Geometric realizations of Lusztig's symmetries

Abstract

Abstract. In this paper, we give geometric realizations of Lusztig's symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztig's symmetries.

On the Monoidality of Saito Reflection Functors

“…Proof. By [12, §9.4], the functor T i induces an isomorphism described in [11,Lemma 38.1.3] (see also [18]). Hence, [11,Theorem 39.4.3] (cf.…”

Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

“…Proposition 5.7 ( [21,22]). There exists an isomorphism of A-algebras i λ 0,A : K( i Q 0 ) → i f A such that the following diagram is commutative…”

“…In this section, we shall recall the geometric realization of T i : i f → i f in [21,22]. By using this geometric realization and the structure of K(Q) in last section, we shall give a geometric realization of Lusztig's symmetry T i : U → U.…”

“…So i is a source of Q ′ = σ i Q = (I, H ′ , s, t), where σ i Q is the quiver by reversing the directions of all arrows in Q containing i. For any ν, ν ′ ∈ NI such that ν ′ = s i ν and I-graded K-vector spaces V, V ′ such that dim V = ν, dim V ′ = ν ′ , there exists a functor ( [21,22]) 21,22]). We have the following commutative diagram…”

“…By using the method of Lusztig, Kato gave geometric realizations of Lusztig's symmetries T i : i f → i f in the case of finite type for all i ( [7]). Then his constructions were generalized by Xiao and Zhao to all symmetrizable Cartan datum ( [21,22]).…”