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

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.

References

Pavel Etingof and David Kazhdan, Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996), no. 1, 1–41. MR 1403351, DOI 10.1007/BF01587938
Pavel Etingof, Travis Schedler, and Olivier Schiffmann, Explicit quantization of dynamical $r$-matrices for finite dimensional semisimple Lie algebras, J. Amer. Math. Soc. 13 (2000), no. 3, 595–609. MR 1758755, DOI 10.1090/S0894-0347-00-00333-7
Pavel Etingof and Alexander Varchenko, Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys. 192 (1998), no. 1, 77–120. MR 1612160, DOI 10.1007/s002200050292
Pavel Etingof and Alexander Varchenko, Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys. 196 (1998), no. 3, 591–640. MR 1645196, DOI 10.1007/s002200050437
Gizem Karaali, Constructing $r$-matrices on simple Lie superalgebras, J. Algebra 282 (2004), no. 1, 83–102. MR 2095573, DOI 10.1016/j.jalgebra.2004.07.005
Gizem Karaali, A new Lie bialgebra structure on $\textrm {sl}(2,1)$, Representations of algebraic groups, quantum groups, and Lie algebras, Contemp. Math., vol. 413, Amer. Math. Soc., Providence, RI, 2006, pp. 101–122. MR 2262367, DOI 10.1090/conm/413/07842
Gizem Karaali, Super solutions of the dynamical Yang-Baxter equation, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2521–2531. MR 2213729, DOI 10.1090/S0002-9939-06-08495-4
Gizem Karaali, Dynamical quantum groups—the super story, Hopf algebras and generalizations, Contemp. Math., vol. 441, Amer. Math. Soc., Providence, RI, 2007, pp. 19–52. MR 2381534, DOI 10.1090/conm/441/08498
Gizem Karaali, On Hopf algebras and their generalizations, Comm. Algebra 36 (2008), no. 12, 4341–4367. MR 2473333, DOI 10.1080/00927870802182424
Olivier Schiffmann, On classification of dynamical r-matrices, Math. Res. Lett. 5 (1998), no. 1-2, 13–30. MR 1618367, DOI 10.4310/MRL.1998.v5.n1.a2