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2009•Algebra ColloquiumRequires access

Generalized Verma Modules over Lie Algebras of Weyl Type

B. Xin, Yuezhu Wu

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Abstract

For a field 𝔽 of characteristic 0 and an additive subgroup Γ of 𝔽, there corresponds a Lie algebra [Formula: see text] of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma [Formula: see text]-module M(Λ, ≺) is defined. In this paper, the irreducibility of M(Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the [Formula: see text]-infinity algebra [Formula: see text] is quasifinite if and only if it is a proper quotient of a Verma module.

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What this paper is about

For a field 𝔽 of characteristic 0 and an additive subgroup Γ of 𝔽, there corresponds a Lie algebra [Formula: see text] of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma [Formula: see text]-module M(Λ, ≺) is defined. In this paper, the irreducibility of M(Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the [Formula: see text]-infinity algebra [Formula: see text] is quasifinite if and only if it is a proper quotient of a Verma module.

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Available abstract

For a field 𝔽 of characteristic 0 and an additive subgroup Γ of 𝔽, there corresponds a Lie algebra [Formula: see text] of generalized Weyl type. Given a total order of Γ and a weight Λ, a generalized Verma [Formula: see text]-module M(Λ, ≺) is defined. In this paper, the irreducibility of M(Λ, ≺) is completely determined. It is also proved that an irreducible highest weight module over the [Formula: see text]-infinity algebra [Formula: see text] is quasifinite if and only if it is a proper quotient of a Verma module.

Key concepts: Verma module, Mathematics, Pure mathematics, Irreducibility, Type (biology), Quotient, Lie algebra, Affine Lie algebra

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