Yiqiang Li

We construct the entire generalized Kac–Moody Lie algebra as a quotient of the positive part of another generalized Kac–Moody Lie algebra. The positive part of a generalized Kac–Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac–Moody Lie algebra. This idea arises from the affine Lie algebra construction and evaluation maps. In [ 16 Li , Y. , Lin , Z. ( 2010 ). A realization of quantum groups via product valued quivers . Algebra and Representation Theory 13 ( 4 ): 427 – 444 . [Crossref] [Google Scholar] ], we give a quantum version of this construction after analyzing Nakajima's quiver variety construction of integral highest weight representations of the quantized enveloping algebras in terms of the irreducible components of quiver varieties.

We study the canonical basis for the negative part U^- of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras U^- associated to two different matrices satisfying certain conditions may coincide (6.3). We show that the canonical bases coincide provided that the algebras U^- coincide (Theorem 6.3.5). We also answer partially a question by Lusztig in [L3] (Theorem 7.1.1).

We study the canonical basis for the negative part of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras associated to two different matrices satisfying certain conditions may coincide. We show that the canonical bases coincide provided that the algebras coincide. We also answer partially a question by Lusztig.

We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.