Zhu algebra

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

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

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

Main properties

• The multiplication of the C₂-algebra is commutative and the additional binary operation [math]\displaystyle{ \{a,b\} = a_{0}b\mod C_2(V) }[/math] is a Poisson bracket on [math]\displaystyle{ R_V }[/math]which gives the C₂-algebra the structure of a Poisson algebra.
• (Zhu's C₂-cofiniteness condition) If [math]\displaystyle{ R_V }[/math]is finite dimensional then [math]\displaystyle{ V }[/math] is said to be C₂-cofinite. There are two main representation theoretic properties related to C₂-cofiniteness. A vertex operator algebra [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ V }[/math] 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 [math]\displaystyle{ V }[/math] induces a filtration [math]\displaystyle{ A(V) = \bigcup_{p \ge 0} A_p(V) }[/math] where [math]\displaystyle{ A_p(V) = \operatorname{im}(\oplus_{j = 0}^p V_p\to A(V)) }[/math]so that [math]\displaystyle{ A_p(V) \ast A_q(V) \subset A_{p+q}(V). }[/math] There is a surjective morphism of Poisson algebras [math]\displaystyle{ R_V \to \operatorname{gr}(A(V)) }[/math].

Associated variety

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

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