Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces
By S. Rao. 2001
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The purpose of this paper is to classify irreducible integrable modules for multi-loop algebras. Let V be a vector space over the complex numbers . Let A = t± 1 t± n be the Laurent polynomials ring in n variables. Let VA = V ⊗A and let v m = v ⊗ t for m = m1 mn ∈ n and t = t1 1 t2 2 · · · tn n . Let be a simple finite-dimensional Lie algebra over the complex numbers . Then A can be given a Lie algebra structure (see (1.1)). Let d1 dn be derivations defined by di X m = miX m and let D be the linear span of d1 dn. Then A = A ⊕ is a Lie algebra which we call a multi-loop algebra. The universal central externion of A is called a toroidal Lie algebra and is studied in [BB, BC, E3, E4, EM]. So our multiloop algebras are quotients of toroidal Lie algebras by centers. Fix a Cartan subalgebra h of . Let ψ be a -graded homomorphism of U hA → A (see Section 1 for details). Then we can define the universal highest weight module M ψ for A. Let V ψ be the unique irreducible quotient of M ψ . We then prove in Theorem 3.4 that any irreducible integrable module for A is isomorphic to V ψ by making use of a result from Chari [C]. We associate to ψ a map ψ from U hA → by evaluating at 1 and an irreducible module V ψ π for A (derivations do not act on V ψ , and therefore it cannot be extended to A). Now consider V ψ ⊗A as a
Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics
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Citations in the corpus
- Weight modules over exp-polynomial Lie algebras (2003, methodology): “Such modules were studied by Chari [C] and Rao [ E ].”
- Representations of Multiloop Algebras (2008, background): “(See [17, 19] for partial results.”
- Irreducible finite-dimensional representations of equivariant map algebras (2009, background): “Generalizations and more precise descriptions of their work have appeared in many papers, for example in Batra [Bat04], Chari-Fourier-Khandai [CFK], Chari-Fourier-Senesi [CFS08], Chari-Moura [CM04], Feigin-Loktev [FL04], Lau [Lau], Li [Li04], and Rao [Rao93, Rao01].”
- Classification of irreducible integrable representations for the full toroidal lie algebras (2005, background): “The proof of our result depends heavily on [14,15,9].”