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.

Definitions

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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  .[1]

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

Main properties

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  • 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.[1]
  • (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. [2][3][4] Various weaker versions of this conjecture are known, including that regularity implies C2-cofiniteness[2] and that for C2-cofinite   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   induces a filtration   where  so that   There is a surjective morphism of Poisson algebras  .[6]

Associated variety

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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 an affine algebraic variety respectively. [7] Moreover, since   acts as a derivation on  [1] 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  .[8]

References

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  1. ^ a b c d Zhu, Yongchang (1996). "Modular invariance of characters of vertex operator algebras". Journal of the American Mathematical Society. 9 (1): 237–302. doi:10.1090/s0894-0347-96-00182-8. ISSN 0894-0347.
  2. ^ a b Li, Haisheng (1999). "Some Finiteness Properties of Regular Vertex Operator Algebras". Journal of Algebra. 212 (2): 495–514. arXiv:math/9807077. doi:10.1006/jabr.1998.7654. ISSN 0021-8693. S2CID 16072357.
  3. ^ Dong, Chongying; Li, Haisheng; Mason, Geoffrey (1997). "Regularity of Rational Vertex Operator Algebras". Advances in Mathematics. 132 (1): 148–166. arXiv:q-alg/9508018. doi:10.1006/aima.1997.1681. ISSN 0001-8708. S2CID 14942843.
  4. ^ Adamović, Dražen; Milas, Antun (2008-04-01). "On the triplet vertex algebra W(p)". Advances in Mathematics. 217 (6): 2664–2699. doi:10.1016/j.aim.2007.11.012. ISSN 0001-8708.
  5. ^ Abe, Toshiyuki; Buhl, Geoffrey; Dong, Chongying (2003-12-15). "Rationality, regularity, and 𝐶₂-cofiniteness". Transactions of the American Mathematical Society. 356 (8): 3391–3402. doi:10.1090/s0002-9947-03-03413-5. ISSN 0002-9947.
  6. ^ Arakawa, Tomoyuki; Lam, Ching Hung; Yamada, Hiromichi (2014). "Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras". Advances in Mathematics. 264: 261–295. doi:10.1016/j.aim.2014.07.021. ISSN 0001-8708. S2CID 119121685.
  7. ^ Arakawa, Tomoyuki (2010-11-20). "A remark on the C 2-cofiniteness condition on vertex algebras". Mathematische Zeitschrift. 270 (1–2): 559–575. arXiv:1004.1492. doi:10.1007/s00209-010-0812-4. ISSN 0025-5874. S2CID 253711685.
  8. ^ Arakawa, T. (2015-02-19). "Associated Varieties of Modules Over Kac-Moody Algebras and C2-Cofiniteness of W-Algebras". International Mathematics Research Notices. arXiv:1004.1554. doi:10.1093/imrn/rnu277. ISSN 1073-7928.