PBW and duality theorems for quantum groups and quantum current algebras

Abstract

We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the $A_1^{(1)}$ case.

44 pages, mistake in one proof corrected

References in corpus (3)

• Schur duality in the toroidal setting
• Convex Bases of PBW type for Quantum Affine Algebras
• Quantum Kac-Moody Algebras and Vertex Representations

Cited by in corpus (13)

• On the center of quiver-Hecke algebras
• On quantum vertex algebras and their modules
• Shuffle approach towards quantum affine and toroidal algebras
• Quantum loop groups and shuffle algebras via Lyndon words
• Drinfeld type presentations of loop algebras
• $\hslash$-adic quantum vertex algebras and their modules
• Twisted quantum affinizations and quantization of extended affine Lie algebras
• Quantization of Lie bialgebras and shuffle algebras of Lie algebras
• On quantum toroidal algebra of type $A_1$
• The PBW theorem for the affine Yangians
• Quantum toroidal algebras and their representations
• Higgs bundles on weighted projective lines and loop crystals
• Quasi-Hopf algebras associated with semisimple Lie algebras and complex curves