:

$\varrho(f(u,v))=f(\xi u,\bar{\xi}v)=f(u,v)\text{ and
}\tau(f(u,v))=f(v,u)=f(u,v).$

Since

$\varrho([v,u])=[\xi v,\bar{\xi}u]=[v,u]\text{ and }\tau([v,u])=[u,v]=-[v,u],$

we obtain that

$h(u_{1},v_{1},u_{2},v_{2})=\nu(f(u,v))=\nu(\varrho(f(u,v)))$
$=\varrho(h(u_{1},v_{1},u_{2},v_{2}))=h(\xi u_{1},\bar{\xi}v_{1},\xi
u_{2},\bar{\xi}v_{2}),$
$h(u_{1},v_{1},u_{2},v_{2})=\nu(f(u,v))=\nu(\tau(f(u,v)))$
$=-\nu(f(v,u))=-h(v_{1},u_{1},v_{2},u_{2}).$

(ii) We shall describe the image of $A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$
in ${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$ under the action of $\nu$. By
Corollary 2.7 the
${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$-module
${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$ is freely generated by

$u_{1}^{a}u_{2}^{c},u_{1}^{a}v_{2}^{d},v_{1}^{b}u_{2}^{c},v_{1}^{b}v_{2}^{d},\,0\leq
a,c\leq n,1\leq b,d\leq n-1.$

Since

$\varrho(u_{1}^{a}v_{1}^{b}u_{2}^{c}v_{2}^{d})=\xi^{a+c-b-d}u_{1}^{a}v_{1}^{b}u_{2}^{c}v_{2}^{d},\,\tau(u_{1}^{a}v_{1}^{b}u_{2}^{c}v_{2}^{d})=u_{1}^{b}v_{1}^{a}u_{2}^{d}v_{2}^{c}$

and the polynomials in ${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}$ and in
${\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$ are fixed under the action of $\varrho$
and $\tau$, from the action of $\varrho$ we obtain that
$\nu(A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}})$ is spanned by monomials
$u_{1}^{a}v_{1}^{b}u_{2}^{c}v_{2}^{d}$ subject to the condition

$a+c\equiv b+d\text{ (mod }n).$

The action of $\tau$ on ${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$ gives that the
${\mathbb{C}}[u_{1},v_{1}]^{D_{2n}}\otimes_{\mathbb{C}}{\mathbb{C}}[u_{2},v_{2}]^{D_{2n}}$-module
$\nu(A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}})$ is generated by

$u_{1}^{a}u_{2}^{n-a}-v_{1}^{a}v_{2}^{n-a},\,a=0,1,\ldots,n,\,u_{1}^{n}u_{2}^{n}-v_{1}^{n}v_{2}^{n},$
$u_{1}^{a}v_{2}^{a}-v_{1}^{a}u_{2}^{a},\,a=1,\ldots,n-1.$

(Here $v_{i}^{n}$, $i=1,2$, are not in the free generating set of the module
${\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]$ but they can be expressed as
$v_{i}^{n}=1\cdot(u_{i}^{n}+v_{i}^{n})-u_{i}^{n}$.) Since

$A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}=\nu^{-1}({\mathbb{C}}[u_{1},v_{1},u_{2},v_{2}]^{D_{2n}}),$
$\nu^{-1}(u_{1}^{a}u_{2}^{c}-v_{1}^{a}v_{2}^{b})=v^{a}[v,u]u^{c}-v^{a}[v,u]v^{c},$
$\nu^{-1}(u_{1}^{a}v_{2}^{a}-v_{1}^{a}u_{2}^{a})=u^{a}[v,u]v^{a}-v^{a}[v,u]u^{a}$

this completes the proof of (ii). ∎

Now we state the main result on the invariants of the dihedral group $D_{2n}$
acting on the free metabelian algebra $A_{2}({\mathfrak{M}})$.

######  Theorem 3.3.

(i) For $n\geq 3$ the algebra $A_{2}({\mathfrak{M}})^{D_{2n}}$ is generated by
$uv+vu$, $u^{n}+v^{n}$ and

$u^{a}[v,u]u^{n-a}-v^{a}[v,u]v^{n-a},\,a=0,1,\ldots,n,\,u^{n}[v,u]u^{n}-v^{n}[v,u]v^{n},$
$u^{a}[v,u]v^{a}-v^{a}[v,u]u^{a},\,a=1,\ldots,n-1.$

(ii) The Hilbert series of $A_{2}({\mathfrak{M}})^{D_{2n}}$ is

$H(A_{2}({\mathfrak{M}})^{D_{2n}},t)=\frac{1}{(1-t^{2})(1-t^{n})}$
$+\frac{1}{(1-t^{2})^{2}(1-t^{n})^{2}}\left((n+1)t^{n+2}+\frac{t^{4}(1-t^{2n})}{1-t^{2}}\right).$

###### Proof.

(i) We can split the set of generators of the algebra
$A_{2}({\mathfrak{M}})^{D_{2n}}$ in two parts – the generators of the
invariants in the polynomial algebra

${\mathbb{C}}[u,v]^{D_{2n}}\cong
A_{2}({\mathfrak{M}})^{D_{2n}}/A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$

and the generators of $A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$ as a
${\mathbb{C}}[u,v]^{D_{2n}}$-bimodule. Lifting the generators $uv$ and
$u^{n}+v^{n}$ of ${\mathbb{C}}[u,v]^{D_{2n}}$ to
$A_{2}({\mathfrak{M}})^{D_{2n}}$ we obtain the polynomials $uv+vu$,
$u^{n}+v^{n}$. The set of the generators of
$A_{2}^{\prime}({\mathfrak{M}})^{D_{2n}}$ as a
${\mathbb{C}}[u,v]^{D_{2n}}$-bimodule is given in Proposition 3.2 (ii).

(ii) We have

$H(A_{2}({\mathfrak{M}})^{D_{2n}},t)=H(A_{2}({\mathfrak{M}})^{D_{2n}}/A^{\prime}_{2}({\mathfrak{M}})^{D_{2n}},t)+H(A^{\prime}_{2}({\mathfrak{M}})^{D_{2n}},t),$
$H(A_{2}({\mathfrak{M}})^{D_{2n}}/A^{\prime}_{2}({\mathfrak{M}})^{D_{2n}},t)=H({\mathbb{C}}[u,v]^{D_{2n}},t)=\frac{1}{(1-t^{2})(1-t^{n})}.$

Since $A^{\prime}_{2}({\mathfrak{M}})^{D_{2n}}$ is a free
${\mathbb{C}}[u,v]^{D_{2n}}$-module freely generated by a finite set of
elements $z_{i}$,

$H(A^{\prime}_{2}({\mathfrak{M}})^{D_{2n}},t)=H^{2}({\mathbb{C}}[u,v]^{D_{2n}},t)\sum_{i}t^{\text{deg}z_{i}}$
$=\frac{1}{(1-t^{2})^{2}(1-t^{n})^{2}}\sum_{i}t^{\text{deg}z_{i}}.$

By Proposition 3.2 (ii) the set of the free generators of the
${\mathbb{C}}[u,v]^{D_{2n}}$-bimodule
$A^{\prime}_{2}({\mathfrak{M}})^{D_{2n}}$ consists of:

  * •

$n+1$ generators $u^{a}[v,u]u^{n-a}-v^{a}[v,u]v^{n-a}$, $a=0,1,\ldots,n$, of
degree $n+2$;

  * •

One generator $u^{n}[v,u]u^{n}-v^{n}[v,u]v^{n}$ of degree $2n+2$;

  * •

One generator $u^{a}[v,u]v^{a}-v^{a}[v,u]u^{a}$ of degree $2a+2$ for each
$a=1,\ldots,n-1$.

Hence

$\sum_{i}t^{\text{deg}z_{i}}=(n+1)t^{n+2}+\sum_{a=1}^{n}t^{2a+2}=(n+1)t^{n+2}+\frac{t^{4}(1-t^{2n})}{1-t^{2}}$

and this completes the proof. ∎

######  Remark 3.4.

The generating set of the algebra $A_{2}({\mathfrak{M}})^{D_{2n}}$ is not
minimal because the commutator $[uv+vu,u^{n}+v^{n}]$ is of degree $n+2$ and
can be expressed as a linear combination of the $n+1$ generators

$u^{a}[v,u]u^{n-a}-v^{a}[v,u]v^{n-a},\,a=0,1,\ldots,n.$

It is easy to see that if we remove one of these generators, we shall obtain a
minimal generating set of the algebra $A_{2}({\mathfrak{M}})^{D_{2n}}$.

######  Remark 3.5.

The generators of $A_{2}({\mathfrak{M}})^{D_{2n}}$ in Theorem 3.3 are given as
polynomials in $u$ and $v$. We can express them as polynomials in the
generators $x$ and $y$ of $A_{2}({\mathfrak{M}})$. For the action of the
dihedral group on the two-dimensional vector space with basis $\\{x,y\\}$ over
a field $K$ of characteristic 0 we need that
$\displaystyle\cos\left(\frac{2\pi}{n}\right),\displaystyle\sin\left(\frac{2\pi}{n}\right)\in
K$. Then the generators of $A_{2}({\mathfrak{M}})^{D_{2n}}$ in Theorem 3.3
expressed in terms of $x$ and $y$ generate also the $D_{2n}$-invariants of the
two-generated free metabelian associative algebra over $K$.

###  3.2. Lie case

Our considerations will be similar to those in Subsection 2.5 but are much
simpler because by Subsection 2.3 the commutator ideal
$L_{2}^{\prime}({\mathfrak{A}}^{2})$ is a free ${\mathbb{C}}[u,v]$-module
generated by $[v,u]$. Proposition 2.6 immediately gives the following.

######  Proposition 3.6.

The commutator ideal $L_{2}^{\prime}({\mathfrak{A}}^{2})$ is a free
${\mathbb{C}}[u,v]^{D_{2n}}$-module freely generated by

$[v,u]\text{\rm ad}^{a}(u),\,a=0,1,\ldots,n,\,[v,u]\text{\rm
ad}^{b}(v),\,b=1,\ldots,n-1.$

The following theorem is the Lie analogue of Theorem 3.3.

######  Theorem 3.7.

(i) The ${\mathbb{C}}[u,v]^{D_{2n}}$-module
$L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ is freely generated by

$[v,u](\text{\rm ad}^{n}(u)-\text{\rm ad}^{n}(v)).$

(ii) The Hilbert series of $L_{2}({\mathfrak{A}}^{2})^{D_{2n}}$ is

$H(L_{2}({\mathfrak{A}}^{2})^{D_{2n}},t)=\frac{t^{n+2}}{(1-t^{2})(1-t^{n})}.$

###### Proof.

(i) As in Subsection 3.1, we define an isomorphism of
${\mathbb{C}}[u,v]$-modules

$\nu:LA_{2}^{\prime}({\mathfrak{A}}^{2})\longrightarrow{\mathbb{C}}[u,v]\text{
by }\nu([v,u]\text{ad}^{a}(u)\text{ad}^{b}(v))=u^{a}v^{b}.$

Again $\nu$ is an isomorphism also of ${\mathbb{C}}[u,v]^{D_{2n}}$-modules. As
in Proposition 3.2 $f(u,v)\in L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ if
and only if it is fixed under the action of the rotation $\varrho$ and the
reflection $\tau$. The polynomial $h(u,v)=\nu(f(u,v))\in{\mathbb{C}}[u,v]$
belongs to ${\mathbb{C}}[u,v]^{D_{2n}}$ if and only if

$\varrho(h(u,v))=h(\xi u,\bar{\xi}v)=h(u,v),\,\tau(h(u,v))=h(v,u)=-h(u,v).$

Taking into account the action of $\varrho$, the image
$\nu(L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}})$ of
$L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ is spanned as a
${\mathbb{C}}[u,v]^{D_{2n}}$-module by monomials

$u^{a},v^{b},\,a,b\equiv 0\text{ (mod }n),\,a=0,1,\ldots,n,\,b=1,\ldots,n-1.$

Now the action of $\tau$ gives that the only possibility is

$u^{n}-v^{n}(=2u^{n}-1\cdot(u^{n}+v^{n})).$

Hence $L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ is generated as a
${\mathbb{C}}[u,v]^{D_{2n}}$-module by

$\nu^{-1}(u^{n}-v^{n})=[v,u](\text{ad}^{n}(u)-\text{ad}^{n}(v)).$

(ii) Since all $D_{2n}$-invariants of $L_{2}({\mathfrak{A}}^{2})$ are in the
commutator ideal, and $L_{2}^{\prime}({\mathfrak{A}}^{2})^{D_{2n}}$ is
generated as a free ${\mathbb{C}}[u,v]^{D_{2n}}$-module by one polynomial of
degree $n+2$, we obtain that

$H(L_{2}({\mathfrak{A}}^{2})^{D_{2n}},t)=t^{n+2}H({\mathbb{C}}[u,v]^{D_{2n}},t)=\frac{t^{n+2}}{(1-t^{2})(1-t^{n})}.$

∎

######  Remark 3.8.

As in Remark 3.5, Theorem 3.7 can be restated for the $D_{2n}$-invariants of
the two-generated free metabelian Lie algebra over a sufficiently large field
$K$ of characteristic 0.

## References

  * [1] A.Z. Anan’in, Locally finitely approximable and locally representable varieties of algebras (Russian), Algebra i Logika 16 (1977), 3-23. Translation: Algebra Logic 16 (1977), 1-16.
  * [2] Y. Bahturin, Identical Relations in Lie Algebras, 2nd edition, De Gruyter Expositions in Mathematics 68. Berlin: De Gruyter, 2021.
  * [3] R.M. Bryant, A.I. Papistas, On the fixed points of a finite group acting on a relatively free Lie algebra, Glasg. Math. J. 42 (2000), No. 2, 167-181.
  * [4] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 67 (1955), 778-782.
  * [5] M. Domokos, V. Drensky, Noether bound for invariants in relatively free algebras, J. Algebra 463 (2016), 152-167.
  * [6] V. Drensky, Fixed algebras of residually nilpotent Lie algebras, Proc. Amer. Math. Soc. 120 (1994), No. 4, 1021-1028.
  * [7] V. Drensky, Commutative and noncommutative invariant theory, Math. and Education in Math., Proc. of the 24-th Spring Conf. of the Union of Bulgar. Mathematicians, Svishtov, April 4-7, 1995, 14-50. arXiv: 2302.10052
  * [8] V.Drensky, Free Algebras and PI-Algebras, Graduate Course in Algebra, Springer-Verlag Singapore, 1999.
  * [9] V. Drensky, Ş. Fındık, N.Ş. Öğüşlü, Symmetric polynomials in the free metabelian Lie algebras, Mediterr. J. Math. 17 (2020), No. 5, Paper No. 151, 11 p.
  * [10] E. Formanek, Noncommutative invariant theory, Contemp. Math. 43 (1985), 87-119.
  * [11] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge, Cambridge University Press, 1992.
  * [12] A.R. Kemer, Nonmatrix varieties (Russian), Algebra i Logika 19 (1980), 255-283. Translation: Algebra Logic 19 (1980), 157-178.
  * [13] V.K. Kharchenko, Noncommutative invariants of finite groups and Noetherian varieties, J. Pure Appl. Alg. 31 (1984), 83-90.
  * [14] O.G. Kharlampovich, M.V. Sapir, Algorithmic problems in varieties, Int. J. Algebra Comput. 5 (1995), No. 4-5, 379-602.
  * [15] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Leipzig, Teubner, 1884.
  * [16] F. Klein, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, New York, Dover Publications, Inc., 1956.
  * [17] V.N. Latyshev, Complexity of nonmatrix varieties of associative algebras. I (Russian), Algebra i Logika 16 (1977), 149-183. Translation: Algebra Logic 16 (1977), 48-122.
  * [18] I. V. Lvov, Maximality conditions in algebras with identity relations (Russian), Algebra i Logika 8 (1969), 449-459. Translation: Algebra Logic 8 (1969), 258-263.
  * [19] E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89-92. Reprinted in “Gesammelte Abhandlungen. Collected Papers”, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1983, 181-184.
  * [20] G.C. Shephard, J.A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304.
  * [21] T.A. Springer, Invariant Theory, Lect. Notes in Math. 585, Springer-Verlag, Berlin–Heidelberg–New York, 1977.
  * [22] E.I. Zelmanov, On Engel Lie algebras (Russian), Sibirsk. Mat. Zh. 29 (1988), No. 5, 112-117. Translation: Siberian Math. J. 29 (1988), 777-781.