Noncommutative marked surfaces (with V. Retakh), preprint.
The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface Σ. This is a noncommutative algebra AΣ generated by noncommutative geodesics" between marked points subject to certain triangular relations and noncommutative analogues of Ptolemy-Plucker relations. It turns out that the algebra AΣ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of Σ, which confirms its cluster nature". As a surprising byproduct, we obtain a new topological invariant of Σ, which is a free or a 1-relator group easily computable in terms of any triangulation of Σ.   Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.

Generalized adjoint actions (with V. Retakh to appear in Journal of Lie Theory
The aim of this paper is to generalize the classical formula e
­­­­x­y e-x­=Σk≥0­­ 1/k! (ad x)­­­­k­(y). We also obtain combinatorial applications to q-exponentials, q-binomials, and Hall-Littlewood polynomials.

),   Comptes Rendus Mathematique, Doi : 10.1016/j.crma.2015.07.002.
We generalize the decomposition of
Uq(g) introduced by A. Joseph and relate it, for g semisimple, to the celebrated computation of central elements due to V. Drinfeld. In that case we construct a natural basis in the center of Uq(g) whose elements behave as Schur polynomials and thus explicitly identify the center with the ring of symmetric functions.

), Comptes Rendus Mathematique,Vol 353, 5 (2015) Pages 387–390.
The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel and M. Kontsevich.

Double canonical bases (with J. Greenstein), submitted.
We introduce a new class of bases for quantized universal enveloping algebras
Uq(g) and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of Uq(g) and are invariant under many symmetries including all Lusztig's symmetries if g is semisimple. It also turns out that a part of a double canonical basis of Uq(g) spans its center.

Mystic reflection groups (with Y. Bazlov), SIGMA 10 (2014), 040, 11 pages.
This paper aims to systematically study mystic reflection groups that emerged independently in a paper by the authors and in a paper by Kirkman, Kuzmanovich, and Zhang. A detailed analysis of this class of groups reveals that they are in a nontrivial correspondence with complex reflection groups G(m,p,n). We also prove that the group algebras of corresponding groups are isomorphic and classify all such groups up to isomorphism.

Quantum cluster characters of Hall algebras (with D. Rupel), Selecta Mathematica, DOI: 10.1007/s00029-014-0177-3.
The aim of the paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category
C over a finite field F
q and any sequence i of simple objects in C the element XV,i of the corresponding algebra PC,i of q-polynomials. We prove that if C was hereditary, then the assignments V->
XV,i define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the PC,i, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q,d) and i=(i0,i0), where i0 is a repetition-free source-adapted sequence, then we prove that the i-character XV,i equals the quantum cluster character XV introduced earlier by the second author in [29] and [30]. Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper [5] of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in [6].

Cocycle twists and extenstions of braided doubles (with Y. Bazlov)   Contemp. Math. 592 (2013), 1970.
It is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how to define the second cohomology group of an abstract monoidal category C, generalising the Schur multiplier of a finite group and the lazy cohomology of a Hopf algebra, recently studied by Schauenburg, Bichon, Carnovale and others. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in our context. In the second part of the paper we propose an approach to twisting the multiplication in braided doubles, which are a class of algebras with triangular decomposition over G. Braided doubles are not G-graded, but may be embedded in a double of a Nichols algebra, where a twist is carried out. This is a source of new algebras with triangular decomposition. As an example, we show how to twist the rational Cherednik algebra of the symmetric group by the cocycle arising from the Schur covering group, obtaining the spin Cherednik algebra introduced by Wang.

M. Bennett, V. Chari, A. Khoroshkin, S. Loktev), Vol. 20,  2 (2014),  585607.
We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category O of modules for g and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of sln+1.to appear in Pacific Journal of Mathematics.
The aim of the present paper is to demonstrate that Hall algebras of a large class of finitary exact categories behave like quantum nilpotent groups in the sense that they are generated by their primitive elements. Another goal is to construct analogues of quantum enveloping algebras as certain primitively generated subalgebras of the Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

Triangular bases in quantum cluster algebras
(with   A. Zelevinsky), Int. Math. Res. Not. 2012, no. 21, 48214883.
A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a canonical basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig's Lemma that we apply to all acyclic quantum cluster algebras. As a result, we construct the "canonical" basis in every such algebra that we call the canonical triangular basis.

The reciprocal of Σn≥0­­ a­­­­n­bn­ V. Retakh, C. Reutenauer, D. Zeilberger ), Contemp. Math. 592 (2013), 103109.
The aim of this paper is to describe the inversion of the sum Σn≥0­­ a­­­­n­bn where a and b are non-commuting variables as a formal series in a and b. We show that the inversion satisfies a non-commutative quadratic equation and that the number of certain monomials in its homogeneous components equals to a Catalan number. We also study general solutions of similar quadratic equations.
Contemp. Math. 592 (2013) Contemp. Math. 592 (2013), 71102.
The goal of this paper is to construct quantum analogues of Chevalley groups inside completions of quantum groups or, more precisely, inside completions of Hall algebras of finitary categories. In particular, we obtain pentagonal and other identities in the quantum Chevalley groups which generalize their classical counterparts and explain Faddeev-Volkov quantum dilogarithmic identities and their recent generalizations due to Keller.
Advances in Mathematics, Vol 284 (2015), 54–111.
In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra  H*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of G. In particular, our formula gives a combinatorial proof of positivity of the Littlewood-Richardson coefficients in the cases when off-diagonal Cartan matrix entries are less than or equal to -2. Moreover, all our results for the Littlewood-Richardson coefficients extend to the structure coefficients of the T-equivariant cohomology algebra H*T(G/P).

A short proof of Kontsevich cluster conjecture
(with V. Retakh), C. R. Math. Acad. Sci., Paris 349 (2011), no. 3-4, 119–122.
We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K
r:(x,y)→(xyx-1,(1+y-r)x-1) are given by noncommutative Laurent polynomials.
), Transformation Groups, Vol.  16, Issue 4 (2011),  955-1007.
The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each "sufficiently rich'' spherical building Y of type W we associate a certain cohomology theory and verify that, first, it depends only on W (i.e., all such buildings are "homotopy equivalent'') and second, the cohomology ring is the associated graded of the coinvariant algebra of W under certain filtration. We also construct the dual homology "pre-ring'' of Y. The convex "stability'' cones defined via these (co)homology theories of Y are then shown to solve the problem of classifying weighted semistable m-tuples on Y in the sense of Kapovich, Leeb and Millson equivalently, they are cut out by the generalized triangle inequalities for thick Euclidean buildings with the Tits boundary Y. Quite remarkably, the cohomology ring is obtained from a certain universal algebra A by a kind of "crystal limit'' that has been previously introduced by Belkale-Kumar for the cohomology of flag varieties and Grassmannians. Another degeneration of A leads to the homology theory of Y.
Int. Math. Res. Not. 2011, no. 21, 4821-4883.
In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra g
σ along a Dynkin diagram automorphism~sigma of g. For each quantum folding we replace gσ by its Langlands dual (gσ)v and construct a nilpotent Lie algebra n which interpolates between the nilpotent parts of g and (gσ)v, together with its quantized enveloping algebra Uq(n) and a Poisson structure on S(n). Remarkably, for the pair (g, (gσ)v)=(so2n+2,sp2n), the algebra Uq(n) admits an action of the Artin braid group Brn and contains a new algebra of quantum n x n matrices with an adjoint action of Uq(sln), which generalizes the algebras constructed by K. Goodearl and M. Yakimov. The hardest case of quantum folding is, quite expectably, the pair (so8,G2) for which the PBW presentation of Uq(n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each.

Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras (with Yu. BurmanTrans. Amer. Math. Soc., 362 (2010), 229–260.
We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for Coxeter groups – the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.

Dunkl Operators and Canonical Invariants of Reflection Groups (with Yu. Burman), SIGMA 5 (2009), 057, 18 pages
Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.

),  Geometriae Dedicata. 156 (2012), 171-207.
We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.

Noncommutative Dunkl operators and braided Cherednik algebras (with Y. Bazlov14, (2009), 325–372.
We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras for which the above operators form a representation. We classify all braided Cherednik algebras using the theory of braided doubles developed in our previous paper. Besides ordinary rational Cherednik algebras, our classification gives new algebras attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators pairwise anti-commute. We explicitly compute these new operators in terms of braided partial derivatives and divided differences.

Braided Doubles and rational Cherednik algebras (with Y. BazlovAdvances in Mathematics, Vol. 220 (2009) 5,  1466–1530.
We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double — this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds into the braided Heisenberg double attached to the corresponding complex reflection group.

Lie algebras and Lie groups over noncommutative rings
(with V. RetakhAdvances in Mathematics, Vol. 218, 6, (2008),  1723–1758.
The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the F-current Lie algebra (g,A)(F), which is the Lie subalgebra of F\otimes A generated by F\otimes g. In most examples A is the universal enveloping algebra of g. Our description of the current algebra has a striking resemblance to the commutator expansions of F used by M. Kapranov in his approach to noncommutative geometry. We also associate with each Lie algebra (g,A)(F) a "noncommutative algebraic" group G which naturally acts on (g,A)(F) by conjugations and conclude the paper with a number of examples of such groups.

Braided symmetric and exterior algebras (with S. Zwicknagl)  Trans. Amer. Math. Soc., 360  (2008), 3429–3472.
We introduce and study symmetric and exterior algebras in braided monoidal categories such as the category O over quantum groups. We relate our braided symmetric algebras and braided exterior algebas with their classical counterparts.

Lecture notes on geometric crystals and their combinatorial analogues (with D. KazhdanCombinatorial aspect of integrable systems, MSJ Memoirs, 17, Mathematical Society of Japan,  Tokyo, 2007.
This is an exposition of the results on Geometric crystals and the associated Kashiwara crystal bases (presented by the first author in RIMS, August 2004).

Geometric and Unipotent Crystals II: From Unipotent Bicrystals to Crystal Bases (with D. KazhdanContemp. Math., 433, Amer. Math. Soc., Providence, RI, 2007, 13–88.
For each reductive algebraic group G, we introduce and study unipotent bicrystals which serve as a regular version of rational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for Gv-modules, where Gv is the Langlands dual groups. In fact, our  analogues of crystal bases (which we refer to as crystals associated to Gv-modules) are associated to Gv-modules directly, i.e., without quantum deformations.

Noncommutative Double Bruhat cells and their factorizations (with V. RetakhInt. Math. Res. Not.8  (2005),  477–516.
In the present paper we study noncommutative double Bruhat cells. Our main results are explicit positive matrix factorizations in the cells via quasiminors of matrices with noncommutative coefficents.

Quantum cluster algebras (with   A. ZelevinskyAdvances in Mathematics, vol. 195, 2 (2005),  405–455.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky; their study continued in a series of papers including Cluster algebras III: Upper bounds and double Bruhat cells. This is a family of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.

Cluster algebras III: Upper bounds and double Bruhat cells (with S. Fomin and   A. ZelevinskyDuke Math. Journal, vol. 126,  1 (2005),  1–52.
We continue the study of cluster algebras. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.

Tensor product multiplicities, canonical bases and totally positive varieties (with  A. Zelevinsky Invent. Math., vol. 143, 1 (2001), 77–128.
We obtain a family of explicit polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of i-trails which resemble Littelmann's paths but seem to be more tractable. We also study  combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: tropicalization" and geometric lifting."

Geometric and unipotent crystals (with D. KazhdanGeom. Funct. Anal., Special Volume, Part I (2000), 188–236.
We introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal torus of G. The structure of a geometric G-crystal on an  algebraic variety X consists of a rational morphism γ:X–>T and a compatible family ei:Gm×X–>X, i in I  of rational actions of the  multiplicative group Gm  satisfying certain braid-like relations.  Such a  structure induces a rational action of W on X. Surprizingly many interesting rational actions of the group W come  from geometric crystals. Also all the known examples of the action of W which  appear in the construction of Gamma-functions for the representations of  the Langlands dual group Gv in the recent work by A. Braverman and D. Kazhdan come from  geometric  crystals. There are many examples of positive geometric crystals on (Gm)l, i.e., those geometric crystals for which the actions ei and the morphism gamma are given by positive rational expressions.  One can associate to each positive geometric crystal X the Kashiwara's  crystal corresponding to the Langlands dual group Gv.  An emergence of Gv in the "crystal world" was observed earlier by G. Lusztig. Another application of geometric crystals is a construction of trivialization which is an W-equivariant isomorhism X-->γ-1(e)×T for any geometric SLn-crystal. Unipotent crystals are geometric analogues of normal Kashiwara crystals. They form a strict monoidal category. To any unipotent crystal built on a variety X we associate a certain gometric crystal.

Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion (with R. Sjamaar), J. Amer. Math. Soc., 13 (2000), no. 2, 433–466.
In this paper we solve of the following problem: Given a reductive group G, and its reductive subgroup H, describe the momentum cone Δo. This is a rational polyhedral cone spanned by all those dominant G-weights λ for which the simple G-module Vλ contains a non-trivial H-invariant. Our result generalizes the result by Klyachko who has solved this problem for G= GLn×GLn×GLn with the subgroup H=GLn embedded diagonally into G. We describe the facets of the cone Δo in terms of the relative'' Schubert calculus of the flag varieties of the two groups. Another formulation of the result is the description of the relative momentum cone Δ, which is spanned by those pairs (λ,λ') for which the  restriction to H of the simple G-module Vλ contains a simple H-module V'λ'.

Domino tableaux, Schutzenberger involution and action of the symmetric group (with Anatol Kirillov), Proceedings of the 10th International Conference on Formal Power Series and Algebraic Combinatorics, Fields Institute, Toronto, 1998,  Discrete Math., vol. 225, 1–3 (2000),  5–24.

Concavity of weighted arithmetic means with applications (with Alex Vainshtein), Arch. Math. (1997) 69, 120–126.

Total positivity in Schubert varieties (with A. Zelevinsky) Comment. Math. Helv. 72 (1997), no. 1, 128–166.
In this paper we further develop the remarkable parallelism discovered by Lusztig between the canonical basis and the variety of totally positive elements in the unipotent group.

Parametrizations of canonical bases and totally positive matrices (with S. Fomin and A. Zelevinsky), Advances in Mathematics 122 (1996), 49–149.
We provide: (i) explicit formulas for Lusztig's transition maps related to the canonical basis of the quantum group of type A; (ii) formulas for the factorizations of a square matrix into elementary Jacobi matrices; (iii) a family of new total positivity criteria.

Group-like elements in quantum groups and Feigin's conjecture, preprint.
In this paper analogue of the Gelfand-Kirillov conjecture for any simple quantum group Gq is proved (here Gq is the q-deformed coordinate ring of a simple algebraic group G). Namely, the field of fractions of Gq is isomorphic to the field of fractions of a certain skew-polynomial ring. The proof is based on a construction of some group-like elements in Gq (which are q-analogs of elements in G).

Canonical bases for the quantum group of type A_r and piecewise-linear combinatorics (with A. Zelevinsky) Duke Math. J. 82 (1996), no. 3, 473–502.
We use  the structure theory of the dual canonical basis B is to obtain a direct representation-theoretic proof of the Littlewood-Richardson rule (or rather, its piecewise-linear versions discussed above).
Another application of string technique  is an explicit formula for the action of the longest element  wo in Sr+1 on the dual canonical basis in each simple slr+1-module. Having been  translated  into the language of Gelfand-Tsetlin patterns and Young tableaux, this involution coincides with the Schützenberger involution.

String bases for quantum groups of type A_r (with A. Zelevinsky) I. M. Gelfand Seminar, 51–89, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993.
We introduce and study a family of string bases for the guantum groups of type Ar (which includes the dual canonical basis). These bases are defined axiomatically and possess many interesting  properties, e.g., they all are good in the sense of  Gelfand and Zelevinsky. For every string basis, we construct a family of combinatorial labelings by strings. These labelings in a different context appeared in more recent works by M. Kashiwara and by P. Littelmann. We expect that B  has a nice multiplicative structure. Namely, we conjecture in [8] that B contains all products of pairwise q-commuting elements of B. The conjecture was  proved in [8] for A2 and A3. In fact, for r< 4, the dual canonical basis B is the only string basis and it consists of all q-commuting products of quantum minors (for r arbitrary, we proved  that any string basis contains all quantum minors).

Groups generated by involutions, Gel'fand-Tsetlin patterns, and combinatorics of Young tableaux (with Anatol Kirillov), Algebra i Analiz 7 (1995), no. 1, 92–152 (Russian). Translation in St. Petersburg Math. J. 7 (1996), no. 1, 77127.
The original motivation of this paper was to understand a rather mysterious action  of the symmetric group Sn on Young tableaux, discovered by Lascoux and Schutzenberger. We introduced an action of Sn by piecewise-linear transformations on the space of Gelfand-Tsetlin patterns. In our approach, this group appears as a subgroup of the infinite group Gn, generated by quite simple piecewise-linear involutions (these involutions are continuous analogues of Bender-Knuth involutions acting on Young tableaux). The structure of Gn is not yet completely understood. Some relations were given in [7]; they involve the famous Schützenberger involution which also belongs to Gn.  Another result of [7] is a conjectural description of Kashiwara's crystal operators for type A, in terms of Gn.

When is the weight multiplicity equal to 1  (Russian) (with  A. Zelevinsky) Funkc. Anal. Pril. 24 (1990), no. 4, 1–13; translation in Funct. Anal. Appl. 24 (1990), no. 4, 259–269.

Tensor product multiplicities and convex polytopes in partition space(with A. Zelevinsky) J. Geom. Phys. 5 (1988), no. 3, 453–472.

A multiplicative analogue of the Bergstrom inequality for a matrix product in the sense of Hadamard (with   Alex Vainshtein) (Russian) Uspekhi Mat. Nauk 42 (1987), no. 6(258), 181–182. Translation: Russian Mathematical Surveys.

The convexity property of the Poisson distribution and its applications in queueing theory(with Alex Vainshtein and A. Kreinin)  (Russian), Stability problems for stochastic models (Varna, 1985), 17–22, VINITI Moscow, 1986. Translation: J. Soviet Math. 47 (1989), no. 1.

Involutions on Gelfand-Tsetlin patterns and multiplicities in skew GL(n)-modules(with A. Zelevinsky) Soviet Math. Dokl. 37 (1988), no. 3, 799–802.