2022/11/05 by Fan, Zhaobing, Han, Shaolong, Kang, Seok-Jin +1
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2211.02859
Let Uq-(\mathfrak g) be the negative half of a quantum Borcherds-Bozec algebra Uq(\mathfrak g) and V(λ) be the irreducible highest weight module with λ∈ P+. In this paper, we investigate the structures, properties and their close connections between crystal bases and canonical bases of Uq-(\mathfrak g) and V(λ). We first re-construct crystal basis theory with modified Kashiwara operators. While going through Kashiwara's grand-loop argument, we prove several important lemmas, which play crucial roles in the later developments of the paper. Next, based on the theory of canonical bases on quantum Bocherds-Bozec algebras, we introduce the notion of primitive canonical bases and prove that primitive canonical bases coincide with lower global bases.