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

Abstract

Majid developed in \cite{majid3} his double-bosonization theory to construct $U_q(\mathfrak{g})$ and expected to generate inductively not just a line but a tree of quantum groups starting from a node. In this paper, the authors confirm the Majid's first expectation (see p. 178 \cite{majid3}) through giving and verifying the full details of the inductive constructions of $U_q(\mathfrak{g})$ for the classical types, i.e., the $ABCD$ series. Some examples in low ranks are given to elucidate that any quantum group of classical type can be constructed from the node corresponding to $U_q({\mathfrak{sl}}_2)$.

arXiv comment22 pages

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• Braided Momentum Structure of the q-Poincare Group
• Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
• Multi-brace cotensor Hopf algebras and quantum groups
• New Quantum Groups by Double-Bosonisation
• Multi-parameter Quantum Groups via Quantum Quasi-Symmetric Algebras
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Cited by in corpus (7)

• Double-bosonization and Majid's Conjecture, (IV): Type-Crossings from $A$ to $BCD$
• Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial
• Double-bosonization and Majid's Conjecture, (II): cases of irregular $R$-matrices and type-crossings of $F_4$, $G_2$
• Double crossed biproducts and related structures
• 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
• Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space