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. Many important representation theoretic properties of the vertex algebra are logically related to properties of its Zhu algebra or C₂-algebra.

Definitions

Let be a graded vertex operator algebra with and let be the vertex operator associated to . Define to be the subspace spanned by elements of the form for . An element is homogeneous with if . There are two binary operations on defined by , for homogeneous elements and extended linearly to all of . Define to be the span of all elements .

The algebra with the binary operation induced by is an associative algebra called the Zhu algebra of .

The algebra with multiplication is called the C₂-algebra of .

Main properties

• The multiplication of the C₂-algebra is commutative and the additional binary operation is a Poisson bracket on which gives the C₂-algebra the structure of a Poisson algebra.
• (Zhu's C₂-cofiniteness condition) If is finite dimensional then is said to be C₂-cofinite. There are two main representation theoretic properties related to C₂-cofiniteness. A vertex operator algebra 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 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 induces a filtration where so that There is a surjective morphism of Poisson algebras .

Associated variety

Because the C₂-algebra is a commutative algebra it may be studied using the language of algebraic geometry. The associated scheme and associated variety of are defined to be , which are an affine scheme and an affine algebraic variety respectively. Moreover, since acts as a derivation on there is an action of on the associated scheme making ' a conical Poisson scheme and a conical Poisson variety. In this language, C₂-cofiniteness is equivalent to the property that is a point.

Example: If is the affine W-algebra associated to affine Lie algebra at level and nilpotent element then is the Slodowy slice through .