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We
introduce the symplectic group

**Geometric
multiplicities**** **(with Y. Li),
preprint

In this paper, we introduce geometric multiplicities, which
are positive varieties with potential fibered over the Cartan subgroup *H*
of a reductive group *G*. They form a monoidal category and we construct a
monoidal functor from this category to the representations of the Langlands
dual group *G ^{}v*
of

**Langlands
Duality and Poisson-Lie Duality via Cluster Theory and Tropicalization****
**(with A. Alekseev,
B. Hoffman, Y.
Li) Selecta Mathematica,
**27** (4) DOI:
10.1007/s00029-021-00682-x.

Let* G* be a
connected semisimple Lie group. There are two natural duality constructions that
assign to it the Langlands dual group *G ^{v}* and the
Poisson-Lie dual group

In the present
work we study actions of various groups generated by involutions on the
category *O** _{q}^{int}*(

We introduce a
notion of weakly log-canonical Poisson structures on positive varieties with potentials.
Such a Poisson structure is log-canonical up to terms dominated by the
potential. To a compatible real form of a weakly log-canonical Poisson variety
we assign an integrable system on the product of a certain real convex
polyhedral cone (the tropicalization of the variety) and a compact torus. We apply this theory to the dual Poisson-Lie group
G* of a simply-connected semisimple complex Lie group G. We define a positive
structure and potential on *G** and show that the natural Poisson-Lie
structure on *G** is weakly log-canonical with respect to this positive structureand potential. For *K*⊂*G* the compact real form, we show that the
real form *K**⊂*G** is compatible and prove that the corresponding integrable
system is defined on the product of the decorated string cone and the compact torus of dimension 1/2(dim *G -* rank *G*).

**Noncommutative Catalan numbers****
**(with V. Retakh) Annals of Combinatorics,
Vol. 23, Issue 3–4 (2019), 527–547.

The goal
of this paper is to introduce and study noncommutative Catalan numbers *C _{n}*
which belong to the free Laurent polynomial algebra in

**Factorizable
module algebras**** **(with K. Schmidt) * **Int. Math.
Res. Not.* **2019**
(21), 6711–6764 (2019).

The aim of this paper is to introduce and study a large class
of ** g**-module algebras which we call factorizable by generalizing
the Gauss factorization of (square or rectangular) matrices. This class
includes coordinate algebras of corresponding reductive groups

Let

**Canonical bases of quantum
Schubert cells and their symmetries** (with J. Greenstein),
*J. Sel.
Math. **N**ew Ser.*,
doi:10.1007/s00029-017-0316-8.

The goal
of this work is to provide an elementary construction of the canonical basis
*B**(**w)* in each quantum
Schubert cell *U** _{q}*(w) and to establish its invariance
under modified Lusztig’s symmetries. To that effect, we obtain a direct
characterization of the upper global basis

**Noncommutative marked
surfaces **(with V.
Retakh), *Advances in
Mathematics*, Vol
328 (2018), 1010–1087.

The aim of the paper is to attach a noncommutative
cluster-like structure to each marked surface Σ. This is a noncommutative
algebra A_{Σ} 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 A_{Σ}
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), *Journal of Lie Theory*, **26** (2016), No. 1, 219–225.

The aim of this paper is to generalize
the classical formula e^{x}ye^{-x}=Σ_{k≥0}
1/k! (ad x)^{k}(y). We also obtain
combinatorial applications to ** q**-exponentials,

We generalize the decomposition of

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), *Advances in Mathematics*,Vol. 316 (2017), 54–111.

We introduce a new class of bases for quantized universal
enveloping algebras *U _{q}*

**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

**Cocycle
twists and extenstions of braided doubles**
(with Y. Bazlov), *Contemp**. Math.*, **592** (2013), 19–70.

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.

**Macdonald Polynomials and BGG
reciprocity for current algebras **(with M. Bennett,** **V. Chari, A. Khoroshkin, S. Loktev), *Selecta Mathematica*,
Vol. 20, **2** (2014), 585–607.

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

**Primitively
generated Hall algebras** (with J. Greenstein),**
***Pacific Journal of Mathematics*, Vol. 281, No. 2, 2016.

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, 4821–4883.

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*_{n≥0}*a ^{n}_{}b^{n}*

The aim of this paper is to describe the
inversion of the sum *Σ _{n≥0}*

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.

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

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group

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**** (**with M. Kapovich), *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. Bazlov) * **Selecta
Mathematica*, **14**, (2009),
325–372.

We introduce braided Dunkl operators that are acting on a ** q**-polynomial algebra and

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.

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

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 algebras 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).

For each reductive algebraic group

**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
coefficients.

**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 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**àT* 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

**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 *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 *string bases*
for the quantum groups of type *A _{r}*
(which includes the dual canonical basis). These bases are defined
axiomatically and possess many interesting properties, e.g., they all are

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.