On the quantization of zero-weight super dynamical 𝑟-matrices

Karaali, Gizem

doi:10.1090/S0002-9939-2011-10873-6

On the quantization of zero-weight super dynamical $r$-matrices
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by Gizem Karaali

Proc. Amer. Math. Soc. 140 (2012), 7-20

DOI: https://doi.org/10.1090/S0002-9939-2011-10873-6

Published electronically: May 5, 2011

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Abstract:

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical $r$-matrices. A super dynamical $r$-matrix $r$ satisfies the zero weight condition if \begin{equation*} [h\otimes 1 + 1 \otimes h, r(\lambda )] = 0 \text { for all } h \in \mathfrak {h}, \lambda \in \mathfrak {h}^*. \end{equation*} In this paper we explicitly quantize zero-weight super dynamical $r$-matrices with zero coupling constant for the Lie superalgebra $\mathfrak {gl}(m,n)$. We also answer some questions about super dynamical $R$-matrices. In particular, we prove a classification theorem and offer some support for one particular interpretation of the super Hecke condition.

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