Quantum Deformations of SLn/B and its Schubert Varieties

  • V. Lakshmibai
  • N. Reshetikhin
Conference paper


In this paper, we prove the results announced in [L-R] for G = SLn. Let G be a simple algebraic group over the base field k. Let M be a maximal torus in G and B, a Borel subgroup, B ⊃ M. Let W be the Weyl group of G. For w ∈ W, let X(w) = \(\overline {BwB} \) (mod B) be the Schubert variety in G/B associated to w. Let L be an ample line bundle on G/B. We shall denote the restriction of L to X(w) also by just L. Let k[X(w)] = ⊙ H0(X(w),L). In this paper, we construct an algebra kq [X(w)] over k(q), where q is a parameter taking values in k*, as a quantization of k[X(w)], G being SLn. The algebra k q [SL n ]: Let G = SLn. Let T = (tij), 1 ≤ i, j ≤ n. Let Let
$$R = \sum\limits_{\mathop {i \ne j}\limits_{i,j = 1} }^n {{e_{ii}}} { \otimes _{jj}} + q\sum\limits_{i \ne i}^n {{e_{ii}}} \otimes {e_{ii}} + \left( {q - {q^{ - 1}}} \right)\sum\limits_{1j < in} {{e_{ij}}} \otimes {e_{ji}}$$
(here, eij’s are the elementary matrices). Let A(R) be the associative algebra (with 1) generated by {tij, 1 ≤ i, j ≤ n}, the relations being given by RT1T2 = T2T1R, where T1 = T ⊗ Id, T2 = Id ⊗ T (cf. [F-R-T]). Then A(R) gives a quantization of k[Mn], Mn being the space of n × n matrices and k[Mn], the coordinate ring of Mn. Now A(R) has a bialgebra structure, then comultiplication being given by Δ: A(R) → A(R) ⊗ A(R), \(\Delta = \left( {{t_{ij}}} \right) = \sum\limits_{k = 1}^n {{t_{ik}}} \otimes {t_{kj}}\) In the sequel, we shall denote A(R) by kq [Mn]. Let
$$D = \sum\limits_{\sigma \in {s_n}}^n {{t_{ik}}} {\left( { - q} \right)^{ - l{{\left( \sigma \right)}_t}}}1\sigma {\left( 1 \right)^t}2\sigma {\left( 2 \right)^{...}}n\sigma \left( n \right)$$
. we shall refer to D as the q-determinant (or the quantum determinant) of (tij).


Young Diagram Linear Independence Quantum Deformation Schubert Variety Ample Line Bundle 
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  1. [D]
    V. Drinfeld, Quantum Groups, Proc. of the ICM, Berkeley, 1986.Google Scholar
  2. [D-M-M-Z]
    E.E. Demidov, Yu. I. Manin, E.E. Mukhin, and D. V. Zhdanowich, Nonstandard Quantum Deformations of GL(n) and Constant Solutions of the Yang-Baxter Equation (preprint).Google Scholar
  3. [F-R-T]
    L. Faddeev, N. Reshetikhin, and L. Takhtajan, Quantization of Lie Groups and Lie Algebras, preprint, LOMI -14–87, 1987; Algebra and Analysis, vol.1, no: 1 (1989).Google Scholar
  4. [H]
    M. Höchster, Grassmannians and their Schubert varieties are arithmetically Cohen-Macaulay, J. Alg., Vol. 25 (1973), 40–57.CrossRefGoogle Scholar
  5. [Ho]
    W.V.D. Hodge, Some enumerative results in the theory of forms, Proc. Camb. Phil. Soc., 39 (1943), 22–30.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [J]
    M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10, (1985), 63–69.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [K-R]
    A. Kirillov and N. Reshetikhin, q-Weyl group and multiplicative formula for R-matrices, preprint of Harvard University, January 1990.Google Scholar
  8. [L-R]
    V. Lakshmibai and N. Reshetikhin, Quantum deformations of Flag and Schubert schemes, to appear in Comptes Rendus, Paris.Google Scholar
  9. [L-S]
    V. Lakshmibai and C.S. Seshadri, Geometry of G/P-V, J. Alg. 100 (1986), 462–557.MathSciNetzbMATHGoogle Scholar
  10. [Lo]
    G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math., 70, 237–249 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [M]
    C. Musili, Postulation formula for Schubert varieties, J. Indian Math. Soc. 36 (1972), 143–171.MathSciNetzbMATHGoogle Scholar
  12. [R]
    N. Reshetikhin, Quantized universal enveloping algebras, Yang-Baxter equation and invariants of links, LOMI-preprint, E-4–87, E-17–87.Google Scholar
  13. [Ro]
    M. Rosso, Finite Dimensional Representations of Quantum Analog of the Enveloping Algebra of a Complex Simple Lie Algebra, Comm. Math. Phys. 117 (1988), 581–593.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [S]
    Ya. Soibelman, Algebra of functions on compact quantum group and its applications, Alg. and Anal., Vol. 2, No: 1 190–212 (1990).zbMATHGoogle Scholar
  15. [T-T]
    E. Taft, J. Towber, “Quantum deformation of flag schemes and Grassmann schemes,” 1989-preprint.Google Scholar

Copyright information

© Springer-Verlag Tokyo 1991

Authors and Affiliations

  • V. Lakshmibai
    • 1
  • N. Reshetikhin
    • 2
  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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