2021/06/01 by Weicai Wu, Wu, Weicai
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2106.00552
In this paper we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, quantum Nichols bicharacter algebras, etc. We obtain explicit bases for \mathfrak L(V)\tiny R and \mathfrak L(V)\tiny L over (i) the quantum linear space V with dim V=2; (ii) a connected braided vector V of diagonal type with dim V=2 and p1,1=p2,2= -1. We give the sufficient and necessary conditions for \mathfrak L(V)\tiny R= \mathfrak L(V), \mathfrak L(V)\tiny L= \mathfrak L(V), \mathfrak B(V) = F⊕ \mathfrak L(V)\tiny R and \mathfrak B(V) = F⊕ \mathfrak L(V)\tiny L, respectively. We show that if \mathfrak B(V) is a connected Nichols algebra of diagonal type with dim V>1, then \mathfrak B(V) is finite-dimensional if and only if \mathfrak L(V)\tiny L is finite-dimensional if and only if \mathfrak L(V)\tiny R is finite-dimensional.