Degree Cones and Monomial Bases of Lie Algebras and Quantum Groups

Abstract

Abstract. We provide N-filtrations on the negative part U q (n − ) of the quantum group associated to a finite-dimensional simple Lie algebra g, such that the associated graded algebra is a skew-polynomial algebra on n − . The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an N-filtration on any finite dimensional simple g-module. We prove for type A n , C n , B 3 , D 4 and G 2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any g.

Cones from quantum groups to tropical flag varieties

“…The quantum degree cones are defined in [2], motivated by studying the quantum PBW filtration on quantum groups. We keep the notations in LS formula.…”

“…Our link between these two processes is provided by a class of polyhedral cones D q w 0 (g), called quantum degree cones [2,9], depending on the complex simple Lie algebra g and a choice of w 0 , a reduced decomposition of the longest element in the Weyl group of g. For such a fixed reduced decomposition, the negative part U q (n − ) of the quantized enveloping algebra is generated by the quantum PBW root vectors with respect to some non-commutative straightening relations. The quantum degree cone is a kind of Gröbner fan in this non-commutative setup, where the monomial ordering is encoded in w 0 .…”

On toric degenerations of flag varieties

“…In general, varying the function w will change the monoid Γ(S, >), yet in [5] we give a family of such functions such that the global monoid stays constant, i.e., we have always Γ(S, >) = Γ P .…”

“…One obtains ( [5]) that the global essential monoid Γ 1 := Γ(S, >) is finitely generated and saturated.…”

“…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]. As both have a Gröbner background, it might be interesting to study the relation between the quantum degree cone [5] and the tropical Graßmannian [83], or, in general, tropical flag varieties.…”