On infinite dimensional grassmannians and their quantum deformations

TL;DR

This paper develops an algebraic framework for infinite dimensional grassmannians and their quantum deformations, including the Sato grassmannian, and explores their structure as homogeneous spaces with quantum group actions.

Contribution

It introduces a novel algebraic approach to define and quantize infinite dimensional grassmannians and extends invariant theory to the quantum infinite dimensional setting.

Findings

Quantum deformations of infinite dimensional grassmannians are constructed.

Both classical and quantum grassmannians are shown to be homogeneous spaces.

An infinite dimensional invariant theory is formulated for quantum groups.

Abstract

An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians, and the Sato grassmannian introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite dimensional version of the first theorem of invariant theory is discussed for both the infinite dimensional special linear group and its quantization.