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.

We introduce a new class of bases for quantized universal enveloping algebras $U_q(\mathfrak g)$ and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of U

Mystic reflection groups (with Y. Bazlov),

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

Quantum cluster characters of Hall algebras

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 Fq 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.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.

Primitively generated Hall algebras

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

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 Σ

The aim of this paper is to describe the inversion of the sum Σ

Quantum Chevalley groups

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.

Littlewood-Richardson coefficients for reflection groups

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*

A short proof of Kontsevich cluster conjecture

We give an elementary proof of the Kontsevich conjecture that asserts that the iterations of the noncommutative rational map K

Stability inequalities and universal Schubert calculus of rank 2

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.

Quantum folding

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

**Quasiharmonic
polynomials for Coxeter groups and representations of Cherednik
algebras **(with Yu. Burman)
Trans. 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.

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.

Affine buildings for dihedral groups

We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group.

Noncommutative Dunkl operators and braided Cherednik algebras (with Y. Bazlov) Selecta Mathematica, 14, (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. Bazlov) Advances 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

The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra

**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.

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. Kazhdan)
Contemp. 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 *birational
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 *G** ^{v}*-modules,
where

**Noncommutative
Double Bruhat cells and their factorizations **(with V. Retakh) Int. 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.
Zelevinsky)
Advances 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.
Zelevinsky)
Duke 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. Kazhdan) Geom. 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 *e** _{i}*:

**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*
Δ_{o}.
This is a rational polyhedral cone spanned by all those dominant *G*-weights
λ for
which the simple *G*-module *V*_{λ} contains a non-trivial *H*-invariant.
Our result generalizes the result by Klyachko who has solved this
problem
for *G*=* **GL _{n}*×

**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**, to appear
in *J.
Algebra*.

In this paper analogue of the Gelfand-Kirillov conjecture for any
simple quantum group *G _{q}*
is proved (here

**Canonical bases for
the
quantum group of type A_r and piecewise-linear combinatorics**
(with A.
Zelevinsky)

We use the structure theory of the dual canonical basis

Another application of string technique is an explicit formula for the action of the longest element

**String bases for
quantum
groups of type A_r** (with A.
Zelevinsky)

We introduce and study a family of

**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, 77-127

The original motivation of this paper was to understand a rather
mysterious action of the symmetric group *S _{n}* on Young
tableaux, discovered by
Lascoux and Schutzenberger. We introduced an action of

**Triple
multiplicities
for sl(r+1) and the spectrum of the exterior algebra of the
adjoint
representation** (with A.Zelevinsky)

**When is the weight
multiplicity
equal to 1 **(Russian) (with A.
Zelevinsky)

**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)