Remarks on Yangian-Type Algebras and Double Poisson Brackets

Abstract

In a recent paper, one of the authors proposed a construction of associative algebras which share a number of properties of the Yangians of series A but are more massive. We show that this construction admits a generalization which reveals a direct connection with a large family of double Poisson brackets on free associative algebras, which was described by Pichereau and Van de Weyer (in 2008).

1. Introduction

The present paper is a continuation of [5]. In that paper, a two-parameter family $\{Y_{d,L}:d,L=1,2,\dots\}$ of algebras was introduced and studied. For the particular value $L=1$, these algebras are the well-known Yangians $Y( \mathfrak{gl} (d,\mathbb{C}))$ of the general Lie algebras $\mathfrak{gl} (d,\mathbb{C})$. For $L\ge2$, the algebras $Y_{d,L}$ are new objects.

It is well known that the Yangian $Y( \mathfrak{gl} (d,\mathbb{C}))$ is a deformation of $\mathcal{U}( \mathfrak{gl} (d,\mathbb{C}[u]))$, the universal enveloping algebra of the Lie algebra $\mathfrak{gl} (d,\mathbb{C}[u])= \mathfrak{gl} (d,\mathbb{C})\otimes\mathbb{C}[u]$; the latter is a polynomial current Lie algebra.

As shown in [5], a similar fact holds for the algebra $Y_{d,L}$: it is a deformation of $\mathcal{U}( \mathfrak{gl} (d, \mathrm R _L))$, where $\mathrm R _L$ is a certain $\mathbb Z _{\ge0}$-graded associative algebra depending on the parameter $L$. However, in contrast to the Yangian case, the algebra $\mathrm R _L$ with $L\ge2$ is noncommutative. Further, it is much more massive than $\mathbb{C}[u]$; namely, the dimension of its $n$th homogeneous component ($n=0,1,2,\dots$) is $d^2 L^{n+1}$ and hence grows exponentially in $n$. Nevertheless, the algebras $Y_{d,L}$ with $L\ge2$ share a number of properties of the Yangian $Y( \mathfrak{gl} (d,\mathbb{C}))$. For this reason, these new algebras were called Yangian-type algebras.

The purpose of the present note is to show that the results of [5] in fact hold in greater generality, namely, for a family $\{Y_d( \Omega )\}$ of Yangian-type algebras, which depend (in addition to the previous parameter $d=1,2,3,\dots$) on an arbitrary associative $\mathbb{C}$-algebra $\Omega$. In this picture, the algebras $Y_{d,L}$ correspond to the particular case of $\Omega =\mathbb{C}^{\oplus L}$, the direct sum of $L$ copies of the field $\mathbb{C}$.

We also establish a connection between the algebras $Y_d( \Omega )$ and a class of double Poisson brackets (in the sense of Van den Bergh [8]), which was introduced in the work [6] by Pichereau and Van de Weyer.

Note that the correspondence $\Omega \rightsquigarrow Y_d( \Omega )$ is functorial in the sense that any algebra morphism $\Omega \to \Omega '$ entails an algebra morphism $Y_d( \Omega )\to Y_d( \Omega ')$ for each fixed $d$. On the other hand, for fixed $\Omega$ and varying $d$, the algebras form a nested chain:

$$Y_1( \Omega )\subset Y_2( \Omega )\subset Y_3( \Omega )\subset\dots\,.$$

As in [5], the algebra $Y_d( \Omega )$ is extracted from a larger algebra $A_d( \Omega )$, which in turn is constructed as a projective limit of filtered algebras: $A_d( \Omega )=\varprojlim A_d( \Omega ;N)$, where $A_d( \Omega ;N)$ are certain centralizers in the universal enveloping algebras $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$. This is a generalization of the so-called centralizer construction (for the old version related to the Yangians, see [4, Sec. 2.1] and [2, Chap. 8]).

The structure of the paper is as follows.

$\bullet$ In Section 2 we introduce a centralizer construction, which leads to the algebras $A_d( \Omega )$.

$\bullet$ In Section 3 we construct some special elements of $A_d( \Omega )$ (Theorem 3.4).

$\bullet$ In Section 4 we define the algebra $Y_d( \Omega )$ as the subalgebra of $A_d( \Omega )$ generated by these special elements. Then we formulate five theorems, which we consider to be our main results.

— Theorem 4.2 states that, as a vector space, $A_d( \Omega )$ splits into the tensor product of the subalgebras $A_0( \Omega )$ and $Y_d( \Omega )$; this shows to what extent $Y_d( \Omega )$ differs from $A_d( \Omega )$. Note that, in the Yangian case, where $\Omega =\mathbb{C}$, these two subalgebras commute and the first subalgebra, $A_0(\mathbb{C})$, is commutative. But in the general case, this is no longer true.

— Theorem 4.3 is an analogue of the Poincaré–Birkhoff–Witt theorem for $Y_d( \Omega )$; it specifies the “size” of the algebra.

— Theorem 4.4 claims that the algebra $Y_d( \Omega )$ can be defined by quadratic-linear commutation relations for generators.

— Theorem 4.5 provides a one-parameter group of automorphisms of $Y_d( \Omega )$. In the Yangian case, these automorphisms are a useful tool for constructing representations.

— Theorem 4.7 describes the structure of the graded algebra $\operatorname{gr} A_0( \Omega )$ associated with $A_0( \Omega )$.

The rest of the paper is essentially a set of comments on these results.

$\bullet$ In Section 5 we deal with Poisson brackets. By the very definition of the algebra $A_d( \Omega )$, it has a canonical filtration such that the associated graded algebra $\operatorname{gr} A_d( \Omega )$ is commutative and carries a Poisson bracket of degree $-1$. Consequently, the algebras $\operatorname{gr} Y_d( \Omega )$ and $\operatorname{gr} A_0( \Omega )$ are also graded Poisson algebras. Their structure is described in Propositions 5.3 and 5.4, and it turns out that it fits into the general formalism of Van den Bergh [8].

In more detail, recall that a double Poisson bracket $\{\!\!\{ -,- \}\!\!\}$ on an associative algebra $\mathcal A$ is a bilinear map $\mathcal A \times \mathcal A \to \mathcal A \otimes \mathcal A$ subject to three axioms [8] (some versions of the skew-symmetry condition, the Leibniz rule, and the Jacobi identity). The paper [8] contains (among many other things) two general constructions. First, $\{\!\!\{ -,- \}\!\!\}$ determines a Poisson structure on the space of finite-dimensional representations of the algebra $\mathcal A$. Second, $\{\!\!\{ -,- \}\!\!\}$ determines a Lie algebra structure on the vector space $\mathcal A /[ \mathcal A , \mathcal A ]$.

We observe that the Poisson brackets computed in Propositions 5.3 and 5.4 are linked with these two constructions, and the corresponding double Poisson brackets (we denote them by $\{\!\!\{ -,- \}\!\!\} _ \Omega$) are precisely the “linear” double brackets on free associative algebras; a classification of such kind of brackets was given by Pichereau and Van de Weyer [6]. As shown in [6], they are specified by an arbitrary collection of structure constants of an associative algebra; in our context this is the algebra $\Omega$.

$\bullet$ In Section 6 to the algebra $\Omega$ we assign a graded vector space

$$ \mathrm R ( \Omega )=\bigoplus_{m=0}^\infty \mathrm R _m( \Omega ), \qquad \mathrm R _m( \Omega ):= \Omega ^{\otimes (m+1)},$$

(1.1)

and endow it with an associative multiplication

$$ \mathrm R _m( \Omega )\times \mathrm R _n( \Omega )\to \mathrm R _{m+n}( \Omega ), \qquad m,n\in \mathbb Z _{\ge0},$$

(1.2)

which turns $\mathrm R ( \Omega )$ into a graded associative algebra. Let us emphasize that the operation (1.2) differs from the standard product in the tensor algebra $\mathrm T ( \Omega )$. For the standard multiplication of tensors on the space $\Omega$, we do not need the algebra structure on $\Omega$, whereas the operation (1.2) uses this structure. It is also important that the grading in (1.1) is shifted by $1$ with respect to the standard grading in $\mathrm T ( \Omega )$. A lucid interpretation of the algebra $\mathrm R ( \Omega )$ is given in Proposition 6.2.

In the particular case of $\Omega =\mathbb{C}^{\oplus L}$, one obtains the algebra $\mathrm R _L$ mentioned at the beginning of the introduction (note that it is isomorphic to the path algebra of a quiver).

We show (Proposition 6.3) that $Y_d( \Omega )$ is a deformation of the universal enveloping algebra $\mathcal{U}( \mathfrak{gl} (d, \mathrm R ( \Omega ))$. This extends the result of [5] (mentioned above) that $Y_{d,L}$ is a deformation of $\mathcal{U}( \mathfrak{gl} (d, \mathrm R _L))$. Thus, the algebras $\mathrm R ( \Omega )$ may be viewed as a noncommutative analogue of the algebra of polynomials in the construction of current algebras.

2. The Centralizer Construction

Throughout the whole paper $\Omega$ stands for an arbitrary (not necessarily unital) associative algebra of finite or countable dimension over $\mathbb{C}$.

2.1. The Lie algebra $\mathfrak{gl} (N, \Omega )$

Let $\operatorname{Mat} (N,\mathbb{C})$ denote the associative algebra of $N\times N$ matrices over $\mathbb{C}$. Likewise, $\operatorname{Mat} (N, \Omega )$ is the algebra of $N\times N$ matrices with entries in $\Omega$. Equivalently,

$$\operatorname{Mat} (N, \Omega ):= \operatorname{Mat} (N,\mathbb{C})\otimes \Omega .$$

The Lie algebra $\mathfrak{gl} (N, \Omega )$ coincides with $\operatorname{Mat} (N, \Omega )$ as a vector space, and the bracket is given by the commutator.

We denote by $E_{ij}$ the matrix units in $\operatorname{Mat} (N,\mathbb{C})$ ($1\le i,j\le N$). They form a basis of $\mathfrak{gl} (N,\mathbb{C})$. For $x\in \Omega$, we set

$$E_{ij}(x):=E_{ij}\otimes x \in \mathfrak{gl} (N, \Omega ).$$

The following relations hold:

$$[E_{ij}(x),E_{kl}(y)]= \delta _{kj}E_{il}(xy)- \delta _{il} E_{kj}(yx), \qquad x,y\in \Omega , \; i,j,k,l\in\{1,\dots,N\}.$$

The elements $E_{ij}(x)$ with $i,j\in\{1,\dots,N\}$ and $x$ ranging over a given basis of $\Omega$ form a basis in $\mathfrak{gl} (N, \Omega )$.

2.2. The action of $\mathfrak{gl} (N,\mathbb{C})$ on $\mathfrak{gl} (N, \Omega )$

If $\Omega$ has a unit $1$, then $\mathfrak{gl} (N,\mathbb{C})$ is embedded into $\mathfrak{gl} (N, \Omega )$ via the map $E_{ij}\mapsto E_{ij}(1)$. Then we can restrict the adjoint action of $\mathfrak{gl} (N, \Omega )$ to the subalgebra $\mathfrak{gl} (N,\mathbb{C})$ and obtain in this way an action of $\mathfrak{gl} (N,\mathbb{C})$ by derivations of the Lie algebra $\mathfrak{gl} (N, \Omega )$.

However, this action is well defined even if $\Omega$ does not have a unit. Indeed, we define the action as $\operatorname{ad}_{ \mathfrak{gl} (N,\mathbb{C})}\otimes\operatorname{Id}$, where $\operatorname{ad}_{ \mathfrak{gl} (N,\mathbb{C})}$ denotes the adjoint representation of the Lie algebra $\mathfrak{gl} (N,\mathbb{C})$, and we identify $\mathfrak{gl} (N, \Omega )$ with $\mathfrak{gl} (N,\mathbb{C})\otimes \Omega$.

We agree to denote the action of $E_{ij}$ as $[E_{ij}, -]$, even if $\Omega$ does not contain a unit. In this notation, we have

$$[E_{ij}, E_{kl}(y)]= \delta _{kj} E_{il}(y)- \delta _{il}E_{kj}(y), \qquad y\in \Omega .$$

2.3. The subalgebras $\mathfrak{gl} _d(N,\mathbb{C})$ and centralizers

For $d=0,1,\dots,N$, we set

$$\mathfrak{gl} _d(N,\mathbb{C}):=\text{linear span of the matrix units $E_{ij}$ with $d+1\le i,j\le N$}.$$

This is a Lie subalgebra of $\mathfrak{gl} (N,\mathbb{C})$ isomorphic to $\mathfrak{gl} (N-d,\mathbb{C})$.

Consider the universal enveloping algebra $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$ of the Lie algebra $\mathfrak{gl} (N, \Omega )$. We set

$$A_d( \Omega ;N):= \mathcal{U}( \mathfrak{gl} (N, \Omega ))^{ \mathfrak{gl} _d(N,\mathbb{C})}.$$

That is, $A_d( \Omega ;N)\subset \mathcal{U}( \mathfrak{gl} (N, \Omega ))$ is the subalgebra of invariants with respect to the action of the Lie subalgebra $\mathfrak{gl} _d(N,\mathbb{C})\subset \mathfrak{gl} (N,\mathbb{C})$.

When $\Omega$ is unital, $\mathfrak{gl} _d(N,\mathbb{C})$ becomes a subalgebra of $\mathfrak{gl} (N, \Omega )$, and then $A_d( \Omega ;N)$ is the centralizer of this subalgebra. With a slight abuse of terminology, we call $A_d( \Omega ;N)$ a centralizer even when $\Omega$ is nonunital.

2.4. The canonical projections $\pi_{N,N-1}$

We denote by $I^+(N)$ the left ideal in $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$ generated by the elements $E_{iN}(x)$, where $i=1,\dots,N$ and $x\in \Omega$. Likewise, $I^-(N)$ is the right ideal in $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$ generated by the elements $E_{Nj}(x)$, where $j=1,\dots,N$ and $x\in \Omega$.

Lemma 2.1.

Let $\mathcal{U}( \mathfrak{gl} (N, \Omega ))^{E_{NN}}=\mathcal{U}( \mathfrak{gl} (N, \Omega ))^{ \mathfrak{gl} _{N-1}(N,\mathbb{C})}$ be the centralizer of $E_{NN}$, that is, the subalgebra of elements annihilated by $[E_{NN},-]$.

(i) The following relation holds:

$$\mathcal{U}( \mathfrak{gl} (N, \Omega ))^{E_{NN}}\cap I^+(N)=\mathcal{U}( \mathfrak{gl} (N, \Omega ))^{E_{NN}}\cap I^-(N).$$

(ii) Let $L(N)$ denote this intersection. This is a two-sided ideal in $\mathcal{U}( \mathfrak{gl} (N, \Omega ))^{E_{NN}}$, and the following direct sum decomposition holds:

$$ \mathcal{U}( \mathfrak{gl} (N, \Omega ))^{E_{NN}}=\mathcal{U}( \mathfrak{gl} (N-1, \Omega ))\oplus L(N).$$

(2.1)

Let

$$\pi_{N,N-1}\colon\mathcal{U}( \mathfrak{gl} (N, \Omega ))^{E_{NN}}\to \mathcal{U}( \mathfrak{gl} (N-1, \Omega ))$$

be the projection onto the first summand in (2.1). Since $L(N)$ is a two-sided ideal, it follows that $\pi_{N,N-1}$ is an algebra homomorphism. It is consistent with the canonical filtration of universal enveloping algebras.

2.5. The algebras $A_d( \Omega )$ as projective limits

Lemma 2.2.

Let $d=0,1,2,\dots$ and $N>d$. Then

$$\pi_{N,N-1}(A_d( \Omega ;N)) \subseteq A_d( \Omega ;N-1).$$

This lemma makes it possible to give the following definition.

Definition 2.3 (the centralizer construction).

For $d=0,1,2,\dots$, we set

$$A_d( \Omega ):=\varprojlim (A_d( \Omega ;N), \pi_{N,N-1}),$$

where the projective limit is taken in the category of filtered algebras. Thus, the limit algebra $A_d( \Omega )$ is a filtered algebra, too.

By the very definition, every element $\mathbf a\in A_d( \Omega )$ is given by a sequence $\{\mathbf a(N): N\ge d\}$ such that

$$\text{$\mathbf a(N)\in A_d( \Omega ;N)$ for $N\ge d$, $\quad\pi_{N,N-1}(\mathbf a(N))=\mathbf a(N-1)$ for $N>d$},$$

and $\deg \mathbf a(N)$ is bounded by a constant independent of $N$.

We denote by $\pi_{\infty,N}$ the natural projection $A_d( \Omega )\to A_d( \Omega ;N)$. In this notation, $\pi_{\infty,N}( \mathbf a )= \mathbf a (N)$.

There are natural embeddings

$$ A_0( \Omega )\subset A_1( \Omega )\subset A_2( \Omega )\subset\dots\,.$$

(2.2)

In particular, $A_0( \Omega )$ is a subalgebra of $A_d( \Omega )$ for any $d\in \mathbb Z _{\ge1}$.

3. Special elements of $A_d( \Omega )$

3.1. The elements $e_{ij}( \mathbf{x} ;N)$

For $m=1,2,\dots$, let $\Omega ^m$ be the $m$-fold direct product $\underbrace{ \Omega \times\dots\times \Omega }_m$. Thus, an element of $\Omega ^m$ is an ordered $m$-tuple $\mathbf{x} =(x_1,\dots,x_m)$ of elements of $\Omega$.

Given an element $\mathbf{x} =(x_1,\dots,x_m)\in \Omega ^m$, a pair of indices $i,j$, and an integer $N\ge\max(i,j)$, we define an element $e_{ij}( \mathbf{x} ;N)$ of $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$ by

$$ e_{ij}( \mathbf{x} ;N):=\sum_{a_1,\dots,a_{m-1}=1}^N E_{ia_1}(x_1) E_{a_1a_2}(x_2)\cdots E_{a_{m-1}j}(x_m),$$

(3.1)

assuming that

$$e_{ij}( \mathbf{x} ;N)=E_{ij}(x_1),\quad \text{when $m=1$ and $ \mathbf{x} =(x_1)$.}$$

3.2. The coagulation transforms $\mathbf{x} \mapsto \mathbf{x} \rhd \nu$

Recall that a composition is an ordered tuple $\nu=(\nu_1,\dots,\nu_n)$ of positive integers. We set

$$|\nu|:=\nu_1+\dots+\nu_n, \qquad \ell(\nu)=n,$$

and denote by $\operatorname{Comp} (m)$ the set of compositions $\nu$ with $|\nu|=m$.

Definition 3.1.

To an $m$-tuple $\mathbf{x} =(x_1,\dots,x_m)\in \Omega ^m$ and a composition $\nu\in \operatorname{Comp} (m)$ with $\ell(\nu)=n$ we assign an $n$-tuple

$$y:= \mathbf{x} \rhd \nu=(y_1,\dots,y_n)\in \Omega ^n, \qquad n:=\ell(\nu),$$

as follows:

$$y_1=x_1\cdots x_{\nu_1}, \quad y_2=x_{\nu_1+1}\cdots x_{\nu_1+\nu_2}, \quad\dots,\quad y_n=x_{\nu_1+\cdots+\nu_{n-1}+1}\cdots x_m,$$

where the products are taken in the algebra $\Omega$. Let us call this transform coagulation.

For instance, for $m=3$, there are four compositions,

$$\nu=(1,1,1),\,(2,1),\,(1,2),\,(3),$$

and the corresponding coagulations are

$$\mathbf{y} =(x_1,x_2,x_3),\, (x_1x_2, x_3), \,(x_1, x_2x_3),\,x_1x_2x_3.$$

3.3. The elements $t_{ij}( \mathbf{x} ;N;s)$ and connection coefficients

We set

$$ W( \Omega ):=\bigsqcup_{m=1}^\infty \Omega ^m.$$

(3.2)

For $\mathbf{x} \in \Omega ^m\subset W( \Omega )$, we set $\ell( \mathbf{x} ):=m$.

Definition 3.2.

Let $s\in\mathbb{C}$ be an auxiliary parameter. As before, let $i$, $j$, and $N$ be positive integers, $1\le i,j\le N$. Next, let $\mathbf{x} \in W( \Omega )$, and let $m:=\ell( \mathbf{x} )$. Given these data, we define elements $t_{ij}( \mathbf{x} ;N;s)$ of the algebra $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$ as

$$t_{ij}( \mathbf{x} ;N;s):=\sum_{\nu\in \operatorname{Comp} (m)} (-N-s)^{\ell( \mathbf{x} )-\ell( \mathbf{x} \rhd \nu)}e_{ij}( \mathbf{x} \rhd \nu;N).$$

Here we agree that $0^0:=1$. By virtue of this convention,

$$t_{ij}( \mathbf{x} ;N;s)\big|_{s=-N}=e_{ij}( \mathbf{x} ;N).$$

Note that $\mathbf{x} \rhd \nu= \mathbf{x}$ for the composition $\nu=(1,\dots,1)$, while for all other compositions, $\ell( \mathbf{x} \rhd \nu)<\ell( \mathbf{x} )$. It follows that

$$t_{ij}( \mathbf{x} ;N;s)=e_{ij}( \mathbf{x} ;N)+(\cdots),$$

where the dots stand for a linear combination of elements of the form $e_{ij}( \mathbf{y} ;N)$ with $\ell( \mathbf{y} )<\ell( \mathbf{x} )= m$.

The following lemma connects elements corresponding to different values of the parameter $s$.

Lemma 3.3.

For any $s,s'\in\mathbb{C}$,

$$t_{ij}( \mathbf{x} ;N;s)=\sum_{\nu\in \operatorname{Comp} (m)}(s'-s)^{\ell( \mathbf{x} )-\ell( \mathbf{x} \rhd \nu)} t_{ij}( \mathbf{x} \rhd \nu;N;s').$$

3.4. The elements $t_{ij}( \mathbf{x} ;s)\in A_d( \Omega )$

Theorem 3.4.

For fixed $\mathbf{x} \in W( \Omega )$ and $s\in\mathbb{C}$,

$$\pi_{N,N-1}(t_{ij}( \mathbf{x} ;N;s))=t_{ij}( \mathbf{x} ;N-1;s), \qquad N>\max(i,j).$$

Corollary 3.5.

Let $d\in \mathbb Z _{\ge1}$ be fixed. For any $\mathbf{x} \in W( \Omega )$, $s\in\mathbb{C}$, and $i,j\in\{1,\dots,d\}$, there exists an element $t_{ij}( \mathbf{x} ;s)\in A_d( \Omega )$ uniquely determined by the property that

$$\pi_{\infty,N}(t_{ij}( \mathbf{x} ;s))=t_{ij}( \mathbf{x} ;N;s)\qquad \textit{for all }\, N\ge d.$$

Together with Lemma 3.3 this in turn implies the following assertion.

Corollary 3.6.

For any $s,s'\in\mathbb{C}$,

$$t_{ij}( \mathbf{x} ;s)=\sum_{\nu\in \operatorname{Comp} (m)}(s'-s)^{\ell( \mathbf{x} )-\ell( \mathbf{x} \rhd \nu)} t_{ij}( \mathbf{x} \rhd \nu;s').$$

4. Main Results

Definition 4.1.

Given $d\in \mathbb Z _{\ge1}$, we denote by $Y_d( \Omega )$ the unital subalgebra of $A_d( \Omega )$ generated by the elements of the form $t_{ij}( \mathbf{x} ;s)$, where $i,j\in\{1,\dots,d\}$, $\mathbf{x} \in W( \Omega )$, and $s\in\mathbb{C}$ is an arbitrary fixed number. By Corollary 3.6, the definition does not depend on the choice of $s$.

We fix a basis $X$ in $\Omega$ and set (cf. (3.2))

$$W(X):=\bigsqcup_{m=1}^\infty X^m \subset W( \Omega ).$$

Each element $t_{ij}( \mathbf{x} ; s)$ with $\mathbf{x} =(x_1,\dots,x_m)$ depends multilinearly on $x_1,\dots,x_m$; therefore, this is a linear combination of some elements $t_{ij}( \mathbf{y} ;s)$ with $\mathbf{y} \in X^m$. It follows that in the definition above we may assume that $\mathbf{x}$ ranges over $W(X)$.

Pick an arbitrary total order $\prec$ on the set of triples $(i,j, \mathbf{x} )$, where $i,j\in\{1,\dots,d\}$ and $\mathbf{x} \in W(X)$. We say that a product

$$ t_{i_1j_1}( \mathbf{x} ^1; s)\cdots t_{i_rj_r}( \mathbf{x} ^r;s), \quad \text{where $ \mathbf{x} ^1,\dots, \mathbf{x} ^r\in W(X)$},$$

(4.1)

is an ordered monomial if the corresponding triples $(i_1,j_1, \mathbf{x} ^1), \dots, (i_r,j_r, \mathbf{x} ^r)$ weakly increase with respect to $\prec$.

Theorem 4.2 (splitting of $A_d( \Omega )$).

Let $m\colon A_d( \Omega )\times A_d( \Omega )\to A_d( \Omega )$ denote the multiplication map. The subalgebras $A_0( \Omega )$ and $Y_d( \Omega )$ split $A_d( \Omega )$ in the sense that

$$m\colon A_0( \Omega )\otimes Y_d( \Omega )\to A_d( \Omega )$$

is an isomorphism of vector spaces.

Recall that, by the very definition, the algebra $A_d( \Omega )$ is endowed with a canonical grading coming from the filtration of the prelimit algebras $\mathcal{U}( \mathfrak{gl} (N, \Omega ))$. Thus, the subalgebra $Y_d( \Omega )$ is also equipped with a filtration. With respect to it

$$\deg t_{ij}( \mathbf{x} ;N;s) =\ell( \mathbf{x} ).$$

Theorem 4.3 (analogue of the Poincaré–Birkhoff–Witt theorem for $Y_d( \Omega )$).

For any choice of a basis $X\subset \Omega$, a total order $\prec$, and an $s\in\mathbb{C}$, the ordered monomials (4.1) together with $1$ form a basis of the algebra $Y_d( \Omega )$.

This fact is connected with the following result (cf. [5, Sec. 7]).

Theorem 4.4 (defining commutation relations).

Fix an arbitrary $s\in\mathbb{C}$. For an arbitrary quadruple of indices $1\le i,j,k,l\le d$ and arbitrary $\mathbf{x} , \mathbf{y} \in W( \Omega )$, the commutator $[t_{ij}( \mathbf{x} ;s), t_{kl}( \mathbf{y} ;s)]$ can be represented as a quadratic-linear expression in some generators of the form

$$t_{kj}( \,\boldsymbol\cdot\, ;s),\,t_{il}( \,\boldsymbol\cdot\, ;s),\,t_{ij}( \,\boldsymbol\cdot\, ;s),\,t_{kl}( \,\boldsymbol\cdot\, ;s),$$

and the degree of this expression is strictly less than

$$\deg t_{ij}( \mathbf{x} ;s) + \deg t_{kl}( \mathbf{y} ;s)=\ell( \mathbf{x} )+\ell( \mathbf{y} ).$$

Next, the commutation relations that can be obtained in this way are defining relations of the algebra $Y_d( \Omega )$.

Theorem 4.5 (shift automorphisms of the algebra $Y_d( \Omega )$).

The additive group $\mathbb{C}$ acts on the algebra $Y_d( \Omega )$ by automorphisms $\operatorname{Shift} _c$, $c\in\mathbb{C}$, such that

$$\operatorname{Shift} _c\colon t_{ij}( \mathbf{x} ;s)\mapsto t_{ij}( \mathbf{x} ;s+c)$$

for any $i,j\in\{1,\dots,d\}$, $\mathbf{x} \in W( \Omega )$, and $s\in\mathbb{C}$.

The adjoint graded algebra $\operatorname{gr} A_d( \Omega )$ is commutative, because this is so for the prelimit algebras. Therefore, the algebras $\operatorname{gr} Y_d( \Omega )$ and $\operatorname{gr} A_0( \Omega )$ are also commutative. Now we describe their structure.

We introduce the following notation:

$$\mathrm T ^+( \Omega ):=\bigoplus_{m=1}^\infty \mathrm T ^m( \Omega ), \quad \mathrm T ^m( \Omega ):= \Omega ^{\otimes m},$$

that is, $\mathrm T ^+( \Omega )$ is the space of tensors over $\Omega$ of strictly positive degree;

$$\mathrm M _d( \Omega ):= \operatorname{Mat} (d,\mathbb{C})\otimes \mathrm T ^+( \Omega );$$

$\mathcal S( \mathrm M _d( \Omega ))$ is the symmetric algebra over the vector space $\mathrm M _d( \Omega )$. We equip $\mathrm M _d( \Omega )$ with the grading induced by the grading of the tensor space $\mathrm T ^+( \Omega )$ and then extend the grading from $\mathrm M _d( \Omega )$ to the symmetric algebra $\mathcal S( \mathrm M _d( \Omega ))$.

Theorem 4.3 has the following corollary.

Corollary 4.6.

There is an isomorphism of graded algebras

$$ \operatorname{gr} Y_d( \Omega )\simeq \mathcal S( \mathrm M _d( \Omega )).$$

(4.2)

Now we denote by $\mathrm T _{ \operatorname {cycl}} ^m( \Omega )$ the space of coinvariants for the action of the group $\mathbb Z /m \mathbb Z$ by cyclic permutations on the space $\mathrm T ^m( \Omega )$ ($m=1,2,\dots$). Next, we introduce the graded vector space

$$\mathrm T _{ \operatorname {cycl}} ^+( \Omega ):=\bigoplus_{m=1}^\infty \mathrm T _{ \operatorname {cycl}} ^m( \Omega )$$

and denote by $\mathcal S( \mathrm T _{ \operatorname {cycl}} ^+( \Omega ))$ the symmetric algebra over this space.

Theorem 4.7.

There is an isomorphism of graded algebras

$$ \operatorname{gr} A_0( \Omega )\simeq \mathcal S( \mathrm T _{ \operatorname {cycl}} ^+( \Omega )).$$

(4.3)

5. Relationship with Double Poisson Brackets

Since the algebras $\operatorname{gr} Y_d( \Omega )$ and $\operatorname{gr} A_0( \Omega )$ are commutative, they possess canonical Poisson brackets. We will provide an explicit description of these brackets using the isomorphisms (4.2) and (4.3). This will reveal a connection with double Poisson brackets in the sense of Van den Bergh [8].

5.1. Double Poisson brackets [8]

Let $\mathcal A$ be an arbitrary associative algebra. We need its tensor square, which will be denoted by the symbol $\mathcal A ^{\boxtimes 2}$. Here we do not use the standard symbol $\otimes$ for tensor multiplication, because our algebra $\mathcal A$ will itself consist of tensors later on.

Recall that a double Poisson bracket $\{\!\!\{ -,- \}\!\!\}$ on $\mathcal A$ is a bilinear map

$$\mathcal A \times \mathcal A \to \mathcal A ^{\boxtimes2}, \qquad (a,b) \mapsto \{\!\!\{ a,b \}\!\!\} ,$$

subject to three axioms, which are analogues of the skew-symmetry condition, the Leibniz rule, and the Jacobi identity (see Van den Bergh’s paper [8]).

Note that if $\mathcal A$ contains a unit $1$, then $\{\!\!\{ 1,a \}\!\!\} = \{\!\!\{ a,1 \}\!\!\} =0$ for all $a\in \mathcal A$.

5.2. Linear double Poisson brackets [6]

Assume for a moment that $\Omega$ is simply a vector space without any additional structure. We denote by $\mathrm T ( \Omega )$ the tensor algebra over $\Omega$, or, which is the same thing, the free algebra generated by the vector space $\Omega$. From now on we set $\mathcal A := \mathrm T ( \Omega )$.

Let $\{\!\!\{ -,- \}\!\!\}$ be a double Poisson bracket on $\mathrm T ( \Omega )$. By one of the axioms (the analogue of the Leibniz rule) $\{\!\!\{ -,- \}\!\!\}$ is uniquely determined by its values on $\Omega \times \Omega$, where we identify $\Omega$ with $\mathrm T ^1( \Omega )$. Following [6], we say that $\{\!\!\{ -,- \}\!\!\}$ is linear if its restriction to $\Omega \times \Omega$ takes values in $1\boxtimes \Omega + \Omega \boxtimes 1$. Then we necessarily obtain

$$ \{\!\!\{ x,y \}\!\!\} = 1\boxtimes \mu(x,y) - \mu(y,x)\boxtimes 1, \qquad x,y\in \Omega ,$$

(5.1)

for a certain bilinear map $\mu: \Omega \times \Omega \to \Omega$. However, $\mu$ cannot be arbitrary (because the analogue of the Jacobi identity must hold).

Proposition 5.1 (Pichereau and Van de Weyer [6, Proposition 10]).

In order that formula (5.1) define a double Poisson bracket, it is necessary and sufficient that $\mu$ be an associative multiplication on the vector space $\Omega$.

(See also Odesskii, Rubtsov, and Sokolov’s paper [3, (2.9)].) Thus, for an arbitrary vector space $\Omega$, there is a one-to-one correspondence between the linear double Poisson brackets on the free algebra $\mathrm T ( \Omega )$ and the associative algebra structures on $\Omega$. (The papers [6] and [3] deal with finite-dimensional vector spaces, but in Proposition 5.1 finite dimensionality is not required.)

5.3. The double Poisson bracket $\{\!\!\{ -,- \}\!\!\} _ \Omega$ on $\mathrm T ( \Omega )$

We again assume that $\Omega$ is an associative algebra. Let $\{\!\!\{ -,- \}\!\!\} _ \Omega$ denote the double Poisson bracket on $\mathrm T ( \Omega )$ provided by Proposition 5.1. Below we give an explicit expression for this bracket.

First, we introduce some new notation. Let $\mathbf{x} =x_1\otimes\dots\otimes x_m\in \mathrm T ^m( \Omega )$ and $\mathbf{y} =y_1\otimes\dots\otimes y_n\in \mathrm T ^n( \Omega )$ be two nonscalar decomposable tensors. For $r=1,\dots,m$ and $s=1,\dots,n$, we set

$$\begin{alignedat}2 \mathbf{x} ^{<r}&=x_1\otimes \dots \otimes x_{r-1},&\quad \mathbf{x} ^{>r}&=x_{r+1}\otimes \dots\otimes x_m, \\ \mathbf{y} ^{<s}&=y_1\otimes\dots \otimes y_{s-1},&\quad \mathbf{y} ^{>s}&=y_{s+1}\otimes \dots\otimes y_n, \end{alignedat}$$

assuming that $\mathbf{x} ^{<1}$, $\mathbf{x} ^{>m}$, $\mathbf{y} ^{<1}$, and $\mathbf{y} ^{>n}$ are interpreted as the unit of the algebra $\mathrm T ( \Omega )$.

Proposition 5.2.

The double Poisson bracket $\{\!\!\{ -,- \}\!\!\} _ \Omega$ on the free algebra $\mathrm T ( \Omega )$ defined according to the general recipe of Proposition 5.1 is given by the formula

$$\begin{gathered} \, \{\!\!\{ \mathbf{x} , \mathbf{y} \}\!\!\} _ \Omega =\sum_{r=1}^m\sum_{s=1}^n (( \mathbf{y} ^{<s}\otimes \mathbf{x} ^{>r})\boxtimes ( \mathbf{x} ^{<r}\otimes (x_r y_s)\otimes \mathbf{y} ^{>s})) \\ -\sum_{r=1}^m\sum_{s=1}^n (( \mathbf{y} ^{<s}\otimes (y_s x_r)\otimes \mathbf{x} ^{>r})\boxtimes ( \mathbf{x} ^{<r}\otimes \mathbf{y} ^{>s})). \end{gathered}$$

(5.2)

Finally, $\{\!\!\{ -,- \}\!\!\} _ \Omega$ is equal to $0$ if at least one of the arguments is in $\mathrm T ^0( \Omega )=\mathbb{C}$.

5.4. The Poisson bracket on $\operatorname{gr} Y_d( \Omega )$

Given $\mathbf{x} \in \mathrm T ^+( \Omega )$ and a bi-index $ij$, where $i,j\in\{1,\dots,d\}$, we set

$$p_{ij}( \mathbf{x} ):= E_{ij}\otimes \mathbf{x} \in \mathrm M _d( \Omega ).$$

On the algebra $\mathcal S( \mathrm M _d( \Omega ))$ there exists a Poisson bracket $\{-,-\}_{d, \Omega }$ uniquely determined by the condition that, for decomposable tensors $\mathbf{x} \in \mathrm T ^m( \Omega )$ and $\mathbf{y} \in \mathrm T ^n( \Omega )$ (where $m,n\ge1$) and indices $i,j,k,l\in\{1,\dots,d\}$,

$$\begin{gathered} \, \{p_{ij}( \mathbf{x} ), p_{kl}( \mathbf{y} )\}_{d, \Omega }=\sum_{r=1}^m\sum_{s=1}^n p_{kj} ( \mathbf{y} ^{<s}\otimes \mathbf{x} ^{>r})p_{il} ( \mathbf{x} ^{<r}\otimes (x_r y_s)\otimes \mathbf{y} ^{>s}) \\ -\sum_{r=1}^m\sum_{s=1}^n p_{kj}( \mathbf{y} ^{<s}\otimes (y_s x_r)\otimes \mathbf{x} ^{>r})p_{il} ( \mathbf{x} ^{<r}\otimes \mathbf{y} ^{>s}), \end{gathered}$$

where the symbol $p_{ab}(\cdot)$ corresponding to the empty product is assumed to be the Kronecker delta $\delta _{ab}$.

Writing the element $\{\!\!\{ \mathbf{x} , \mathbf{y} \}\!\!\} _ \Omega \in \mathrm T ( \Omega )^{\boxtimes2}$ in Sweedler’s notation as $\sum \{\!\!\{ \mathbf{x} , \mathbf{y} \}\!\!\} '\boxtimes \{\!\!\{ \mathbf{x} , \mathbf{y} \}\!\!\} ''$, we can rewrite the formula as

$$ \{p_{ij}( \mathbf{x} ), p_{kl}( \mathbf{y} )\}_{d, \Omega }=\sum p_{kj}( \{\!\!\{ \mathbf{x} , \mathbf{y} \}\!\!\} '_ \Omega )p_{il}( \{\!\!\{ \mathbf{x} , \mathbf{y} \}\!\!\} ''_ \Omega ).$$

(5.3)

The transition from the double Poisson bracket $\{\!\!\{ -,- \}\!\!\} _ \Omega$ on the free algebra $\mathrm T ( \Omega )$ (formula (5.2)) to the Poisson bracket $\{-,-\}_{d, \Omega }$ on the symmetric algebra $\mathcal S( \mathrm M _d( \Omega ))$ (formula (5.3)) “almost” fits into Van den Bergh’s formalism [8, Proposition 7.5.1]. The reservation “almost” is due to the fact that in our situation we do not need to impose the relation $\sum_q a_{pq}b_{qr}=(ab)_{pr}$, written at the bottom of page 5751 in [8] (about this point see [5, Sec. 5.2]).

Proposition 5.3.

Under the isomorphism (4.2) the canonical Poisson bracket on $\operatorname{gr} Y_d( \Omega )$ coincides with the bracket $\{-,-\}_{d, \Omega }$ given by the formula (5.3).

Thus, the algebra $Y_d( \Omega )$ is a filtered deformation of the Poisson algebra $(\mathcal S( \mathrm M _d( \Omega )), \{-,-\}_{d, \Omega })$.

5.5. The Poisson bracket on $\operatorname{gr} A_0( \Omega )$

Assume first that $\mathcal A$ is an arbitrary associative algebra and let $[ \mathcal A , \mathcal A ]$ be its subspace spanned by the commutators $[a,b]$, where $a,b\in \mathcal A$. We denote the quotient space $\mathcal A /[ \mathcal A , \mathcal A ]$ by $\mathcal N_{ \mathcal A }$. As shown by Van den Bergh [8, Proposition 1.4 and Corollary 2.4.6], a double Poisson bracket $\{\!\!\{ -,- \}\!\!\}$ on $\mathcal A$ gives rise to a Lie algebra structure on the space $\mathcal N_{ \mathcal A }$ and hence to a Poisson algebra structure on the symmetric algebra $\mathcal S( \mathcal N_{ \mathcal A } )$.

Now consider the case where $\mathcal A$ is the free algebra $\mathrm T ( \Omega )$ with the double Poisson bracket $\{\!\!\{ -,- \}\!\!\} _ \Omega$ (see (5.2)). Then the space $\mathcal N_{ \mathcal A }$ can be identified with $\mathbb{C}\oplus \mathrm T _{ \operatorname {cycl}} ^+( \Omega )$. Thus, $\mathbb{C}\oplus \mathrm T _{ \operatorname {cycl}} ^+( \Omega )$ acquires a Lie algebra structure. This Lie algebra is the direct sum of the trivial one-dimensional Lie algebra $\mathbb{C}$ and the Lie subalgebra $\mathrm T _{ \operatorname {cycl}} ^+( \Omega )$; we are interested in the latter subalgebra. The Lie algebra structure on $\mathrm T _{ \operatorname {cycl}} ^+( \Omega )$ determines a Poisson bracket on $\mathcal S( \mathrm T _{ \operatorname {cycl}} ^+( \Omega ))$; let us denote it by $\{-,-\}_ \Omega$.

Proposition 5.4.

As usual, let $\Omega$ be an arbitrary associative algebra. Under the isomorphism (4.3) the Poisson bracket on $\operatorname{gr} A_0( \Omega )$ coincides with the Poisson bracket $\{-,-\}_ \Omega$ on $\mathcal S( \mathrm T _{ \operatorname {cycl}} ^+( \Omega ))$ coming from the general Van den Bergh construction described above.

Thus, the algebra $A_0( \Omega )$ serves as a filtered quantization of the Poisson algebra $(\mathcal S( \mathrm T _{ \operatorname {cycl}} ^+( \Omega )), \{-,-\}_ \Omega )$.

In the works [7] of Schedler and [1] of Ginzburg and Schedler the quantization problem for Poisson algebras of the form $\mathcal S( \mathcal N_{ \mathcal A } )$ was studied in another context, related to quivers.

6. The Degeneration $Y_d( \Omega )\rightsquigarrow \mathcal{U}( \mathfrak{gl} (d, \mathrm R ( \Omega )))$

As above, we work with an arbitrary associative algebra $\Omega$. Let $\mathrm R ( \Omega )$ denote the vector space $\mathrm T ^+( \Omega )$ endowed with a new grading starting from zero:

$$\mathrm R ( \Omega ):=\bigoplus_{n=0}^\infty \mathrm R ( \Omega )_n, \qquad \mathrm R ( \Omega )_n:= \mathrm T ^{n+1}( \Omega ).$$

Next, we endow $\mathrm R ( \Omega )$ with a multiplication denoted by the symbol $\odot$: given decomposable tensors

$$\mathbf{x} =x_1\otimes\dots\otimes x_{m+1}\in \mathrm R _m( \Omega ), \quad \mathbf{y} =y_1\otimes\dots\otimes y_{n+1}\in \mathrm R _n( \Omega ),$$

where $m,n\in\{0,1,2,\dots\}$, their product is

$$\mathbf{x} \odot \mathbf{y} :=x_1\otimes\dots\otimes x_{m}\otimes (x_{m+1}y_1)\otimes y_2\otimes \dots\otimes y_{n+1}\in \mathrm R _{m+n}( \Omega ).$$

This operation is associative and consistent with the new grading. Thus, $\mathrm R ( \Omega )$ is a graded algebra.

Remark 6.1.

(i) The $0$th component of $\mathrm R ( \Omega )$ is the initial algebra $\Omega$.

(ii) $\mathrm R ( \Omega )$ has a unit if and only if so does $\Omega$.

(iii) $\mathrm R ( \Omega )$ is noncommutative unless either $\Omega =\mathbb{C}$ or the product in $\Omega$ is null.

(iv) For $\Omega =\mathbb{C}$, the corresponding algebra $\mathrm R (\mathbb{C})$ is isomorphic to the algebra of polynomials $\mathbb{C}[u]$: the isomorphism is given by the assignment $1^{\otimes (r+1)}\mapsto u^r$.

(v) More generally, if $\Omega =\mathbb{C}^{\oplus L}$ (the direct sum of $L$ copies of $\mathbb{C}$), then $\mathrm R ( \Omega )$ is isomorphic to the path algebra of the quiver $Q_L$ with $L$ vertices indexed by $1,\dots,L$ and $L^2$ edges $i\to j$, one for each ordered pair of vertices $(i,j)$, including $i=j$.

Remark 6.2.

The vector space $M:= \Omega \otimes_{\mathbb{C}} \Omega$ possesses a natural structure of a $\Omega$-bimodule, so that we can built the graded tensor algebra over the bimodule $M$:

$$\mathrm T _ \Omega (M):= \Omega \oplus M \oplus (M\otimes_ \Omega M) \oplus (M \otimes_ \Omega M\otimes_ \Omega M) \oplus\dots,$$

where all tensor products are taken over $\Omega$. If $\Omega$ contains a unit, then the algebra $\mathrm R ( \Omega )$ is isomorphic to the algebra $\mathrm T _ \Omega (M)$. We are grateful to Boris Feigin for this remark. It shows that the algebra $\mathrm R ( \Omega )$ is not something exotic.

Consider now the Lie algebra $\mathfrak{gl} (d, \mathrm R ( \Omega ))$, where $d\in\{1,2,3,\dots\}$. As a vector space, it coincides with the associative algebra $\operatorname{Mat} (d,\mathbb{C})\otimes \mathrm R ( \Omega )$, and the Lie bracket is the commutator $[-,-]$ in this algebra, that is,

$$[X\otimes \mathbf{x} ,Y\otimes \mathbf{y} ]=(XY)\otimes ( \mathbf{x} \odot \mathbf{y} )-(YX)\otimes( \mathbf{y} \odot \mathbf{x} ), \quad X,Y\in \operatorname{Mat} (d,\mathbb{C}), \; \mathbf{x} , \mathbf{y} \in \mathrm R ( \Omega ).$$

The grading of $\mathrm R ( \Omega )$ induces, in a natural way, a grading of the Lie algebra $\mathfrak{gl} (d, \mathrm R ( \Omega ))$, from which we obtain a grading of the universal enveloping algebra $\mathcal{U}( \mathfrak{gl} (d, \mathrm R ( \Omega )))$.

Along with the canonical filtration of $Y_d( \Omega )$, which we have dealt with so far, the algebra $Y_d( \Omega )$ also admits another filtration, which we will call the “shifted filtration”: it is defined by the condition that the degree of a generator $t_{ij}( \mathbf{x} ;s)$, where $\mathbf{x} \in \Omega ^{\otimes m}$, is set to $m-1$ rather than to $m$, as before.

Proposition 6.3.

The graded algebra associated with the shifted filtration of $Y_d( \Omega )$ is isomorphic to the algebra $\mathcal{U}( \mathfrak{gl} (d, \mathrm R ( \Omega )))$.