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Alcove path and Nichols-Woronowicz model of the equivariant $K$-theory of generalized flag varieties

Cristian Lenart·Toshiaki Maeno
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Abstract

Fomin and Kirillov initiated a line of research into the realization of the cohomology and $K$-theory of generalized flag varieties $G/B$ as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the $T$-equivariant $K$-theory of a generalized flag variety $K_T(G/B)$ in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for $K_T(G/B)$ due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach.

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Cristian Lenart, Toshiaki Maeno. 2006-08-16. Alcove path and Nichols-Woronowicz model of the equivariant $K$-theory of generalized flag varieties. https://arxiv.org/abs/math/0607136

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