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We
introduce the symplectic group _{}2_{}2 over various
noncommutative algebras, which provides new insights into their structure theory.We construct several
geometric spaces, on which the groups _{}2(_{}2(_{}2(_{}2(

**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 *Gv*
of *G*. Using this, we explicitly compute various multiplicities in *G** ^{v}*-modules
in many ways. In particular, we recover the formulas for tensor product
multiplicities of Berenstein-Zelevinsky and generalize them in several
directions. In the case when our geometric multiplicity

**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**e**w**
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 Mathematic**s*, 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. No**t**.* 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}_{}
for non-commuting a and b, Catalan numbers and non-commutative
quadratic equations** (with V. Retakh, C. Reutenauer, D. Zeilberger )

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 *w**o *in *S _{r}_{+1}* on the dual
canonical basis in each simple

**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

**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). Translation: *J. Soviet Math.* **47**
(1989), no. 1.

**Involutions
on Gelfand-Tsetlin patterns and multiplicities in
skew GL(n)-modules**