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Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions

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Abstract

Boundary solutions to the quantum Yang–Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ‘modified’ qYB equation, the latter being analogous to the modified classical Yang–Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit ‘boundary quantizations’, i.e., boundary solutions to the qYB equation of the form I + tr + t2r2 +⋯. In the last section we list and give quantizations for all classical r-matrices in sl(3) ∧ sl(3).

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References

  1. Chari, V. and Pressley, A.: A Guide to Quantum Groups, Cambridge University Press, New York, 1994.

    Google Scholar 

  2. Gerstenhaber, M. and Giaquinto, A.: Boundary soulutions of the classical Yang–Baxter equation, Lett. Math. Phys. 40 (1997), 337–353.

    Google Scholar 

  3. Gerstenhaber, M., Giaquinto, A. and Schack, S. D.: Construction of quantum groups from Belavin–Drinfel'd infinitesimals, in A. Joseph and S. Shnider (eds), Quantum Deformation of Algebras and their Representations, Amer. Math. Soc., Providence, 1993, pp. 45–64.

    Google Scholar 

  4. Giaquinto, A. and Zhang, J. J.: Bialgebra actions, twists, and universal deformation formulas, J. Pure Appl. Algebra 129 (1998), to appear.

  5. Hietarinta, J.: All solutions to the constant quantum Yang–Baxter equation in two dimensions, Phys. Lett. A 165 (1992), 245–251.

    Google Scholar 

  6. Stolin, A.: Constant solutions of Yang–Baxter equation for sl(2) and sl(3), Math. Scand. 69 (1991), 81–88.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104-6395, U.S.A. e-mail

    Murray Gerstenhaber

  2. Department of Mathematics, Texas A & M University, College Station, TX, 77843-3368, U.S.A. e-mail

    Anthony Giaquinto

Authors
  1. Murray Gerstenhaber
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  2. Anthony Giaquinto
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Correspondence to Anthony Giaquinto.

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Gerstenhaber, M., Giaquinto, A. Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions. Letters in Mathematical Physics 44, 131–141 (1998). https://doi.org/10.1023/A:1007404917266

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  • DOI: https://doi.org/10.1023/A:1007404917266

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