Communications in Mathematical Physics

, Volume 301, Issue 1, pp 131–174 | Cite as

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

Article

Abstract

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.

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References

  1. 1.
    Andersen H.H., Polo P., Wen K.X.: Representations of quantum algebras. Invent. Math. 104(1), 1–59 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Atiyah M., Bott R., Patodi V.K.: On the heat equation and the index theorem. Invent. Math. 19, 279–330 (1973)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Berenstein A., Zwicknagl S.: Braided symmetric and exterior algebras. Trans. Amer. Math. Soc. 360(7), 3429–3472 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brown, K.A., Goodearl K.R.: Lectures on algebraic quantum groups. Advanced Courses in Mathematics. CRM Barcelona, Basel: Birkhäuser Verlag, 2002Google Scholar
  5. 5.
    Brundan J.: Dual canonical bases and Kazhdan-Lusztig polynomials. J. Algebra 306(1), 17–46 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Connes A.: “Noncommutative geometry”. Academic Press, London-NewTork (1994)Google Scholar
  7. 7.
    de Concini C., Procesi C.: A characteristic free approach to invariant theory. Adv. Math. 21, 330–354 (1976)MATHCrossRefGoogle Scholar
  8. 8.
    Drinfeld, V.G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798–820, Providence, RI: Amer. Math. Soc., 1987, pp. 798–820Google Scholar
  9. 9.
    Du J., Scott L., Parshall B.: Quantum Weyl reciprocity and tilting modules. Commun. Math. Phys. 195(2), 321–352 (1998)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Goodearl K.R., Lenagan T.H., Rigal L.: The first fundamental theorem of coinvariant theory for the quantum general linear group. Publ. Res. Inst. Math. Sci. 36(2), 269–296 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goodearl, K.R., Lenagan, T.H.: Quantized coinvariants at transcendental q. In: Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., 239, New York: Dekker, 2005, pp. 155–165Google Scholar
  12. 12.
    Gover A.R., Zhang R.B.: Geometry of quantum homogeneous vector bundles and representation theory of quantum groups. I. Rev. Math. Phys. 11, 533–552 (1999)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gurevich, D.I., Pyatov, P.N., Saponov, P.A.: Quantum matrix algebras of GL(m|n)-type: the structure of the characteristic subalgebra and its spectral parametrization. (Russian) Teoret. Mat. Fiz. 147(1), 14–46 (2006); translation in Theoret. Math. Phys. 147(1), 460–485 (2006)Google Scholar
  14. 14.
    Howe R.: Transcending classical invariant theory. J Amer. Math. Soc. 2, 535–552 (1989)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Jantzen, J.C.: Lectures on quantum groups. Graduate Studies in Mathematics, 6, Providence, RI: Amer. Math. Soc., 1996Google Scholar
  16. 16.
    Jimbo M.: A q-analogue of \({U({\mathfrak{gl}}(N+1))}\) , Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys. 11(3), 247–252 (1986)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Leduc R., Ram A.: A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras. Adv. Math. 125, 1–94 (1997)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lehrer G.I., Zhang R.B.: Strongly multiplicity free modules for Lie algebras and quantum groups. J. Alg. 306(1), 138–174 (2006)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lehrer, G.I., Zhang, R.B.: A Temperley-Lieb analogue for the BMW-algebra, Progress in Mathematics, Basel-Boston: Birkhäuser, in pressGoogle Scholar
  20. 20.
    Loday, J.-L.: Cyclic homology, Appendix E by Mara O. Ronco. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 301, Berlin: Springer-Verlag, 1992Google Scholar
  21. 21.
    Manin, Yu.: Quantum groups and noncommutative geometry. Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988Google Scholar
  22. 22.
    Montgomery, S.: Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Vol. 82, Providence, RI: Amer. Math. Soc., 1993Google Scholar
  23. 23.
    Müller E.F., Schneider H.-J.: Quantum homogeneous spaces with faithfully flat module structures. Israel J. Math. 111, 157–190 (1999)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Podleś P.: Differential calculus on quantum spheres. Lett. Math. Phys. 18(2), 107–119 (1989)MATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Procesi C.: The invariant theory of n × n matrices. Adv. Math. 19, 306–381 (1976)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Ram A., Wenzl H.: Matrix units for centralizer algebras. J. Alg. 145, 378–395 (1992)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Reshetikhin N.Yu., Turaev V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127(1), 1–26 (1990)MATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Reshetikhin N., Turaev V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Rossi-Doria O.: A Uq(sl(2))-representation with no quantum symmetric algebra. Rend. Mat. Acc. Lincei s., 9 10, 5–9 (1999)MATHMathSciNetGoogle Scholar
  30. 30.
    Strickland E.: Classical invariant theory for the quantum symplectic group. Adv. Math. 123(1), 78–90 (1996)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wenzl H.: On tensor categories of Lie type E N, N ≠ 9. Adv. Math. 177(1), 66–104 (2003)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Weyl, H.: The classical groups. Their invariants and representations. Fifteenth printing. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton, NJ: Princeton University Press, 1997Google Scholar
  33. 33.
    Woronowicz S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987)MATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Zhang R.B., Gould M.D., Bracken A.J.: From representations of the braid group to solutions of the Yang-Baxter equation. Nucl. Physics B 354(2-3), 625–652 (1991)CrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Zhang R.B.: Structure and representations of the quantum general linear supergroup. Commun. Math. Phys. 195, 525–547 (1998)MATHCrossRefADSGoogle Scholar
  36. 36.
    Zhang R.B.: Howe duality and the quantum general linear group. Proc. Amer. Math. Soc. 131(9), 2681–2692 (2003)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Zwicknagl S.: R-matrix Poisson algebras and their deformations. Adv. Math. 220(1), 1–58 (2009)MATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina

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