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CLIFFORD AND WEYL SUPERALGEBRAS AND SPINOR REPRESENTATIONS

  • JONAS T. HARTWIGEmail author
  • VERA SERGANOVA
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  • 19 Downloads

Abstract

We construct a family of twisted generalized Weyl algebras which includes Weyl–Clifford superalgebras and quotients of the enveloping algebras of \( \mathfrak{gl}\left(m|n\right) \) and \( \mathfrak{osp}\left(m|2n\right) \). We give a condition for when a canonical representation by differential operators is faithful. Lastly, we give a description of the graded support of these algebras in terms of pattern-avoiding vector compositions.

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References

  1. [1]
    В. В. Бавула, Обобщенные алгебры Вейля и их представления, Алгебра и анализ 4 (1992), вып. 1, 75–97. Engl. transl.: V. V. Bavula, Generalized Weyl algebras and their representations, St. Petersburg Math. J. 4 (1993), no. 1, 71–92.Google Scholar
  2. [2]
    V. V. Bavula, D. A. Jordan, Isomorphism problems and groups of automorphisms for generalized Weyl algebras, Trans. Amer. Math. Soc. 353 (2000), 769–794.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    V. V. Bavula, F. Van Oystaeyen, The simple modules of certain generalized crossed products, J. Algebra 194 (1997), no. 2, 521–566.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    G. Benkart, D. J. Britten, F. Lemire Modules with bounded weight multiplicities for simple Lie algebras, Math. Zeit. 225 (1997), 333–353.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    R. Block The irreducible representations of the Lie algebra \( \mathfrak{sl} \)(2) and of the Weyl algebra, Adv. Math. 39 (1981), 69–110.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. J. Britten, F. W. Lemire, A classiffication of simple Lie modules having a 1-dimensional weight space, Trans. Amer. Math. Soc. 299 (1987), 683–697.MathSciNetzbMATHGoogle Scholar
  7. [7]
    T. Brzeziński, Circle and line bundles over generalized Weyl algebras, Algebr. Represent. Theory 19 (2016), no. 1, 57–69.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Cassidy, B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279 (2004), 402–421.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    K. Coulembier, On a class of tensor product representations for the orthosymplectic superalgebra, J. Pure Appl. Algebra 217 (2013), no. 5, 819–837.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    V. Futorny, J. T. Hartwig, On the consistency of twisted generalized Weyl algebras, Proc. Amer. Math. Soc. 140 (2012), no. 10, 3349–3363.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J. Hartwig, V. Futorny, E. Wilson, Irreducible completely pointed modules for quantum groups of type A, J. Algebra 432 (2015), 252–279.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    J. T. Hartwig, Locally finite simple weight modules over twisted generalized Weyl algebras, J. Algebra 303 (2006), 42–76.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. T. Hartwig, Twisted generalized Weyl algebras, polynomial Cartan matrices and Serre-type relations, Comm. Algebra 38 (2010), 4375–4389.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. T. Hartwig, Noncommutative singularities and lattice models, arXiv:1612.08125 (2016).Google Scholar
  15. [15]
    J. T. Hartwig, J. Öinert, Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras, J. Algebra 373 (2013), 312–339.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    J. T. Hartwig, V. Serganova, Twisted generalized Weyl algebras and primitive quotients of enveloping algebras, Algebr. Represent. Theory 19 (2016), no. 2, 277–313.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96.CrossRefzbMATHGoogle Scholar
  18. [18]
    O. Mathieu, Classification of irreducible weight modules, Ann. l’institut Fourier 50 (2000), 537–592.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    V. Mazorchuk, M. Ponomarenko, L. Turowska, Some associative algebras related to U(\( \mathfrak{g} \)) and twisted generalized Weyl algebras, Math. Scand. 92 (2003), 5–30.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    V. Mazorchuk, L. Turowska, Simple weight modules over twisted generalized Weyl algebras, Comm. Algebra 27 (1999), 2613–2625.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    V. Mazorchuk, L. Turowska, *-Representations of twisted generalized Weyl constructions, Algebr. Represent. Theory 5 (2002), no. 2, 163–186.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Nauwelaerts, F. Van Oystaeyen, Introducing crystalline graded algebras, Algebr. Represent. Theory 11 (2008), no. 2, 133–148.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    K. Nishiyama, Oscillator representations for orthosymplectic algebras, J. Algebra 129 (1990), no. 1, 231–262.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Sergeev, Enveloping algebra U(gl(3)) and orthogonal polynomials, in: Noncommutative Structures in Mathematics and Physics (Kiev, 2000), NATO Sci. Ser. II Math. Phys. Chem., Vol. 22, Kluwer Acad. Publ., Dordrecht, 2001, pp. 113–124.Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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