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On self-adjoint representations of the sklyanin algebras

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Abstract

Self-adjoint representations in Hilbert space of a family of quadratic algebras constructed by E. K. Sklyanin are studied. Complete sets of classes of unitary equivalence of irreducible representations are obtained for algebras that correspond to second-order points of elliptic curves.

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Literature cited

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    E. K. Sklyanin, “On some algebraic structures connected with the Yang-Baxter equation,” Funkts. Anal. Prilozhen.,16, No. 4, 27–34 (1982).

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    E. K. Sklyanin, “On some algebraic structures connected with the Yang-Baxter equation. Representations of the quantum algebra,” Funkts. Anal. Prilozhen.,17, No. 4, 34–48 (1983).

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    A. M. Vershik, “Algebras with quadratic relations,” in: Spectral Theory of Operators and Infinite-Dimensional Analysis [in Russian], Inst. Mat. Akad. Nauk Ukr. SSR, Kiev (1984), pp. 32–57.

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    S. A. Kruglyak, “Representations of a quantum algebra connected with the Yang-Baxter equation,” in: Spectral Theory of Operators and Infinite-Dimensional Analysis [in Russian], Inst. Mat. Akad. Nauk Ukr. SSR, Kiev (1984), pp. 111–120.

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1567–1574, November, 1991.

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Bespalov, Y.N. On self-adjoint representations of the sklyanin algebras. Ukr Math J 43, 1458–1465 (1991). https://doi.org/10.1007/BF01067287

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Keywords

  • Hilbert Space
  • Irreducible Representation
  • Elliptic Curf
  • Quadratic Algebra
  • Sklyanin Algebra