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**Transitive and Gallai colorings **(with R. M. Adin , J.
Greenstein, J-R Li, A. Marmor, Y. Roichman), preprint.

Gallai coloring of the complete graph is an
edge-coloring with no rainbow triangle. This concept first appeared in the
study of comparability graphs and anti-Ramsey theory. We introduce a transitive
analogue for acyclic directed graphs, and generalize both notions to Coxeter
systems, matroids and commutative algebras.

It is shown that for any finite matroid (or
oriented matroid), the maximal number of colors is equal to the matroid rank.
This generalizes a result of Erdős-Simonovits-Sós
for complete graphs. The number of Gallai (or
transitive) colorings of the matroid that use at most

We count Gallai and
transitive colorings of the root system of type A using the maximal number of
colors, and show that, when equipped with a natural descent set map, the
resulting quasisymmetric function is symmetric and
Schur-positive.

**Twists on rational Cherednik
algebras** (with Y.
Bazlov, E.
Jones-Healey, and A. McGaw), Quarterly Journal
of Mathematics, **74** (2), 2022.

We show that braided Cherednik algebras introduced by the
first two authors are cocycle twists of rational Cherednik algebras of the
imprimitive complex reflection groups *m*,*p*,*n*)* m* is even. This gives a new construction of mystic
reflection groups which have Artin-Schelter regular rings of quantum polynomial
invariants. As an application of this result, we show that a braided Cherednik
algebra has a finite-dimensional representation if and only if its rational
counterpart has one.

**Symplectic groups over noncommutative algebras** (with D. Alessandrini, V. Retakh, E. Rogozinnikov, A. Wienhard) Selecta Mathematica, **28**, 82 (2022)

We
introduce the symplectic group _{}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**, 69 (2021)

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),
*Selecta Mathematica*, **23**, 2017).

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*, **21**, 2015).

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
extensions 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.

**Primitively generated
Hall algebras** (with J. Greenstein),** ***Pacific Journal of Mathemati**c**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. N**o**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.

Dunkl Operators and Canonical Invariants of Reflection 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),

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

We introduce braided Dunkl operators that are acting on a

Braided Doubles and rational Cherednik algebras

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

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.

Lecture notes on geometric crystals and their combinatorial analogues

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

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 Alek 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

**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 Alek 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 Alek 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**