Conformal algebras over linear
algebraic groups
Pavel Kolesnikov
Conformal algebras and similar
structures usually appear in the context of operator product expansion (OPE)
[K1, GKK]. In algebra, these structures are related with vertex operator
algebras (see, e.g., [FLM]) and their numerous applications.
In [BDK], a generalization of the
notion of a conformal algebra was proposed in terms of pseudo-tensor categories
associated with a Hopf algebra.
The natural source of Hopf algebras is provided by
linear algebraic groups. Hence, it is natural to expect that conformal algebras
and some of their generalizations have a natural interpretation in terms of
linear algebraic groups.
The purpose of this talk is to
present the corresponding construction explicitly: given a linear algebraic
group $G$ we build a category of $(G)$-conformal
algebras. In particular, if $G$ is the affine line then $(G)$-conformal
algebra is the same as conformal algebra in the sense of the definition in [
The main result presents a
description of irreducible subalgebras of the $(G)$-conformal algebra of translation-invariant
transformations of the space of vector-valued regular functions on $G$. This is
a generalization of the classical Burnside theorem as well as its analogue for
conformal algebras~[K].
[BDK] B.~Bakalov,
A.~D'Andrea, V.G.~Kac,
Theory of finite pseudoalgebras, Adv. Math. 162 (1) (2001).
[FLM] I.B.~Frenkel,
J.~Lepowsky, A.~Meurman,
Vertex operator algebras and the Monster, Pure and Applied Mathematics,
vol.~134, Academic Press,
[GKK] M.I.~Golenishcheva-Kutuzova,
V.G.~Kac, $\Gamma$-conformal algebras, J. Math. Phys. 39 (1998), no. 4, 2290--2305.
[K1] V.G. Kac, Vertex
algebras for beginners, University Lecture Series, vol.~10.
AMS,
[
[K] P.S.~Kolesnikov,
Associative conformal algebras with finite faithful representation, Adv. Math.
202 (2006), no.~2, 602--637.