nLab Radford-Majid bosonisation

Idea

Suppose $H$ is a cocommutative Hopf algebra in a braided monoidal category $C$ and suppose $H$ acts on another Hopf algebra $B$ in $C$. Then a smash product and smash coproduct of the two has a structure of an ordinary Hopf algebra called a Radford biproduct or Majid bosonisation of $H$. In an intuitive description, it turns the braided statistics into a bosonic statistics. A particular case is the case of super-Hopf algebras related to the more traditional physical concept of bosonization.

Majid uses English spelling bosonisation rather than bosonization, so it is a tradition that the following literature uses the same spelling for the algebraic concept.

Examples

One can perform bosonisation on Nichols algebras.

Literature

• David E. Radford, Hopf algebras with projection, J. Algebra 92 (1985) 322–347 doi
• Shahn Majid, Cross products by braided groups and bosonization, Journal of Algebra 163:1 (1994) 165–190 doi
• Shahn Majid, Double-bosonization of braided groups and the construction of $U_q(g)$, Math. Proc. Cambridge Philos. Soc. 125 (1999) 151–192

category: algebra