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Determinantal Ideals and the Canonical Commutation Relations: Classically or Quantized

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We construct homomorphic images of \(su(n,n)^{{\mathbb {C}}}\) in Weyl Algebras \({{\mathcal {H}}}_{2nr}\). More precisely, and using the Bernstein filtration of \({{\mathcal {H}}}_{2nr}\), \(su(n,n)^{{\mathbb {C}}}\) is mapped into degree 2 elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of \({{\mathcal {H}}}_{2nr}\), these homomorphisms give all unitary highest weight representations of \(su(n,n)^{{\mathbb {C}}}\) thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of \(u(r)^{{\mathbb {C}}}\) into \({{\mathcal {H}}}_{2nr}\) whose image commutes with the image of \(su(n,n)^{{\mathbb {C}}}\), and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of \((r+1)\times (r+1)\) minors is prime. Here, of course, \(r\le n-1\) and for a fixed such r, the space of any irreducible representation of \(su(n,n)^{{\mathbb {C}}}\) is annihilated by this ideal. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for \({{\mathcal {U}}}_q(su(n,n)^{{\mathbb {C}}})\). The Weyl Algebras are replaced by the Hayashi–Weyl Algebras \({{\mathcal {H}}}{{\mathcal {W}}}_{2nr}\) and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of \({{\mathcal {U}}}_q(u(r)^{{\mathbb {C}}})\) into \({{\mathcal {H}}}{{\mathcal {W}}}_{2nr}\) is constructed with analogous properties. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.

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Notes

  1. If \(x=n-1\) or \(y=n-1\) this is then not of the form we usually insist on. We refrain from changing it since it is utterly clear how to do it.

References

  1. Artin, E.: Galois Theory, 2nd edn. Notre Dame (1948)

  2. Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: Differential operators on the base affine space and a study of \({{\mathfrak{g} }}\)-modules. In: Gelfand, I.M. (ed.) Lie Groups Represent., pp. 21–64. Wiley, New York (1975)

    Google Scholar 

  3. Boe, B.D.: Homomorphisms between generalized Verma modules. Trans. Am. Math. Soc. 288, 791–799 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bruns, W., Vetter, U.: Determinantal Rings. Lecture Notes in Mathematics, vol. 1327. Springer-Verlag, Berlin (1988)

    Book  MATH  Google Scholar 

  5. De Concini, C., Eisenbud, D., Procesi, C.: Young diagrams and determinantal varieties. Invent. Math. 56, 129–165 (1980)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Davidson, M.G., Enright, T.J., Stanke, R.J.: Differential Operators and Highest Weight Representations. Memoirs of the American Mathematical Society, vol. 455 (1991)

  7. Drinfeld, V. G.: Quantum Groups. In: Proceedings International Congress of Mathematicians, Berkeley, pp. 798–820 (1986)

  8. Enright, T.J., Joseph, A.: An intrinsic analysis of unitarizable highest weight modules. Math. Ann. 288, 571–594 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Enright, T.J., Willenbring, J.F.: Hilbert series, Howe duality and branching for classical groups. Ann. Math. 159, 337–375 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. In: Algebraic Analysis. Academic Press, pp. 120–140 (1988)

  11. Fioresi, R.: Commutation relations among generic quantum minors in \(O_q(M_n(k))\). J. Algebra 280(2), 655–682 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goodearl, K., Lenagan, T.: Quantum determinantal ideals. Duke Math. J. 103(1), 165–190 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gross, K., Kunze, R.: Fourier Bessel transforms and holomorphic discrete series. In: Proceedings of the Maryland Conference on Harmonic Analysis. Lecture Notes in Mathematics, vol. 266. Springer Verlag, Berlin (1977)

  14. Gross, K., Kunze, R.: Bessel functions and representation theory II: holomorphic discrete series and metaplectic representations. J. Funct. Anal. 25, 1–49 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Harish-Chandra: On some applications of the universal enveloping algebra of a semisimple Lie algebra. Trans. Am. Math. Soc. 70, 28–96 (1951)

  16. Harish-Chandra. Representations of semi-simple Lie groups IV, V, VI. Am. J. Math. 77, 743–777 (1955); 78, 1–41 and 564–628 (1956)

  17. Harrish, M., Jakobsen, H.P.: Covariant differential operators. In: Group Theoretical Methods in Theoretical Physics, Proceedings, Istanbul 1982. Lecture Notes in Physics, vol. 180, pp. 16–34. Springer Verlag (1983)

  18. Hayashi, T.: \(q\)-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys. 127, 129–144 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Jakobsen, H.P.: Intertwining differential operators for Mp(n, R) and SU(n, n). Trans. Am. Math. Soc. 246, 311–337 (1978)

    MathSciNet  MATH  Google Scholar 

  20. Jakobsen, H.P.: The last possible place of unitarity for certain highest weight modules. Math. Ann. 256, 439–447 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jakobsen, H.P.: Hermitian symmetric spaces and their unitary highest weight modules. J. Funct. Anal. 52, 385–412 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jakobsen, H.P.: Basic covariant differential operators on Hermitian symmetric spaces. Ann. Scient. Èc. Norm. Sup. 18, 421–436 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jakobsen, H.P.: An intrinsic classification of the unitarizable highest weight modules as well as their associated varieties. Compositio Math. 101, 313–352 (1996)

    MathSciNet  MATH  Google Scholar 

  24. Jakobsen, H.P.: Unitarity of highest weight modules for quantum groups. Lett. Math. Phys. 42, 119–133 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jakobsen, H.P.: Algebras of Variable Coefficient Quantized Differential Operators. arXiv:1905.04478v1 [math.QA]

  26. James, G.D.: The Representation Theory of the Symmetric Groups. Springer Lecture Notes in Mathematics, vol. 682, Berlin (1978)

  27. Jantzen, J.C.: Lectures on Quantum Groups, A.M.S. Graduate Studies in Mathematics, vol. 6 (1996)

  28. Jimbo, M.: A \(q\)-difference analogue of \(U({{\mathfrak{g} }})\) and the Yang–Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Kitchin, A.P., Launois, S.: On the automorphisms of quantum Weyl algebras. J. Pure Appl. Algebra 223, 1514–1530 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kashiwara, M., Vergne, M.: On the Segal–Shale–Weil representation and harmonic polynomials. Invent. Math. 44, 1–47 (1978)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Launois, S., Lenagan, T.H., Rigal, L.: Quantum unique factorisation domains. J. Lond. Math. Soc. 74, 321–340 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Leclerc, B.: Fock space representations of \(U_q(\widehat{{{\mathfrak{s}}}{{\mathfrak{l}}}}_n)\). In: Summer School on Geometric Methods in Representation Theory 2008. École thématique, Institut Fourier (2008)

  33. Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70, 237–249 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lusztig, G.: Introduction to Quantum Groups. Progress In Mathematics, vol. 110, Birkhäuser (1993)

  35. Mack, G., Todorov, I.: Irreducibility of the ladder representations of U(2, 2) when restricted to the Poincare subgroup. J. Math. Phys. 10, 2078–2085 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Martens, S.: The characters of the holomorphic discrete series. Proc. Nat. Acad. Sci. USA 72, 3275–3276 (1975)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Mount, K.R.: A remark on determinantal loci. J. Lond. Math. Soc. 42, 595–598 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  38. Noumi, M., Yamada, H., Mimachi, K.: Finite dimensional representations of the quantum group \(GL_q(n;{\mathbb{C} })\) and the zonal spherical functions on \(U_q(n-1)\setminus U_q(n)\). Jpn. J. Math. 19, 31–80 (1993)

    Article  MATH  Google Scholar 

  39. Poulsen, N.S.: On the canonical commutation relations. Math. Scand. 32, 112–122 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rosenberg, A.: Noncommutative Algebraic Geometry and Representations of Quantized Algebras. Kluwer Academic Pblishers, Dordrecht (1995)

    Book  MATH  Google Scholar 

  41. Rossi, H., Vergne, M.: Analytic continuation of the holomorphic discrete series of a semi-simple Lie group. Acta Math. 136, 1–59 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rosso, M.: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Commun. Math. Phys. 117, 581–593 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Segal, I.E.: (reviewer), Math Review MR0045059 (1951)

  44. Segal, I.E.: Foundations of the theory of dynamical systems of infinitely many degrees of freedom. I, Mat. Fys. Medd. Danske Vid. Selsk. 31, 1–39 (1959)

  45. Shale, D.: Linear symmetries of free boson fields. Trans. Am. Math. Soc. 103, 149–167 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shklyarov, D., Sinel’shchikov, S., Stolin, A., Vaksman, L.: On A \(q\)-analogue of the Penrose transform. In: Vaksman, L. (ed.) Lectures on \(q\)-Analogues of Cartan Domains and Associated Harish-Chandra Modules. arXiv:math/0109198

  47. Shklyarov, D., Sinel’shchikov, S., Vaksman, L.: Geometric realizations for some series of representations of the quantum group SU(2,2). Math. Phys. Anal. Geom. 8(1), 90–110 (2001)

    MathSciNet  MATH  Google Scholar 

  48. Shklyarov, D., Zhang, G.: Covariant q-differential operators and unitary highest weight representations for \(U_q (su(n,n))\). J. Math. Phys. 46 no. 6 id.062307 (2005)

  49. Skoda, Z.: Every quantum minor generates an Ore set. Int. Math. Res. Notices 2008; 2008 rnn063. https://doi.org/10.1093/imrn/rnn063

  50. Stone, M.H.: Linear transformations in Hilbert space. III, Operational methods and group theory. Proc. Nat. Acad. Sci 16, 172–175 (1930)

    Article  ADS  MATH  Google Scholar 

  51. van Hove, L.: Sur certaines représentations unitaires d’un groupe infini de transformations, Mémoires de Acad. Roy. de Belg. No. 1618 (1951)

  52. Van Hove, L.: Sur le probléme des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques de la mécanique classique. Acad. Roy. Belgique. Bull. Cl. Sci. 5(37), 610–620 (1951)

    MATH  Google Scholar 

  53. Vaksman, L.L.: Quantum matrix ball: the Cauchy–Szegö kernel and the Shilov boundary. J. Math. Phys. Anal. Geom. 8(4), 366–384 (2001)

    MathSciNet  MATH  Google Scholar 

  54. von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Mathematische Annalen 104, 570–578 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  55. Verma, D.-N.: Structure of certain induced representations of comples semisimple Lie algebras. Bull. Am. Math. Soc. 74, 160–166 (1968)

    Article  MATH  Google Scholar 

  56. Wallach, N.: The analytic continuation of the discrete series II. Trans. Am. Math. Soc. 251, 19–37 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  57. Weil, A.: Sur certains groupes d’operateurs unitaires. Acta Math. 111, 193–211 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  58. Xiao, J.: Drinfeld double and Ringel–Green theory of Hall algebras. J. Alg. 190, 100–144 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhang, R.B.: Howe duality and the quantum general linear group. Proc. Am. Math. Soc. 131, 2681–2692 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Hans Plesner Jakobsen

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Jakobsen, H.P. Determinantal Ideals and the Canonical Commutation Relations: Classically or Quantized. Commun. Math. Phys. 398, 375–438 (2023). https://doi.org/10.1007/s00220-022-04524-5

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