Braided quantum groups and their bosonizations in the $C^{\ast }$-algebraic framework

Abstract

We present a general theory of braided quantum groups in the C*-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of a regular quantum group $\mathbb{G}$ we construct a braided C*-quantum group over $\mathbb{G}$ as a C*-bialgebra in the monoidal category of the $\mathbb{G}$-Yetter-Drinfeld C*-algebras. Furthermore, we establish the one to one correspondence between braided C*-quantum groups and C*-quantum groups with projection. Consequently, we generalise the bosonization construction for braided Hopf-algebras of Radford and Majid to braided C*-quantum groups. Several examples are discussed. In particular, we show that the complex quantum plane admits a the braided C*-quantum group structure over the circle group $\mathbb{T}$ and identify its bosonization with the simplified quantum $E(2)$ group.

25 pages; clarified a few points, corrected Proposition 3.4, included the proof of the one to one correspondence between braided C*-quantum groups and quantum groups with a projection up to isomorphism (Theorem 5.16)

References in corpus (10)

• Quantum group-twisted tensor products of C*-algebras
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Cited by in corpus (2)

• Braided quantum symmetries of graph ${\mathrm{C}}^{\ast }$-algebras
• Anyonic quantum symmetries of finite spaces