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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.MathSciNetMATHCrossRefGoogle Scholar
  6. [J]
    M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10, (1985), 63–69.MathSciNetMATHCrossRefGoogle 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.MathSciNetMATHGoogle Scholar
  10. [Lo]
    G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math., 70, 237–249 (1988).MathSciNetMATHCrossRefGoogle Scholar
  11. [M]
    C. Musili, Postulation formula for Schubert varieties, J. Indian Math. Soc. 36 (1972), 143–171.MathSciNetMATHGoogle 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.MathSciNetMATHCrossRefGoogle 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).MATHGoogle 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

Personalised recommendations