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 $V=\bigoplus _{n\geq 0}V_{(n)}$ be a graded vertex operator algebra with $V_{(0)}=\mathbb {C} \mathbf {1}$ and let $Y(a,z)=\sum _{n\in \mathbb {Z} }a_{n}z^{-n-1}$ 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

$$a*b=\sum _{i\geq 0}{\binom {\operatorname {wt} a}{i}}a_{i-1}b,~~~~~a\circ b=\sum _{i\geq 0}{\binom {\operatorname {wt} a}{i}}a_{i-2}b$$

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 $A(V)=\bigcup _{p\geq 0}A_{p}(V)$ 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 $f$.