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

Abstract

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r r -matrices. A super dynamical r r -matrix r r satisfies the zero weight condition if [ h ⊗ 1 + 1 ⊗ h , r ( λ ) ] = 0 for all h ∈ h , λ ∈ h ∗ . \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 r -matrices with zero coupling constant for the Lie superalgebra g l ( m , n ) \mathfrak {gl}(m,n) . We also answer some questions about super dynamical R R -matrices. In particular, we prove a classification theorem and offer some support for one particular interpretation of the super Hecke condition.