IUMJ

Title: Turaev group coalgebras and twisted Drinfeld double

Authors: Shuanhong Wang

Issue: Volume 58 (2009), Issue 3, 1395-1418

Abstract:

A new method of constructing Turaev group coalgebras with quasitriangular structure is introduced. Starting from a Hopf dual pairing $(A,B,\sigma)$ with appropriate homomorphisms: $\phi:\pi\longrightarrow\Aut(A)$ and $\psi:\pi\longrightarrow\Aut(B)$, we construct a twisted Drinfeld double $D(A,B,\sigma;\phi,\psi)$ which is a Turaev $\mathscr{S}(\pi)$-coalgebra, where the group $\mathscr{S}(\pi)$ is a twisted semi-direct square of $\pi$. Moreover, when $A$ or $B$ is finite-dimensional, we define a non-trivial quasitriangular structure on $D(A,B,\sigma;\phi,\psi)$. This approach allows us to produce new examples of Turaev $\pi$-coalgebras for many infinite groups $\pi$ such as $GL_n(\mathbbm{k})$ and $(\mathbb{C}^{*})^{\ell}$ with $\ell\geq1$.