Quantum Deformations of SL_n/B and its Schubert Varieties

Abstract

In this paper, we prove the results announced in [L-R] for G = SL_n. 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 k_q [X(w)] over k(q), where q is a parameter taking values in k*, as a quantization of k[X(w)], G being SL_n. The algebra k _q [SL _n ]: Let G = SL_n. Let T = (t_ij), 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, e_ij’s are the elementary matrices). Let A(R) be the associative algebra (with 1) generated by {t_ij, 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[M_n], M_n being the space of n × n matrices and k[M_n], the coordinate ring of M_n. 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 k_q [M_n]. 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 (t_ij).

On leave of absence from USSR; LOMI, Fontanka 27, Leningrad, 191011