Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Abstract

Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T _i on U for all i ∈ I. Since T _i (f) is not contained in f, Lusztig considered two subalgebras _i f and ⁱ f of f for any i ∈ I, where _i f={x ∈ f | T _i (x) ∈ f} and ${^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\,|\,\,T^{-1}_{i}(x)\in \mathbf {f}\}$. The restriction of T _i on _i f is also denoted by $T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}$. The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of _i f, ⁱ f and $T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}$ by using the language ’quiver with automorphism’ introduced by Lusztig.