Cones from quantum groups to tropical flag varieties

We relate quantum degree cones, parametrizing PBW degenerations of quantized enveloping algebras, to (negative tight monomial) cones introduced by Lusztig in the study of monomials in canonical bases, to K-theoretic cones for quiver representations, and to some maximal prime cones in tropical flag varieties.


Introduction
The aim of this work is to point out a rather unexpected relation between two substantially different degeneration processes arising in the algebraic Lie theory.
The first is the study of toric degenerations of flag varieties G/B of complex semisimple Lie algebras, the totality of which is encoded in tropical flag varieties with respect to Plücker embeddings. The second is the PBW-type degenerations of quantized enveloping algebras U q (g) associated to complex semisimple Lie algebras.
Our link between these two processes is provided by a class of polyhedral cones D q w 0 (g), called quantum degree cones [2,8], 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 . Interior points of such cones give non-commutative degenerations of U q (n − ) to skew-polynomial algebras. As in the commutative Gröbner theory, due to lacking of knowledge on the straightening relations, the facets of the quantum degree cones are usually hard to describe.
The first main result of the present work is an embedding (conjectured to be an equality) of the quantum degree cone into the negative tight monomial cone of the Langlands dual Lie algebra, which is a variant of a cone introduced by Lusztig [10] for the identification of monomial elements in the canonical basis of U q (n − ). This result is based on a detailed analysis of a result of Levendorskii and Soibelman [11] on commutators of quantum PBW root vectors, for which we identify a certain PBW basis element appearing in this commutator with non-vanishing coefficient.
The second main result is a proof of the conjecture for reduced decompositions adapted to a Dynkin quiver. For the proof, we interpret the cone D q w 0 (g) in terms of the K-theory of quiver representations, and use known results on degenerations and extensions of representations of Dynkin quivers [3].
Date: December 6, 2019. 1 Tropical flag varieties [4] are subfans of Gröbner fans of flag varieties with respect to Plücker embeddings. We observe that, via a result in [7], the quantum degree cone arising from type A and a specific choice of reduced decomposition can be identified with a maximal prime cone in the corresponding tropical flag variety. This observation is quite surprising, as both cones arise from apparently different Gröbner theory: the straightening relations in the quantum setup become trivial after specialisation!
The authors hope that the present observation can be further developed into a more general link allowing the study of tropical flag varieties using the piece-wise linear combinatorics arising in quantized enveloping algebras.
The paper is organized as follows: in Section 2, we first recall the construction of PBW bases in quantized enveloping algebras and state the commutation formula for root elements. Most of this section is devoted to the proof of the non-vanishing result (Theorem 3) stated above. This result allows us to relate the quantum degree cone and the negative tight monomial cone in Section 3, and to formulate our main conjecture (Conjecture 1). In Section 4 we review basic results on the representation theory of Dynkin quivers, and on the relation of their associated Hall algebras to the quantized enveloping algebra U q (n − ). This allows us to give a K-theoretic interpretation of the quantum degree cone, and to prove our conjecture (Theorem 4) in many cases. Finally, in Section 5 we rephrase the main result of [7] in the present language, identifying a maximal prime cone in the tropical flag variety with a particular quantum degree cone.
Acknowledgement. X.F. would like to thank Gleb Koshevoy on discussions related to the tight monomial cones.

Commutation relations of quantum PBW root vectors
2.1. Basics on Lie algebras and quantum groups. Let g be a simple Lie algebra of rank n, with Cartan matrix A = (a ij ) n×n . Let α 1 , . . . , α n be the simple roots and ∆ + be the set of positive roots. The root lattice will be denoted by Q and the weight lattice by P. Let (−, −) be the W -invariant scalar product on the weight lattice such that: for any short root α, (α, α) = 2; when g = B n , C n , F 4 , for any long root β, (β, β) = 4; for g = G 2 and the long root γ, (γ, γ) = 6.
Let U q (g) be the generic quantum group associated to g. It is generated as an algebra by E i , F i and K ±1 Details can be found in Chapter 37 of [9]. The quantum PBW root vector F βt associated to β t is defined by: The PBW theorem of quantum groups says that the set is a linear basis of U q (n − ) ( [9], Corollary 40.2.2).
Since the total order β 1 < β 2 < . . . < β N is convex, by specializing q to 1, we know that in the LS-formula (2.1), if there exists k < r < ℓ such that β r = β k + β ℓ , then the monomial F βr appears on the right hand side of (2.1) with non-zero coefficient.
These are all information that can be read from the specialization q → 1. It is natural to ask: what are the monomials appearing on the right hand side of (2.1) with non-zero coefficients?
The goal of the rest of this section is to show that particular monomials may appear in some situations. This will be applied later to study the quantum degree cones.
For 1 ≤ k ≤ N and s ∈ N, we define k[s] := t if t > k such that i t = i k and the number of i k appearing in {i k+1 , . . . , i t } is exactly s; otherwise we set it to be zero. Hence k[1] = ℓ implies that i k = i ℓ and for any k < p < ℓ, i p = i k .
The rest of this section will be devoted to prove this theorem. First observe that by applying T −1 i k−1 . . . T −1 i 1 , we can suppose that k = 1. By assumption, i ℓ = i 1 .

2.3.
Preparations to the proof. We keep the notations in the statement of the theorem and start deriving some combinatorial formulas on roots. Lemma 1. The following statements hold: Proof.
(1) By the definition of β ℓ , we have: Iterating this computation gives As β 1 = α i 1 , the formula is proved. (2) It suffices to notice that there is no i 1 in the set {i 2 , . . . , i ℓ−2 }, and the expression The same argument as in (1) shows that Since for any 2 ≤ s ≤ ℓ −1, c s ≥ 0, it suffices to show that c 2 β 2 −α i 1 is a positive root. We have As The following lemma will be used later.
Lemma 2. Suppose that c 2 = 0 and let X : (2) In the following expression of X: Proof.
Moreover, we can assume that c 2 = 0 and c ℓ−1 = 0, since otherwise the statement holds by induction hypothesis.
We start with considering the quantum PBW root vector F β ℓ : We suppose that . The commutator of quantum PBW root vectors gives By Theorem 2, we have does not appear with non-zero coefficient in Σ 1 . Indeed, by Lemma 2, writing X in the PBW basis, F β ℓ−1 does not appear in any monomial. Hence commuting F (r) β ℓ−1 with X using Theorem 2 will not producing any element in the PBW basis with the power of F β ℓ−1 higher than r. Again by Theorem 2, commuting F (s) β ℓ−1 and F β 1 will produce elements in the PBW basis with the power of F β ℓ−1 strictly less than s. Combining them together proves the claim.
The same argument shows that, if we write using Theorem 2 β ℓ−1 does not appear in Σ 2 with non-zero coefficient. As a consequence, to study the coefficient of F We simplify the power of q in the bracket: by the proof of Lemma 1 (1), The term in the bracket now reads By induction hypothesis, in this commutator, when written in the PBW basis, there exists a term λF with λ = 0. From this we obtain the following term in the above sum By applying Theorem 2 again, we obtain where µ = ℓ−2 t=2 c t r(β t , β ℓ−1 ), and the monomial F It suffices to show that this term is non-zero. We first simplify the power of q: apply Lemma 1 (1) and replace s = c ℓ−1 − r, where for the third equality, the W -invariance is used. Since both λ and q −(β 1 ,β ℓ−1 ) are non-zero constants, to show that the coefficient (2.4) is non-zero, it suffices to show that (2.5) Here we used the fact that c ℓ−1 = −d −1 i ℓ−1 (α i ℓ−1 , α i 1 ). By the assumption, c ℓ−1 = 0, hence in the case where the rank of g is higher than 2, c ℓ−1 = 1 or 2. If c ℓ−1 = 1, the equation (2.5) reads 1 − q 2 i ℓ−1 . If c ℓ−1 = 2, the equation In both cases, they are non-zero. This terminates the proof of Theorem 3.

Remark 1.
It is clear from the proof that Theorem 3 still holds when q is not a small root of unity.

Quantum degree cone and Lusztig tight monomial cone
3.1. Lusztig tight monomial cones. In [10] Lusztig defined a polyhedral cone in the simply-laced cases in order to study monomials in the canonical basis. The natural generalisations to all Lie types are given in [5].
Let w 0 = s i 1 · · · s i N be a reduced decomposition of w 0 . The Lusztig tight monomial cone L w 0 (g) ⊂ R N is the polyhedral cone defined by the following inequalities: (i) for any two indices 1 ≤ p < p ′ ≤ N with p ′ = p[1], we have (ii) for 1 ≤ p ≤ N, x p ≥ 0. It is shown in [5] that the cone L w 0 (g) is simplicial.. We will denote L − w 0 (g) ⊂ R N , termed a negative tight monomial cone, the polyhedral cone defined by (i)' for any two indices 1 ≤ p < p ′ ≤ N with p ′ = p[1], we have The cone L − w 0 (g) is a product of a linearity space of dimension n and a simplicial cone of dimension N − n by setting the coordinates associated to simple roots to zero.

3.2.
Quantum degree cones. 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. The quantum degree cone D q w 0 (g) associated to a reduced decomposition w 0 is defined by: n k d β k if c(n i+1 , · · · , n j−1 ) = 0}.
We will denote D q w 0 (g) := D q w 0 (g) ∩ R ∆ + ≥0 its non-negative part. Our notation here is slightly different to [2]: the quantum degree cone therein is the interior of the nonnegative cone D q w 0 (g). It is proved in loc.cit. that D q w 0 (g) is non-empty, and any d ∈ D q w 0 (g) defines a filtered algebra structure on U q (n − ) by letting deg(F β i ) = d β i . The associated graded algebra is isomorphic to a skew-polynomial algebra if and only if d is contained in the interior of D q w 0 (g).

3.3.
Relations between two cones. For a fixed w 0 , we identify As a consequence of Theorem 3, for a fixed w 0 , we obtain an embedding of the quantum degree cone to the negative tight monomial cone. We denote g L the Langlands dual of the Lie algebra g. Corollary 1. For any w 0 ∈ R(w 0 ), we have D q w 0 (g) ⊆ L − w 0 (g L ). We conjecture that these two cones coincide.
Conjecture 1. For any w 0 ∈ R(w 0 ), we have D q w 0 (g) = L − w 0 (g L ). In the next section we prove this conjecture in the case where g is simply-laced and w 0 is compatible with a quiver.

Quiver representations and Hall algebras
A basic reference on quiver representations and Auslander-Reiten theory is [1]. For Hall algebras, we refer to [13], [8].
4.1. Quiver representations. Let K be a field. We fix a Dynkin quiver of type A, D, or E with vertices I and the number of arrows between two different vertices i and j (in either direction) equals −a ij .
Let KQ be the path algebra of Q, mod(KQ) be the category of finite dimensional KQ-modules.
We denote K 0 := K 0 (KQ) the Grothendieck group of KQ, which can be identified with Z I by mapping the class of a module to its dimension vector.
Let The simple modules S i in mod(KQ) are parametrized by the vertices i ∈ I. This identifies K 0 = Z I with the root lattice of g, the Lie algebra having the underlying graph of Q as Dynkin diagram, by sending the dimension vector of S i to the simple root α i . The indecomposable modules are parametrized by the positive roots in the root lattice: for each α ∈ ∆ + , there exists a unique (up to isomorphism) indecomposable module U α having dimension vector α.
Notice that taking the dimension vector induces a map dim : We consider the category rep K Q of finite dimensional K-representations of Q, which is an abelian K-linear category of global dimension at most one. We denote by dimM the dimension vector of a representation M, viewed as an element of the root lattice via The Euler form of Q is defined by: We denote its symmetrization by (−, −), which is the bilinear form defined by the Cartan matrix.

4.2.
Auslander-Reiten quivers arising from reduced decompositions. The category mod(KQ) is representation-directed, which means that there exists an enumeration β 1 , · · · , β N of the positive roots such that Hom(U β k , U β l ) = 0 for k > l and Ext 1 (U β k , U β l ) = 0 for k ≤ l.
We will write U k := U β k for short. It is known that there exists a sequence i 1 , · · · , i N in I such that w 0 := s i 1 · · · s i N is a reduced decomposition in R(w 0 ), and such that β k = s i 1 · · · s i k−1 (α i k ) for all k = 1, · · · , N.
Many representation theoretical information can be recovered from this reduced decomposition.
(1) The Auslander-Reiten quiver can be constructed in the following way: -the vertices are 1, 2, · · · , N; -there are arrows k → ℓ and ℓ → k[1] (when k [1] exists) if there exists an arrow i ℓ → i k in Q, and ℓ is minimal along those indices larger then k with this property; -there is a translation i k We define a partial order U V on the indecomposable modules if there exists a path from U to V in the Auslander-Reiten quiver. By the above argument, this partial order is compatible with the directed enumeration, that is to say, if U k U ℓ then k ≤ ℓ.

Hall algebras.
We consider the (generic) Hall algebra H(Q) of Q. It is the Q(q)algebra with basis F [V ] for V ranging over all isomorphism classes of finite dimensional KQ-modules. The multiplication is defined by: where H X V,W is the Hall polynomial counting the number of sub-modules U of X which are isomorphic to W , with quotient X/U isomorphic to V over F q .
We summarises some basic facts about these Hall polynomials: . The algebra H(Q) is isomorphic to U q (n − ) [13]. The isomorphism is given by Given a finite dimensional KQ-module M, we can decompose it as Then in the Hall algebra we have . For two indecomposable modules U V , their q-commutator is given by: = terms coming from non-split extensions of V by U .
(2) The other special case is when U = τ V is the Auslander-Reiten translation of V . In this case there exists exactly one non-split extension X of V by U given by the middle term of the Auslander-Reiten sequence, and the q-commutator can be computed as We translate these results in H(Q) to U q (n − ) using the algebra isomorphism described above. Under this isomorphism, F [U k ] is sent to the quantum PBW root vector F β k corresponding to the reduced decomposition arising from the enumeration of positive roots. Then the above formulae can be rewritten as We describe the terms appearing in the q-commutator.
Lemma 3. The following statements are equivalent: (2) X is a non-split extension of V by U; (3) X properly degenerates to U ⊕ V ; for all indecomposable modules Z such that τ −1 U ′ Z ≺ V , with strict inequality for at least one such Z.
Proof. By Lemma 3, the first and the third conditions are necessary is clear. For the second condition, note that every such Y has to admit a non-zero map from U and to V , otherwise it would split off from the exact sequence 0 → U → X → V → 0, which is impossible since U and V are indecomposable. To

4.4.
Comparison of cones: quiver-compatible case. Taking the dimension vector induces a map dim : K ⊕ 0 → K 0 . Let Λ denote its kernel. Then we have the short exact sequence of free abelian groups 0 → Λ → K ⊕ 0 → K 0 → 0. Dualizing by taking (−) * := Hom(−, Z) we obtain an short exact sequence Notice that Λ * consists of additive functions on K ⊕ 0 modulo those functions depending only on the dimension vectors of modules.  Proof. Recall that  [3], this implies that V degenerates to W . By [3,Corollary 4.2], it is equivalent to V ≤ ext W , which is defined as follows (see [3, Section 1]): there exists a sequence V = V 0 , V 1 , · · · , V s = W of KQ-modules such that for any k = 0, 1, · · · , s − 1, there exists a short exact sequence with middle term V k , whose end terms add up to V k+1 . This shows that in the following expression

5.2.
Tropical complete flag varieties. We fix an integer n > 1. Let SL n be the special linear group defined over C and B ⊂ SL n be the Borel sub-group of upper triangular matrices in SL n . Let F l n := SL n /B be the complete flag variety and C n be the natural representation of SL n . We consider the Plücker embedding of F l n into a product of projective spaces F l n ֒−→ n−1 k=1 P(Λ k C n ).
We fix the standard basis e 1 , · · · , e n of C n and I = {i 1 , · · · , i k } such that 1 ≤ i 1 < · · · < i k ≤ n. The linear function P I := (e i 1 ∧ e i 2 ∧ · · · ∧ e i k ) * ∈ (Λ k C n ) * is called a Plücker coordinate on the product of projective spaces. The complete flag variety F l n is defined by the Plücker relations in this product of projective spaces.
Let trop(F l n ) be the tropical variety associated to the complete flag variety with respect to the Plücker embedding. More details and some small dimension examples of trop(F l n ) can be found in [4].
Then one of the main theorems in [7] can be rephrased as: ). The map ϕ maps D q w max 0 (g) bijectively onto the closure of a maximal cone in trop(F l n ).

Remark 2.
(1) The maximal prime cone obtained as the image of ϕ gives the flat toric degeneration of G/B to the toric variety associated to the Feigin-Fourier-Littelmann-Vinberg polytopes [7]. (2) It is shown in [12] that changing the directions of the inequalities in the example in Section 4.5 identifies other maximal prime cones in tropical flag varieties.