×

zbMATH — the first resource for mathematics

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

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bakalov, B; Kac, V; Voronov, A, Cohomology of conformal algebras, Comm. math. phys., 200, 561-598, (1999) · Zbl 0959.17018
[2] Belavin, A; Polyakov, A; Zamolodchikov, A, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear phys. B, 241, 2, 333-380, (1984) · Zbl 0661.17013
[3] Bloch, S, Zeta values and differential operators on the circle, J. algebra, 182, 476-500, (1996) · Zbl 0868.17017
[4] Borcherds, R, Vertex algebras, kac – moody algebras, and the monster, Proc. natl. acad. sci. USA, 83, 3068-3071, (1986) · Zbl 0613.17012
[5] Cheng, S; Kac, V, Conformal modules, Asian J. math., 1, 1, 181-193, (1997) · Zbl 1022.17018
[6] Cheng, S; Kac, V; Wakimoto, M, Extensions of conformal modules, (), 33-57
[7] D’Andrea, A; Kac, V, Structure theory of finite conformal algebras, Selecta math., 4, 3, 377-418, (1998) · Zbl 0918.17019
[8] De Sole, A; Kac, V, Subalgebras of gc_N and Jacobi polynomials, Canad. math. bull., 45, 4, 567-605, (2002) · Zbl 1037.17025
[9] Djokovic, D.Z, Hermitian matrices over polynomial rings, J. algebra, 43, 2, 359-374, (1976) · Zbl 0343.15006
[10] D.Z. Djokovic, Private communication
[11] Kac, V, Vertex algebras for beginners, (1998), Amer. Math. Soc Providence, RI · Zbl 0924.17023
[12] Kac, V, Formal distributions algebras and conformal algebras, (), 80-97 · Zbl 1253.17002
[13] Kac, V; Radul, A, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. math. phys., 157, 429-457, (1993) · Zbl 0826.17027
[14] Kac, V; Wang, W; Yan, C, Quasifinite representations of classical Lie subalgebras of W1+∞, Adv. math., 139, 56-140, (1998) · Zbl 0938.17018
[15] Knus, M, Quadratic and Hermitian forms over rings, (1991), Springer Berlin · Zbl 0756.11008
[16] Retakh, A, Associative conformal algebras of linear growth, J. algebra, 237, 169-788, (2001) · Zbl 1151.17312
[17] Retakh, A, Unital associative pseudoalgebras and their representations · Zbl 1059.17016
[18] Zelmanov, E, On the structure of conformal algebras, (), 139-153 · Zbl 1039.17031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.