Zhu algebra

In mathematics, 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.^[1] Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or C₂-algebra.

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

The algebra ${\displaystyle A(V):=V/O(V)}$ with the binary operation induced by ${\displaystyle *}$ is an associative algebra called the Zhu algebra of ${\displaystyle V}$ ^[1].

The algebra ${\displaystyle R_{V}:=V/C_{2}(V)}$ with multiplication ${\displaystyle a\cdot b=a_{-1}b\mod C_{2}(V)}$ is called the C₂-algebra of ${\displaystyle V}$.

• The multiplication of the C₂-algebra is commutative and the additional binary operation ${\displaystyle \{a,b\}=a_{0}b\mod C_{2}(V)}$ is a Poisson bracket on ${\displaystyle R_{V}}$which gives the C₂-algebra the structure of a Poisson algebra.^[1]
• (Zhu's C₂-cofiniteness condition) If ${\displaystyle R_{V}}$is finite dimensional then ${\displaystyle V}$ is said to be C₂-cofinite. There are two main representation theoretic properties related to C₂-cofiniteness. A vertex operator algebra ${\displaystyle V}$ 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. ^[2][3][4] Various weaker versions of this conjecture are known, including that regularity implies C₂-cofiniteness^[2] and that for C₂-cofinite ${\displaystyle V}$ the conditions of rationality and regularity are equivalent.^[5] 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 ${\displaystyle V}$ induces a filtration ${\displaystyle A(V)=\bigcup _{p\geq 0}A_{p}(V)}$ where ${\displaystyle A_{p}(V)=\operatorname {im} (\oplus _{j=0}^{p}V_{p}\to A(V))}$ so that ${\displaystyle A_{p}(V)\ast A_{q}(V)\subset A_{p+q}(V).}$ There is a surjective morphism of Poisson algebras ${\displaystyle R_{V}\to \operatorname {gr} (A(V))}$.^[6]

Because the C₂-algebra ${\displaystyle R_{V}}$ is a commutative algebra it may be studied using the language of algebraic geometry. The associated scheme ${\displaystyle {\widetilde {X}}_{V}}$ and associated variety ${\displaystyle X_{V}}$ of ${\displaystyle V}$ are defined to be $${\displaystyle {\widetilde {X}}_{V}:=\operatorname {Spec} (R_{V})}$$, $${\displaystyle X_{V}:=({\widetilde {X}}_{V})_{\mathrm {red} }}$$ which are an affine scheme and an affine algebraic variety respectively. ^[7] Moreover, since ${\displaystyle L(-1)}$ acts as a derivation on ${\displaystyle R_{V}}$^[1] there is an action of ${\displaystyle \mathbb {C} ^{\ast }}$ on the associated scheme making ${\displaystyle {\widetilde {X}}_{V}}$ a conical Poisson scheme and ${\displaystyle X_{V}}$ a conical Poisson variety. In this language, C₂-cofiniteness is equivalent to the property that ${\displaystyle X_{V}}$ is a point.

Example: If ${\displaystyle W^{k}({\widehat {\mathfrak {g}}},f)}$ is the affine W-algebra associated to affine Lie algebra ${\displaystyle {\widehat {\mathfrak {g}}}}$ at level ${\displaystyle k}$ and nilpotent element ${\displaystyle f}$ then ${\displaystyle {\widetilde {X}}_{W^{k}({\widehat {\mathfrak {g}}},f)}={\mathcal {S}}_{f}}$is the Slodowy slice through ${\displaystyle f}$.^[8]