Double-bosonization and Majid's Conjecture, (II): cases of irregular $R$-matrices and type-crossings of $F_4$, $G_2$

Abstract

The purpose of the paper is to build up the related theory of weakly quasitriangular dual pairs suitably for non-standard $R$-matrices (irregular), and establish the generalized double-bosonization construction theorem for irregular $R$, which generalize Majid's results for regular $R$ in \cite{majid1}. As an application, the type-crossing construction for the exceptional quantum groups of types $F_4$, $G_2$ is obtained. This affirms the Majid's expectation that the tree structure of nodes diagram associated with quantum groups can be grown out of the node corresponding to $U_q({\mathfrak{sl}}_2)$ by double-bosonization procedures. Notably from a representation perspective, we find an effective method to get the minimal polynomials for the non-standard $R$-matrices involved.

42 pages

References in corpus (2)

• Double-bosonization and Majid's conjecture, (I): rank-induction of $ABCD$
• Multi-parameter Quantum Groups via Quantum Quasi-Symmetric Algebras

Cited by in corpus (2)

• Double-bosonization and Majid's Conjecture, (III): type-crossing and inductions of $E_6$ and $E_7$, $E_8$
• Quantum (dual) Grassmann superalgebra as ${\mathcal{U}}_q(\mathfrak{gl}(m|n))$-module algebra and beyond