# Geometric realizations of Lusztig's symmetries on the whole quantum groups

## TL;DR

This paper explores the geometric realization of Lusztig's symmetries across entire quantum groups by analyzing the Grothendieck group of Lusztig's perverse sheaves and establishing a decomposition theorem.

## Contribution

It introduces a decomposition theorem for the Grothendieck group and extends Lusztig's symmetries from the positive part to the entire quantum group using geometric methods.

## Key findings

- Decomposition theorem for the Grothendieck group
- Geometric realization of Lusztig's symmetries on the whole quantum group
- Extension of symmetries from positive part to entire quantum group

## Abstract

In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztig's perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztig's symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztig's symmetries on the whole quantum group.