Double-bosonization and Majid's Conjecture, (IV): Type-Crossings from $A$ to $BCD$

Abstract

Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rank-inductive and type-crossing construction for $U_q(\mathfrak{g})$'s is still a remaining open question. In this paper, working with Majid's framework, based on our generalized double-bosonization Theorem proved in \cite{HH2}, we further describe explicitly the type-crossing construction of $U_q(\mathfrak{g})$'s for $(BCD{)}_n$ series direct from type $A_{n-1}$ via adding a pair of dual braided groups determined by a pair of $(R,R^{\prime })$-matrices of type $A$ derived from the respective suitably chosen representations. %which generalize the lower rank cases constructed in \cite{HH1}. Combining with our work in \cite{HH1,HH2,HH3}, this solves Majid's conjecture, that is, any quantum group $U_q(\mathfrak{g})$ associated to a simple Lie algebra $\mathfrak{g}$ can be grown out of $U_q({\mathfrak{sl}}_2)$ inductively by a series of suitably chosen double-bosonization procedures.

26 pages, 1 figure, Sci. China Ser A (2016) (to appear)

References in corpus (1)

• Double-bosonization and Majid's conjecture, (I): rank-induction of $ABCD$

Cited by in corpus (2)

• Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial
• Double crossed biproducts and related structures