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Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras
Author(s) -
Kang SeokJin,
Kwon JaeHoon
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012661
Subject(s) - mathematics , pure mathematics , lie superalgebra , simple lie group , abelian group , representation of a lie group , automorphism , lie group , algebra over a field , affine lie algebra , current algebra
Let Γ be a countable abelian semigroup and A be a countable abelian group satisfying a certain finiteness condition. Suppose that a group G acts on a Γ × A‐graded Lie superalgebra L = ⊕ (α, a ) Γ × A L (α, a ) by Lie superalgebra automorphisms preserving the Γ × A‐gradation. In this paper, we show that the Euler–Poincaré principle yields the generalized denominator identity for L and derive a closed form formula for the supertraces str( g ∣ L (α, a ) for all g ∈ G , where (α, a ) ∈ Γ × A. We discuss the applications of our supertrace formula to various classes of infinite‐dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac–Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible GL( n ) × GL( k )‐modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and irreducible highest‐weight modules over a generalized Kac–Moody superalgebra g corresponding to the Dynkin diagram automorphism σ are the same as the usual characters of Verma modules and irreducible highest‐weight modules over the orbit Lie superalgebra ğ = g(σ) determined by σ. 1991 Mathematics Subject Classification : 17A70, 17B01, 17B65, 17B70, 11F22.