Standard bases for the universal associative conformal envelopes of Kac–Moody conformal algebras

Conformal algebras also known as Lie vertex algebras were introduced in [16] as an algebraic tool to study the singular part of the operator product expansion (OPE) of chiral fields in 2-dimensional conformal field theory coming back to [5]. From the categorical point of view, a conformal algebra is just an algebra in the appropriate (pseudo-tensor) category $\mathcal{M}^{*}(\mathbb{C}[\partial])$ of modules over the polynomial algebra $\mathbb{C}[\partial]$ in one variable [2]. The pseudo-tensor structure (see [4]) reflects the main features of multi-linear maps in the category of linear spaces: composition, identity, symmetric structure. These features are enough to define the basic notions like what is an algebra (associative, commutative, Lie, etc.), homomorphism, ideal, representation, module, cohomology. Therefore, the notion of a conformal algebra is a natural expansion of the notion of an “ordinary” algebra over $\mathbb{C}$ to the pseudo-tensor category $\mathcal{M}^{*}(\mathbb{C}[\partial])$. Namely, as an ordinary algebra is a linear space equipped with a bilinear product, a conformal algebra is a $\mathbb{C}[\partial]$-module $V$ equipped with a $\mathbb{C}[\partial]$-bilinear map (pseudo-product)

A more convenient presentation for the operation $*$ uses the language of a $\lambda$-product or a family of $n$-products for all integer $n\geq 0$ ([16], see also Section 2).

Conformal algebras representing the singular part of OPE in vertex algebras are Lie algebras in the category $\mathcal{M}^{*}(\mathbb{C}[\partial])$, i.e., Lie conformal algebras. For example, if $\mathfrak{g}$ is a Lie algebra then the free module $\mathbb{C}[\partial]\otimes\mathfrak{g}$ equipped with the pseudo-product $a*b=(1\otimes 1)\otimes_{\mathbb{C}[\partial]}[a,b]$, $a,b\in\mathfrak{g}$, is a Lie conformal algebra denoted $\mathop{Cur}\nolimits\mathfrak{g}$ (current conformal algebra). If $\langle\cdot|\cdot\rangle$ is a bilinear symmetric invariant form on $\mathfrak{g}$ then $\mathop{Cur}\nolimits\mathfrak{g}$ has a 1-dimensional central extension $K(\mathfrak{g})$ defined by

where $e$ is a central element and $\partial e=0$. For example, in the Kac–Moody vertex algebra $V(\mathfrak{g})$ [12] the singular part of the OPE on the generating fields is described by this particular structure $K(\mathfrak{g})$ called a Kac–Moody conformal algebra.

As in the case of ordinary algebras, an associative conformal algebra $C$ turns into a Lie one with respect to the commutator $[a*b]=(a*b)-(\tau\otimes_{\mathbb{C}[\partial]}1)(b*a)$, $a,b\in C$, where $\tau$ is the switching map on $\mathbb{C}[\partial]^{\otimes 2}$. However, not all Lie conformal algebras embeds into associative ones in this way [26]. This is an open problem whether every finite (i.e., finitely generated as a $\mathbb{C}[\partial]$-module) Lie conformal algebra embeds into an associative conformal algebra with respect to conformal commutator. Even for the class of quadratic conformal algebras [27] (see also [15]) it remains unknown in general if every such Lie conformal algebra embeds into an appropriate associative one.

A routine way to solve this kind of problems is to construct a universal envelope. In general, such an algebra is defined by generators and relations. For a Lie conformal algebra, there exists a lattice of universal enveloping associative conformal algebras, each related to an (associative) locality bound on the generators ([26], see also Section 3.4). In order to prove (or disprove) the embedding of a Lie conformal algebra into its universal enveloping associative conformal algebra one needs to know the normal form of elements in the last algebra.

A general and powerful method for finding normal forms in an algebra defined by generators and relations is to calculate a standard (or Gröbner–Shirshov) basis of defining relations. The idea goes back to Newmann’s Diamond Lemma [23], see also [7, 6]. In the recent years, the Gröbner–Shirshov bases theory was developed to serve the problem of combinatorial analysis of various algebraic structures, see [9]. For associative conformal algebras it was initially invented in [8], later developed in [24] and [20]. In this paper, we use the last approach exposed in a form convenient for actual computation: we consider defining relations in a conformal algebra as rewriting rules on a module over an appropriate associative algebra (the Gröbner–Shirshov basis of the last algebra is known).

A series of particular observations made in [21], [22] shows that for all considered examples of quadratic Lie conformal algebras $L$ it is enough to consider universal associative conformal envelopes $U$ relative to the locality bound $N=3$ to get an injective mapping $L\curvearrowright U$. This is one of the reasons why we focus on the locality bound $N=3$ for the envelopes of current conformal algebras as they are particular examples of quadratic conformal algebras.

The main purpose of this paper is to find a standard (Gröbner–Shirshov) basis of defining relations for the universal enveloping associative conformal algebra of a Kac–Moody conformal algebra at locality level $N=3$. As a corollary, we get an analogue of the Poincaré–Birkhoff–Witt Theorem (PBW-Theorem) stating that the associated graded conformal algebra obtained from the universal envelope of a current Lie conformal algebra with respect to the natural filtration is isomorphic to the free commutative conformal algebra. Note that the classical PBW-Theorem may be interpreted as a conformal one at the locality level $N=1$: for a Lie algebra $\mathfrak{g}$, its “ordinary” universal envelope $U(\mathfrak{g})$ gives rise to the conformal algebra $\mathop{Cur}\nolimits U(\mathfrak{g})_{0}$ which is exactly the universal enveloping conformal algebra of $\mathop{Cur}\nolimits\mathfrak{g}$ with $N=1$ (here $U(\mathfrak{g})_{0}$ is the augmentation ideal of $U(\mathfrak{g})$).

There are several reasons for studying the universal envelopes of $\mathop{Cur}\nolimits\mathfrak{g}$ at higher locality than $N=1$.

First, the $N=1$ envelope $\mathop{Cur}\nolimits U(\mathfrak{g})$ does not reflect the homological properties of $\mathop{Cur}\nolimits\mathfrak{g}$. For example, if $\mathfrak{g}$ is a simple finite-dimensional Lie algebra then the second cohomology group $H^{2}(\mathop{Cur}\nolimits\mathfrak{g},\Bbbk)$ is one-dimensional [3]. The corresponding central extension is the Kac–Moody conformal algebra $K(\mathfrak{g})$ representing the singular part of the Kac–Moody vertex algebra [12]. On the other hand, it is easy to find that the second Hochschild cohomology group of $\mathop{Cur}\nolimits U(\mathfrak{g})_{0}$ with coefficients in the trivial 1-dimensional module is zero: there are no nontrivial central extensions. Our results show that the universal enveloping associative conformal algebras for $\mathop{Cur}\nolimits\mathfrak{g}$ at locality level $N=2,3$ do have a nontrivial central extension which is exactly the universal envelope of $K(\mathfrak{g})$.

The second reason is related with Poisson algebras. Assume $P$ is an ordinary commutative algebra with a Poisson bracket $\{\cdot,\cdot\}$. Then $\mathop{Cur}\nolimits P$ may be considered as a Lie conformal algebra since $P$ is a Lie algebra relative to the Poisson bracket. There is a conformal representation of $\mathop{Cur}\nolimits P$ on itself given by the rule

The study of this representation provides a way to get new results in (quadratic) conformal algebras as well as in Poisson algebras [21, 22]. The conformal linear operators $\rho(a)\in\mathop{Cend}\nolimits(\mathop{Cur}\nolimits P)$, $\rho(a)_{\lambda}:f\mapsto(a\mathrel{{}_{(\lambda)}}f)$, are local to each other, and the locality bound is $N=3$. Indeed, according to the definition of a conformal representation [11, 16] we have

for $a,b,f\in P$. If $abf\neq 0$ then right-hand side is of degree 2 in $\lambda$ that means $N(\rho(a),\rho(b))=3$ in $\mathop{Cend}\nolimits(\mathop{Cur}\nolimits P)$. Therefore, the corresponding associative envelope belongs to the class of envelopes with locality $N=3$.

The third reason to study the case $N=3$ comes from the following relation between commutative conformal and Novikov algebras. Suppose $C$ is a commutative conformal algebra and $M$ is a subset of $C$ such that $N_{C}(a,b)\leq 3$ for all $a,b\in M$. Then $M$ generates an ordinary (nonassociative) subalgebra $N(M)$ in the space $C$ considered relative to the single product $x\circ y=x\mathrel{{}_{(1)}}y$. Indeed, all elements of $N(M)$ are local to each other with locality bound $3$. Moreover, the following relations hold:

for all $x,y,z\in N(M)$. These identities are known to define the variety of Novikov algebras initially appeared in [1], [13]. In order to perform a systematic study of this relation, one needs to know the structure of the universal object in the category of commutative conformal algebras with locality bound $N=3$ on the generators.

For all these reasons, we study the universal enveloping associative conformal algebras for Kac–Moody conformal algebras $K(\mathfrak{g})$ relative to the locality level $N=3$. The corollaries of the main result of the paper (Theorem 2) allow us to get the structure of the universal envelopes for current Lie conformal conformal algebras at $N=3$ and also describe the free commutative conformal algebra at the same locality level. Practically, we find a standard (Gröbner–Shirshov) basis of defining relations for these conformal algebras and derive an analogue of the PBW-Theorem.

The definition of a conformal algebra as an algebra in an appropriate pseudo-tensor category [2] corresponds to the convenient algebraic approach using $\lambda$-brackets [16] if it is presented in terms of operads associated with linear algebraic groups [18].

Let $G$ be a linear algebraic group over a field $\Bbbk$ of characteristic zero, and let $H_{G}=\Bbbk[G]$ be the Hopf algebra of regular functions on $G$. For every $H_{G}$-module $V$ there is a non-symmetric operad (let us denote it $V_{G}$) defined as follows. Given $n\in\{1,2,\dots\}$, set

The condition of regularity means that $f$ may be presented by a polynomial function with coefficients in the space $\mathop{Hom}\nolimits(V^{\otimes n},V)$ of $\Bbbk$-polylinear maps on $V$, and the 3/2-linearity (sesqui-linearity) may be expressed as

for $v=f(\lambda_{1},\dots,\lambda_{n-1})(v_{1},\dots,v_{n})$, $\lambda_{i}\in G$, $v_{i}\in V$, $h(x)\in H_{G}$ (here $x$ is a variable ranging in $G$). In particular, $V_{G}(1)$ is the space of all $H_{G}$-linear transformations of $V$ thus contains the identity map $\mathop{id}\nolimits$.

The composition rule in $V_{G}$ is defined by the following partial composition. If $f\in V_{G}(n)$, $g\in V_{G}(m)$, $i\in\{1,\dots,n\}$ then

acts as

for $\lambda_{i},\mu_{j}\in G$, where $\circ_{i}$ in the right-hand side stands for the ordinary partial composition of polylinear maps. In particular, for $i=n$ the partial composition is equal to

It is easy to see that the resulting maps are indeed regular and 3/2-linear.

One may easily check that the partial composition in $V_{G}$ defined above meets the sequental, parallel, and unit axioms [10, Definition 3.2.2.3] and thus this is indeed an non-symmetric operad with a well-defined composition rule

Suppose the group $G$ is abelian. Then $V_{G}(n)$ has a natural action of the symmetric group $S_{n}$ defined in the following way. If $f\in V_{G}(n)$ and $(1i)$, $i=2,\dots,n$, is a transposition in $S_{n}$ then

for $i<n$ (here the action of $(1i)$ in the right-hand side is just the permutation of arguments in a polylinear map). For $i=n$, the definition is slightly more complicated: if $f$ is presented by a polynomial function

then $f^{(1n)}(\lambda_{1},\dots,\lambda_{n-1})$ is given by

where the each $f^{\prime}_{i}(x)\in H_{G}$ is the regular function $f_{i}((\lambda_{1}\dots\lambda_{n-1}x)^{-1},\lambda_{2},\dots,\lambda_{n-1})$.

The composition rule $\gamma^{r}_{m_{1},\dots,m_{r}}$ is equivariant (see, e.g., [10, Definition 5.2.1.1]) since the structure obtained is equivalent to the structure of an $H_{G}$-module operad defined over a cocommutative Hopf algebra [2]. Namely, one may identify a map

with

Recall that if $\mathcal{O}$ is a (symmetric) operad then a morphism $\mathcal{O}\to V_{G}$ defines an algebraic structure on the space $V$. In the trivial case $G=\{e\}$, $H_{G}=\Bbbk$, the operad $V_{G}$ coincides with the operad of polylinear maps on the linear space $V$, and thus a morphism $\mathcal{O}\to V_{G}$ defines the ordinary notion of an $\mathcal{O}$-algebra over $\Bbbk$, a space equipped with a family of polylinear operations.

The classes of conformal [16] and $\mathbb{Z}$-conformal [14] algebras naturally appear in the next step, if we choose $G$ to be a connected linear algebraic group of dimension 1 (the affine line and $GL_{1}$, respectively). For a non-connected group $G$ with the identity component denoted $G^{0}$, the structure of a conformal algebra over $G$ is naturally interpreted as a $G/G^{0}$-graded conformal algebra over $G^{0}$ [19]. If $G=\mathbb{A}_{1}=(\Bbbk,+)$, $H_{G}=\Bbbk[\partial]$ (the variable is traditionally denoted by $\partial$), then a morphism $\mathcal{O}\to V_{G}$ defines a $\mathcal{O}$-conformal algebra structure on a $\Bbbk[\partial]$-module $V$.

For example, if $\mathcal{O}=\mathrm{As}$ is the operad governing the variety of associative algebras (generated by $\mu=x_{1}x_{2}\in\mathrm{As}(2)$ modulo the relation $\mu\circ_{1}\mu=\mu\circ_{2}\mu$) then an associative conformal algebra structure on a $\Bbbk[\partial]$-module $V$ is given by an image of $\mu$, a map $f=(\cdot\mathrel{{}_{(\lambda)}}\cdot):V\otimes V\to\Bbbk[\lambda]\otimes V$ which is 3/2-linear

for $u,v\in V$, and associative in the sense that

By (1), the latter means

(to compute the right-hand side, put $\mu_{1}=\lambda$ and $\lambda_{1}=\mu$ in (1)).

According to the same scheme, a Lie conformal algebra structure on a $\Bbbk[\partial]$-module $V$ is a morphism from the operad $\mathrm{Lie}$ governing the variety of Lie algebras to $V_{\mathbb{A}_{1}}$. To define such a morphism, it is enough to fix a 3/2-linear map $\mu\in V_{\mathbb{A}_{1}}(2)$

such that $\mu^{(12)}=-\mu$ and $(\mu\circ_{2}\mu)-(\mu\circ_{2}\mu)^{(12)}=(\mu\circ_{1}\mu)$. The last two relations represent anti-commutativity and Jacobi identity, respectively:

$u,v,w\in V$.

In the sequel, we will use the notation $(\cdot\mathrel{{}_{(\lambda)}}\cdot)$ for the operation on an associative conformal algebra and $[\cdot\mathrel{{}_{(\lambda)}}\cdot]$ for Lie conformal algebras.

Since there is a morphism of operads $(-):\mathrm{Lie}\to\mathrm{As}$ sending $\mu$ to $f-f^{(12)}$, every associative conformal algebra turns into a Lie conformal algebra relative to the operation

For an associative conformal algebra $V$ defined via a morphism of operads $\mathrm{As}\to V_{\mathbb{A}_{1}}$, let $V^{(-)}$ stand for the Lie conformal algebra obtained as a composition $\mathrm{Lie}\overset{(-)}{\to}\mathrm{As}\to V_{\mathbb{A}_{1}}$.

The property of a commutator to be a derivation on an associative algebra may also be expressed as a relation in $\mathrm{As}(3)$. Being translated to conformal algebras it turns into the following identity on an associative conformal algebra $V$:

As in the case of ordinary algebras, $V\mapsto V^{(-)}$ is a functor from the category of associative conformal algebras to the category of Lie algebras. In contrast to the case of ordinary algebras, this functor does not have a left adjoint one when considered on the entire category of associative conformal algebras. However, if we restrict the class of associative conformal algebras by means of locality on the generators ([26], see Section 3.4 for details) then there is an analogue of the universal enveloping associative algebra for Lie conformal algebras.

In terms of “ordinary” algebraic operations, a conformal algebra is a linear space $V$ equipped with a linear operator $\partial$, the generator of $H_{\mathbb{A}_{1}}=\Bbbk[\partial]$, and a series of bilinear operations $(\cdot\mathrel{{}_{(n)}}\cdot)$, $n\in\mathbb{Z}_{+}$, given by

These operations are called $n$-products. They have to satisfy the following properties:

• (C1)

  For every $u,v\in V$ there exists $N=N(u,v)$ such that $(u\mathrel{{}_{(n)}}v)=0$ for all $n\geq N$;

• (C2)

  $(\partial u\mathrel{{}_{(n)}}v)=-n(u\mathrel{{}_{(n-1)}}v)$;

• (C3)

  $(u\mathrel{{}_{(n)}}\partial v)=\partial(u\mathrel{{}_{(n)}}v)+n(u\mathrel{{}_{(n-1)}}v)$.

The property (C1) is known as the locality axiom, (C2) and (C3) represent 3/2-linearity. For every conformal algebra $V$, the locality function $N_{V}$ is a map $V\times V\to\mathbb{Z}_{+}$ such that $u\mathrel{{}_{(n)}}v=0$ for every $u,v\in V$ and $n\geq N_{V}(u,v)$.

A conformal algebra $V$ is associative if

for all $u,v,w\in V$ and $n,m\in\mathbb{Z}_{+}$. In a similar way, one may rewrite the identities defining the class of Lie conformal algebras.

Given a set $X$ and a function $N:X\times X\to\mathbb{Z}_{+}$, there exists a unique (up to isomorphism) associative conformal algebra denoted $\mathop{Conf}\nolimits(X,N)$ which is universal among all associative conformal algebras $V$ generated by $X$ such that $N_{V}(x,y)\leq N(x,y)$ for all $x,y\in X$ [25]. The details of the construction of $\mathop{Conf}\nolimits(X,N)$ are stated in Section 3.3.