PBW-type filtration on quantum groups of type A

Abstract

Abstract. We will introduce an N-filtration on the negative part of a quantum group of type An, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module. This construction can be viewed as a quantum analog of the classical PBW framework, and in fact, by considering the classical limit, this basis is the monomial basis provided by Feigin, Littelmann and the second author in the classical setup.

On toric degenerations of flag varieties

“…It is shown in [25] that the essential basis associated to these data coincides with the one arising from the classical PBW filtration (Section 6.4).…”

“…Is it possible to extend the methods using birational sequences to affine Kac-Moody groups [29]? Is it possible to generalize the filtration aspect to the setting of quantum groups [25]? Some recent work in this direction can be found in [5].…”

Computing Toric Degenerations of Flag Varieties

“…A similar idea for a more general case is carried out in [23, §7]. For the FFLV polytope we compute weight vectors for Fℓ n with n = 4, 5 (see Example 6) following a construction given in [14].…”

“…, 4} yields the FFLV polytope from Definition 11 with respect to the embedding Fℓ 4 ֒→ P(V (ρ)). Then we define linear forms e min (m J ) = m 1 + 2m 2 + m 3 + 2m 4 + m 5 + m 6 , e reg (m J ) = 3m 1 + 4m 2 + 2m 3 + 3m 4 + 2m 5 + m 6 , according to the degrees defined in [14]. We obtain in analogy to Definition 12 the corresponding weight vectors w min = (0, 2, 2, 1, 0, 1, 1, 2, 1, 2, 0, 1, 1, 1), 3, 4, 3, 0, 2, 2, 4, 3, 5, 0, 1, 2, 3).…”

Degree Cones and Monomial Bases of Lie Algebras and Quantum Groups

“…This important property distinguishes the canonical basis from other bases of U(n − ) and V (λ). In this paper, motivated by [FaFoR,FeFoL1,FeFoL2], we are interested in the existence of monomial bases E of U(n − ) satisfying the following properties: (P1) there exists an N-filtration F on U(n − ) such that the associated graded algebra is the polynomial algebra S(n − ); the set E is a linear basis of the associated graded algebra; (P2) let V F (λ) be the associated graded S(n − )-module with cyclic vector v F λ ,…”