On the classification of subalgebras of \(\text{Cend}_N\) and \(\text{gc}_N\). (English) Zbl 1034.17018

Boyallian, Carina; Kac, Victor G.; Liberati, Jose I.

J. Algebra 260, No. 1, 32-63 (2003).

The authors classify all infinite subalgebras of the conformal algebras \(\text{Cend}_{N}\) and \(\text{gc}_{N}\) over the field \(\mathbb C\) (these algebras are direct ‘conformal’ analogues of the matrix algebras \(M_N(\mathbb C)\) and \(\text{gl}_N (\mathbb C)\), respectively). For the definition of conformal algebra, see V. G. Kac, Vertex algebras for beginners. 2nd ed. University Lecture Series. 10. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0924.17023). They describe all finite irreducible modules over \(\text{Cend}_{N,P}\). The authors also describe all automorphisms of \(\text{Cend}_{N,P}\) and classify all homomorphisms and anti-homomorphisms of \(\text{Cend}_{N,P}\) to \(\text{Cend}_{N}\) .

Reviewer: Yong-Qing Chen (Shenzhen)

MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B68	Virasoro and related algebras
17B81	Applications of Lie (super)algebras to physics, etc.
81R10	Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations

Keywords:

conformal algebras; classification; automorphisms; analogue of Burnside theorem

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