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

Abstract

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 $S{L}_q(2)$ with three generators.

17 pages. Latex errors corrected and references added

References in corpus (2)

• Cartan Calculus for Hopf Algebras and Quantum Groups
• Uniqueness of $U_q(N)$ as a quantum gauge group and representations of its differential algebra