Zhu algebra

In the theory of vertex algebras, the Zhu algebra and the closely related C2-algebra, introduced by Yongchang Zhu in his PhD thesis, are two associative algebras canonically constructed from a given vertex operator algebra. Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or C2-algebra.

Definitions
Let $$V = \bigoplus_{n \ge 0} V_{(n)}$$ be a graded vertex operator algebra with $$V_{(0)} = \mathbb{C}\mathbf{1}$$ and let $$Y(a, z) = \sum_{n \in \Z} a_n z^{-n-1}$$ be the vertex operator associated to $$a \in V. $$ Define $$C_2(V)\subset V$$to be the subspace spanned by elements of the form $$a_{-2} b$$ for $$a,b \in V. $$ An element $$a \in V$$ is homogeneous with $$\operatorname{wt} a = n$$ if $$a \in V_{(n)}.$$ There are two binary operations on $$V$$defined by$$a * b = \sum_{i \ge 0} \binom{\operatorname{wt} a}{i} a_{i-1}b, a \circ b = \sum_{i \ge 0} \binom{\operatorname{wt}a}{i} a_{i-2} b$$for homogeneous elements and extended linearly to all of $$V$$. Define $$O(V)\subset V$$to be the span of all elements $$a\circ b$$.

The algebra $$A(V) := V/O(V)$$ with the binary operation induced by $$*$$ is an associative algebra called the Zhu algebra of $$V$$.

The algebra $$R_V := V/C_2(V)$$ with multiplication $$a\cdot b = a_{-1}b \mod C_2(V)$$ is called the C2-algebra of $$V$$.

Main properties

 * The multiplication of the C2-algebra is commutative and the additional binary operation $$\{a,b\} = a_{0}b\mod C_2(V)$$ is a Poisson bracket on $$R_V$$which gives the C2-algebra the structure of a Poisson algebra.
 * (Zhu's C2-cofiniteness condition) If $$R_V$$is finite dimensional then $$V$$ is said to be C2-cofinite. There are two main representation theoretic properties related to C2-cofiniteness. A vertex operator algebra $$V$$ is rational if the category of admissible modules is semisimple and there are only finitely many irreducibles. It was conjectured that rationality is equivalent to C2-cofiniteness and a stronger condition regularity, however this was disproved in 2007 by Adamovic and Milas who showed that the triplet vertex operator algebra is C2-cofinite but not rational.  Various weaker versions of this conjecture are known, including that regularity implies C2-cofiniteness and that for C2-cofinite $$V$$ the conditions of rationality and regularity are equivalent. This conjecture is a vertex algebras analogue of Cartan's criterion for semisimplicity in the theory of Lie algebras because it relates a structural property of the algebra to the semisimplicity of its representation category.
 * The grading on $$V$$ induces a filtration $$A(V) = \bigcup_{p \ge 0} A_p(V)$$ where $$A_p(V) = \operatorname{im}(\oplus_{j = 0}^p V_p\to A(V))$$so that $$A_p(V) \ast A_q(V) \subset A_{p+q}(V).$$ There is a surjective morphism of Poisson algebras $$R_V \to \operatorname{gr}(A(V))$$.

Associated variety
Because the C2-algebra $$R_V$$ is a commutative algebra it may be studied using the language of algebraic geometry. The associated scheme $$\widetilde{X}_V$$ and associated variety $$X_V$$ of $$V$$ are defined to be $$\widetilde{X}_V := \operatorname{Spec}(R_V), X_V := (\widetilde{X}_V)_{\mathrm{red}}$$which are an affine scheme an affine algebraic variety respectively. Moreover, since $$L(-1)$$ acts as a derivation on $$R_V$$ there is an action of $$\mathbb{C}^\ast$$ on the associated scheme making $$\widetilde{X}_V$$ a conical Poisson scheme and $$X_V$$ a conical Poisson variety. In this language, C2-cofiniteness is equivalent to the property that $$X_V$$ is a point.

Example: If $$W^k(\widehat{\mathfrak g}, f)$$ is the affine W-algebra associated to affine Lie algebra $$\widehat{\mathfrak g}$$ at level $$k$$ and nilpotent element $$f$$ then $$\widetilde{X}_{W^k(\widehat{\mathfrak g}, f)} = \mathcal{S}_f$$is the Slodowy slice through $$f$$.