Poisson-Lie structures on the external algebra of $SL(2)$ and their quantization

Aref'eva, I. Ya.; Arutyunov, G. E.; Medvedev, P. B.

Poisson-Lie structures on the external algebra of $SL(2)$ and their quantization

Authors

Aref'eva, I. Ya.

Arutyunov, G. E.

Medvedev, P. B.

Description

All possible graded Poisson-Lie structures on the external algebra of $SL(2)$ are described. We prove that differential Poisson-Lie structures prolonging the Sklyanin brackets do not exist on $SL(2)$. There are two and only two graded Poisson-Lie structures on $SL (2)$ and neither of them can be obtained by a reduction of graded Poisson-Lie structures on the external algebra of $GL(2)$. Both of them can be quantized and as a result we get a new graded algebra of quantum right-invariant forms on $SL_q(2)$ with three generators.
17 pages. Latex errors corrected and references added