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

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

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arXiv · hep-th/9401127

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

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

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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 $SL_q(2)$ with three generators.

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I. Ya. Aref'eva, G. E. Arutyunov, P. B. Medvedev. 1994-01-28. Poisson-Lie structures on the external algebra of $SL(2)$ and their quantization. https://arxiv.org/abs/hep-th/9401127

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