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Abstract

The ‘plus part’ U + of a quantum group U q (g) has been identified by M. Rosso with a subalgebra G sym of an algebra G which is a quantized version of R. Ree's shuffle algebra. Rosso has shown that G sym and G – and hence also Hopf algebras which are analogues of quantum groups – can be defined in a much wider context. In this paper we study one of Rosso's quantizations, which depends on a family of parameters t ij . G sym is determined by a family of matrices Ωα whose coefficients are polynomials in the t ij . The determinants of the Ωα factorize into a number of irreducible polynomials, and our main Theorem 5.2a gives strong information on these factors. This can be regarded as a first step towards the (still very distant!) goal, the classification of the symmetric algebras G sym which can be obtained by giving special values to the parameters t ij .

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References

  1. Bourbaki, N.: Groupes et algèbres de Lie, Chapitres 4, 5, et 6, Éléments de mathématique, Fascicule XXXIV, Hermann, Paris, 1968.

    Google Scholar 

  2. Curtis, C. W. and Reiner, I.: Methods of Representation Theory, Volume II, Wiley, New York, 1987

    Google Scholar 

  3. Green, J. A.: Shuffle algebras, Lie algebras and quantum groups, Textos de Matemática, Série B, No. 9 (29 pp.), Departamento de Matemática, Universidade de Coimbra, Portugal, 1995.

    Google Scholar 

  4. Green, J. A.: Quantum groups, Hall algebras and quantized shuffles, In: M. Cabanes (ed.), Finite Reductive Groups; Related Structures and Representations, Birkhäuser, Boston, 1997, pp. 273-290.

    Google Scholar 

  5. Kac, V. G.: Infinite Dimensional Lie Algebras, 3rd edn, Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

  6. Lusztig, G.: Introduction to Quantum Groups, Progress in Math. 110, Birkhäuser, Boston, 1993.

    Google Scholar 

  7. Rosso, M.: Quantum groups and braid groups, In: Symétries quantiques, Les Houches, Session LXIV, Elsevier, Amsterdam, 1998, pp. 757-785.

    Google Scholar 

  8. Rosso, M.: Integrals of vertex operators and quantum shuffles, Lett. Math. Phys. 41 (1997), 161-168.

    Google Scholar 

  9. Rosso, M.: Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399-416.

    Google Scholar 

  10. Ree, R.: Lie elements and an algebra associated with shuffles, Ann. Math. 68 (1958), 210-220.

    Google Scholar 

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de Chela, D.F., Green, J.A. Quantum Symmetric Algebras. Algebras and Representation Theory 4, 55–76 (2001). https://doi.org/10.1023/A:1009953611721

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  • DOI: https://doi.org/10.1023/A:1009953611721

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