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On some noncommutative algebras related to K-theory of flag varieties, part I

Anatol N. Kirillov·Toshiaki Maeno
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Abstract

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 {\it Ruijsenaars-Schneider-Macdonald elements}, $RSM$-elements for short, in the corresponding {\it 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 {\it 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 {\it images} of the RSM-elements in the corresponding {\it 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 subalgebra generated by the {\it 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 $A$.

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Anatol N. Kirillov, Toshiaki Maeno. 2006-02-15. On some noncommutative algebras related to K-theory of flag varieties, part I. https://arxiv.org/abs/math/0504290

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