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 $V = \bigoplus_{n \ge 0} V_{(n)}$ be a graded vertex operator algebra with $V_{(0)} = \mathbb{C}\mathbf{1}$ and let $Y(a, z) = \sum_{n \in \Z} a_n z^{-n-1}$ 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 $a * b = \sum_{i \ge 0} \binom{\operatorname{wt} a}{i} a_{i-1}b, ~~~~~ a \circ b = \sum_{i \ge 0} \binom{\operatorname{wt}a}{i} a_{i-2} b$ 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 $V$ .{{Cite journal |last=Zhu |first=Yongchang |date=1996 |title=Modular invariance of characters of vertex operator algebras |journal=Journal of the American Mathematical Society |volume=9 |issue=1 |pages=237–302 |doi=10.1090/s0894-0347-96-00182-8 |doi-access=free |issn=0894-0347}}

The algebra $R_V := V/C_2(V)$ with multiplication $a\cdot b = a_{-1}b \mod C_2(V)$ 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. {{Cite journal |last=Li |first=Haisheng |date=1999 |title=Some Finiteness Properties of Regular Vertex Operator Algebras |journal=Journal of Algebra |volume=212 |issue=2 |pages=495–514 |doi=10.1006/jabr.1998.7654 |doi-access=free |s2cid=16072357 |issn=0021-8693|arxiv=math/9807077 }}{{Cite journal |last1=Dong |first1=Chongying |last2=Li |first2=Haisheng |last3=Mason |first3=Geoffrey |date=1997 |title=Regularity of Rational Vertex Operator Algebras |journal=Advances in Mathematics |volume=132 |issue=1 |pages=148–166 |doi=10.1006/aima.1997.1681 |doi-access=free |s2cid=14942843 |issn=0001-8708|arxiv=q-alg/9508018 }}{{Cite journal |last1=Adamović |first1=Dražen |last2=Milas |first2=Antun |date=2008-04-01 |title=On the triplet vertex algebra W(p) |journal=Advances in Mathematics |volume=217 |issue=6 |pages=2664–2699 |doi=10.1016/j.aim.2007.11.012 |issn=0001-8708|doi-access=free }} 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.{{Cite journal |last1=Abe |first1=Toshiyuki |last2=Buhl |first2=Geoffrey |last3=Dong |first3=Chongying |date=2003-12-15 |title=Rationality, regularity, and 𝐶₂-cofiniteness |journal=Transactions of the American Mathematical Society |volume=356 |issue=8 |pages=3391–3402 |doi=10.1090/s0002-9947-03-03413-5 |doi-access=free |issn=0002-9947}} 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 $A(V) = \bigcup_{p \ge 0} A_p(V)$ where $A_p(V) = \operatorname{im}(\oplus_{j = 0}^p V_p\to A(V))$ so that $A_p(V) \ast A_q(V) \subset A_{p+q}(V).$ There is a surjective morphism of Poisson algebras $R_V \to \operatorname{gr}(A(V))$ .{{Cite journal |last1=Arakawa |first1=Tomoyuki |last2=Lam |first2=Ching Hung |last3=Yamada |first3=Hiromichi |date=2014 |title=Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras |journal=Advances in Mathematics |volume=264 |pages=261–295 |doi=10.1016/j.aim.2014.07.021 |s2cid=119121685 |doi-access=free |issn=0001-8708}}

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 $\widetilde{X}_V := \operatorname{Spec}(R_V), ~~~ X_V := (\widetilde{X}_V)_{\mathrm{red}}$ which are an affine scheme and an affine algebraic variety respectively. {{Cite journal |last=Arakawa |first=Tomoyuki |date=2010-11-20 |title=A remark on the C 2-cofiniteness condition on vertex algebras |url=http://dx.doi.org/10.1007/s00209-010-0812-4 |journal=Mathematische Zeitschrift |volume=270 |issue=1–2 |pages=559–575 |doi=10.1007/s00209-010-0812-4 |s2cid=253711685 |issn=0025-5874|arxiv=1004.1492 }} Moreover, since $L(-1)$ acts as a derivation on $R_V$ there is an action of $\mathbb{C}^\ast$ 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 $W^k(\widehat{\mathfrak g}, f)$ is the affine W-algebra associated to affine Lie algebra $\widehat{\mathfrak g}$ at level $k$ and nilpotent element $f$ then $\widetilde{X}_{W^k(\widehat{\mathfrak g}, f)} = \mathcal{S}_f$ is the Slodowy slice through $f$ .{{Cite journal |last=Arakawa |first=T. |date=2015-02-19 |title=Associated Varieties of Modules Over Kac-Moody Algebras and C2-Cofiniteness of W-Algebras |url=http://dx.doi.org/10.1093/imrn/rnu277 |journal=International Mathematics Research Notices |doi=10.1093/imrn/rnu277 |issn=1073-7928|arxiv=1004.1554 }}

References

Category:Algebras

Category:Algebraic geometry