Zhu algebra explained

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

V = o plus _{n} V _{(n)} be a graded vertex operator algebra with V _{(0)} = C 1 and let Y(a, z) = \sum _{n} 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$$a\; *\; b\; =\; \backslash sum\_\; \backslash binom\; a\_b,\; ~~~~~\; a\; \backslash circ\; b\; =\; \backslash sum\_\; \backslash binom\; a\_\; b$$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 .^[1]

The algebra

R _{V} : = V/C _{2(V)} with multiplication a ⋅ 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. ^{[2] [3] [4]} 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.^[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

V induces a filtration A(V) = c up _{p} A _{p(V)} where A _{p(V)} = \operatorname{im}( ⊕ ^{p} _{j =0} 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)) .^[6]

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 $$\backslash widetilde\_V\; :=\; \backslash operatorname(R\_V),\; ~~~\; X\_V\; :=\; (\backslash widetilde\_V)\_$$which are an affine scheme an affine algebraic variety respectively. ^[7] Moreover, since L( -1 ) acts as a derivation on R _{V} there is an action of 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{akg},} f) is the affine W-algebra associated to affine Lie algebra \widehat{ ak g} at level k and nilpotent element f then \widetilde{X} _{W ^{k(\widehat{akg}} , f)} = l{S} _{f} is the Slodowy slice through f .^[8]

Notes and References

1. Zhu . Yongchang . 1996 . Modular invariance of characters of vertex operator algebras . Journal of the American Mathematical Society . 9 . 1 . 237–302 . 10.1090/s0894-0347-96-00182-8 . free . 0894-0347.
2. Li . Haisheng . 1999 . Some Finiteness Properties of Regular Vertex Operator Algebras . Journal of Algebra . 212 . 2 . 495–514 . 10.1006/jabr.1998.7654 . free . 16072357 . 0021-8693. math/9807077 .
3. Dong . Chongying . Li . Haisheng . Mason . Geoffrey . 1997 . Regularity of Rational Vertex Operator Algebras . . 132 . 1 . 148–166 . 10.1006/aima.1997.1681 . free . 14942843 . 0001-8708. q-alg/9508018 .
4. Adamović . Dražen . Milas . Antun . 2008-04-01 . On the triplet vertex algebra W(p) . Advances in Mathematics . 217 . 6 . 2664–2699 . 10.1016/j.aim.2007.11.012 . 0001-8708. free .
5. Abe . Toshiyuki . Buhl . Geoffrey . Dong . Chongying . 2003-12-15 . Rationality, regularity, and ₂-cofiniteness . Transactions of the American Mathematical Society . 356 . 8 . 3391–3402 . 10.1090/s0002-9947-03-03413-5 . free . 0002-9947.
6. Arakawa . Tomoyuki . Lam . Ching Hung . Yamada . Hiromichi . 2014 . Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras . . 264 . 261–295 . 10.1016/j.aim.2014.07.021 . 119121685 . free . 0001-8708.
7. Arakawa . Tomoyuki . 2010-11-20 . A remark on the C 2-cofiniteness condition on vertex algebras . Mathematische Zeitschrift . 270 . 1–2 . 559–575 . 10.1007/s00209-010-0812-4 . 253711685 . 0025-5874. 1004.1492 .
8. Arakawa . T. . 2015-02-19 . Associated Varieties of Modules Over Kac-Moody Algebras and C2-Cofiniteness of W-Algebras . International Mathematics Research Notices . 10.1093/imrn/rnu277 . 1073-7928. 1004.1554 .