On Some Noncommutative Algebras Related with K - Theory of Flag Varieties , I

For any Lie algebra of classical type or type G 2 we define a K-theoretic analog of Dunkl's elements, the so-called truncated Ruijsenaars-Schneider-Macdonald elements, RSM-elements for short, in the corresponding Yang-Baxter group, which form a commuting family of elements in the latter. For the root systems of type A we prove that the subalgebra of the bracket algebra generated by the RSM-elements is isomorphic to the Grothendieck ring of the flag variety. In general, we prove that the subalgebra generated by the images of the RSM-elements in the corresponding Nichols-Woronowicz algebra is canonically isomorphic to the Grothendieck ring of the corresponding flag varieties of classical type or of type G 2. In other words, we construct the " Nichols-Woronowicz algebra model " for the Grothendieck Calculus on Weyl groups of classical type or type G 2 , providing a partial generalization of some recent results by Y. Bazlov. We also give a conjectural description (theorem for type A) of a commutative subalge-bra generated by the truncated RSM-elements in the bracket algebra for the classical root systems. Our results provide a proof and generalizations of recent conjecture and result by C. Lenart and A. Yong for the root system of type A.