$R$-matrices and the Yang-Baxter equation on GNS representations of C$^{*}$-bialgebras

Details $R$-matrices and the Yang-Baxter equation on GNS representations of C$^{*}$-bialgebras $R$-matrices and the Yang-Baxter equation on GNS representations of C$^{*}$-bialgebras

2009-07-14

arxiv.org/abs/0907.2280v3

Mathematics

Operator Algebras

15 pages

Scientific paper

A new construction method of $R$-matrix is given. Let $A$ be a C$^{*}$-bialgebra with a comultiplication $\Delta$. For two states $\omega$ and $\psi$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(\omega,\psi)$ of the C$^{*}$-bialgebra $(A,\Delta)$ on the tensor product of GNS representation spaces associated with $\omega$ and $\psi$. The set $\{R(\omega,\psi):\omega,\psi\}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such $R$-matrices for a non-quasi-cocommutative C$^{*}$-bialgebra.

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