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of quantum Schubert cells and their symmetries
goal of this work is to provide an elementary construction of the
canonical basis B(w) in each quantum Schubert cell Uq(w)
and to establish its invariance under modified Lusztig's symmetries. To
that effect, we obtain a direct characterization of the upper global
basis Bup in terms of a suitable bilinear
form and show that B(w)
is contained in Bup
and its large part is preserved by modified Lusztig's symmetries.
marked surfaces (with
V. Retakh), submitted.
The aim of the paper is to
attach a noncommutative cluster-like
structure to each marked surface Σ.
This is a noncommutative
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,
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.
(with V. Retakh), Journal
of Lie Theory, 26 (2016), No. 1, 219--225.
The aim of this paper is to
generalize the classical formula exy e-x=Σk≥0
1/k! (ad x)k(y).
obtain combinatorial applications to q-exponentials, q-binomials, and
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.
Greenstein, D. Kazhdan),
Comptes Rendus Mathematique,Vol
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
introduce a new class of bases for quantized universal enveloping
other doubles attached to semisimple
and Kac-Moody Lie algebras. These bases contain dual canonical bases of
upper and lower halves of Uq(g)
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)
(with Y. Bazlov),
SIGMA 10 (2014), 040, 11 pages.
This paper aims to
systematically study mystic reflection groups that
independently in a paper by the authors and in a paper by Kirkman,
and Zhang. A detailed analysis of this class of groups reveals that
are in a nontrivial correspondence with complex reflection groups G(m,p,n).
We also prove that the group algebras of corresponding groups are
and classify all such groups up to isomorphism.
characters of Hall algebras
Mathematica, DOI: 10.1007/s00029-014-0177-3.
The aim of the paper is to introduce a generalized quantum cluster
which assigns to each object V
a finitary Abelian category C
over a finite field Fq
and any sequence i
in C the element XV,i of the
algebra PC,i of q-polynomials. We prove that if C was hereditary, then the
define algebra homomorphisms from the (dual) Hall-Ringel
algebra of C
to the PC,i,
which generalize the well-known Feigin homomorphisms from the
of a quantum group to q-polynomial
C is the
category of an acyclic valued quiver (Q,d) and i=(i0,i0),
is a repetition-free source-adapted sequence,
we prove that the i-character
equals the quantum cluster character XV
earlier by the second author in  and . Using this
we deduce a quantum cluster structure on the quantum unipotent cell
to the square of a Coxeter element. As a corollary, we prove a
from the joint paper  of the first author with A. Zelevinsky for
quantum unipotent cells. As a byproduct, we construct the quantum twist
prove that it preserves the triangular basis introduced by A.
and the first author in .
extenstions of braided doubles (with Y. Bazlov)
Math. 592 (2013), 19–70.
is well known
central extensions of a group G
to 2-cocycles on G. Cocycles
be used to construct extensions of G-graded
algebras via a version of the Drinfeld twist introduced by Majid. We
how to define the second cohomology group of an abstract monoidal
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
of Nichols algebras in C, and
how the recent work on twists of Nichols algebras by Andruskiewitsch,
Garcia and Vendramin fits in our context. In the second part of the
we propose an approach to twisting the multiplication in braided
which are a class of algebras with triangular decomposition over G.
Braided doubles are not G-graded, but may be embedded in a
of a Nichols algebra, where a twist is carried out. This is a source of
algebras with triangular decomposition. As an example, we show how to
the rational Cherednik algebra of the symmetric group by the cocycle
from the Schur covering group, obtaining the spin Cherednik algebra
and BGG reciprocity for current algebras (with M. Bennett,
S. Loktev), Selecta
Mathematica, Vol. 20, 2 (2014), 585–607.
study the category of graded representations with finite--dimensional
pieces for the current algebra associated to a simple Lie algebra. This
has many similarities with the category O of
and in this paper, we use the combinatorics of Macdonald polynomials to
an analogue of the famous BGG duality in the case of sln+1.
generated Hall algebras
Greenstein), Pacific Journal
of Mathematics, Vol. 281, No. 2, 2016.
aim of the present paper is to demonstrate that Hall algebras of a
class of finitary exact categories behave like quantum nilpotent groups
the sense that they are generated by their primitive elements. Another
is to construct analogues of quantum enveloping algebras as certain
generated subalgebras of the Hall algebras and conjecture an analogue
"Lie correspondence" for those finitary categories.
bases in quantum cluster algebras (with A.
Int. Math. Res. Not. 2012, no.
lot of recent activity has been directed towards various constructions
"natural" bases in cluster algebras. We develop a new approach to this
which is close in spirit to Lusztig's construction of a canonical
and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke
The key ingredient of our approach is a new version of Lusztig's Lemma
we apply to all acyclic quantum cluster algebras. As a result, we
the "canonical" basis in every such algebra that we call the canonical
The reciprocal of Σn≥0 anbn for non-commuting a and b, Catalan numbers and
quadratic equations (with
Retakh, C. Reutenauer, D. Zeilberger ), Contemp.
Math. 592 (2013), 103–109.
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
series in a and b.
We show that the inversion satisfies a non-commutative quadratic
and that the number of certain monomials in its homogeneous components
to a Catalan number. We also study general solutions of similar
groups (with J. Greenstein),
Math. 592 (2013) Contemp.
Math. 592 (2013), 71–102.
goal of this paper is to construct quantum analogues of Chevalley
inside completions of quantum groups or, more precisely, inside
of Hall algebras of finitary categories. In particular, we obtain
and other identities in the quantum Chevalley groups which generalize
classical counterparts and explain Faddeev-Volkov quantum dilogarithmic
and their recent generalizations due to Keller.
coefficients for reflection groups (with E. Richmond),
in Mathematics, Vol
284 (2015), 54–111.
In this paper we
compute all Littlewood-Richardson coefficients for semisimple or
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
in enumerative geometry, algebraic combinatorics and representation
Our formula for the Littlewood-Richardson coefficients is purely
and is given in terms of the Cartan matrix and the Weyl group of G. In particular, our formula
a combinatorial proof of positivity of the Littlewood-Richardson
in the cases when off-diagonal Cartan matrix entries are less than or
to -2. Moreover,
our results for the Littlewood-Richardson coefficients extend to the
coefficients of the T-equivariant
cohomology algebra H*T(G/P).
of Kontsevich cluster conjecture (with V. Retakh), C. R. Math. Acad. Sci.,
349 (2011), no. 3-4, 119–122.
We give an elementary proof of the Kontsevich conjecture that asserts
the iterations of the noncommutative rational map Kr:(x,y)→(xyx-1,(1+y-r)x-1)
are given by
and universal Schubert calculus of rank 2 (with M.
Groups, Vol. 16,
Issue 4 (2011), 955-1007.
The goal of the
is to introduce a version of Schubert calculus for each dihedral
group W. That is, to each
rich'' spherical building Y
W we associate a certain
cohomology theory and verify that, first, it depends only on W (i.e., all such buildings are
equivalent'') and second, the cohomology ring is the associated graded
the coinvariant algebra of W
certain filtration. We also construct the dual homology "pre-ring'' of Y. The convex "stability'' cones
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
Millson equivalently, they are cut out by the generalized triangle
for thick Euclidean buildings with the Tits boundary Y. Quite remarkably, the cohomology
is obtained from a certain universal algebra A by a kind of "crystal limit''
has been previously introduced by Belkale-Kumar for the cohomology of
varieties and Grassmannians. Another degeneration of A leads to the homology theory of Y.
(with J. Greenstein), Int. Math. Res. Not. 2011, no.
introduce a quantum analogue of the classical folding of a simply-laced
to the non-simply-laced algebra gσ
along a Dynkin diagram automorphism~sigma of g. For each
folding we replace gσ
by its Langlands dual (gσ)v and construct a
between the nilpotent parts of g and (gσ)v, together with its
a Poisson structure on S(n). Remarkably,
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
n x n matrices with an adjoint
of Uq(sln), which generalizes the algebras
by K. Goodearl and M. Yakimov. The hardest case of quantum folding is,
expectably, the pair (so8,G2) for which the PBW presentation of
and the corresponding Poisson bracket on S(n) contain more than 700 terms each.
polynomials for Coxeter groups and representations of Cherednik
algebras (with Yu. Burman)
Trans. Amer. Math. Soc.,
362 (2010), 229–260.
and Canonical Invariants of Reflection Groups (with Yu. Burman), SIGMA 5 (2009), 057, 18 pages
We introduce and study deformations of finite-dimensional modules over
Cherednik algebras. Our main tool is a generalization of usual harmonic
for Coxeter groups – the so-called quasiharmonic polynomials. A
application of this approach is the construction of canonical
symmetric polynomials and their deformations for all Coxeter groups.
we introduce a continuous family of canonical invariants of finite
groups. We verify that the elementary canonical invariants of the
group are deformations of the elementary symmetric polynomials. We also
the canonical invariants for all dihedral groups as certain
dihedral groups (with M.
Geometriae Dedicata. 156
We construct rank 2
nondiscrete affine buildings associated with an arbitrary finite
operators and braided Cherednik algebras (with Y. Bazlov)
Selecta Mathematica, 14,
We introduce braided Dunkl
operators that are acting on a q-polynomial algebra and q-commute. Generalizing the
Etingof and Ginzburg, we explain the q-commutation
phenomenon by constructing braided Cherednik algebras for which the
operators form a representation. We classify all braided Cherednik
using the theory of braided doubles developed in our previous paper.
Besides ordinary rational Cherednik algebras, our classification gives
attached to an infinite family of subgroups of even elements in complex
groups, so that the corresponding braided Dunkl operators pairwise anti-commute. We
compute these new operators in terms of braided partial derivatives and
rational Cherednik algebras (with Y. Bazlov)
(2009) 5, 1466–1530.
We introduce and study a
large class of algebras with triangular
which we call braided doubles. Braided doubles provide a unifying
for classical and quantum universal enveloping algebras and rational
algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld
modules over Hopf algebras which turn out to be a generalisation of the
Yetter-Drinfeld modules. To each braiding (a solution to the braid
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
any rational Cherednik algebra canonically embeds into the braided
double attached to the corresponding complex reflection group.
and Lie groups over noncommutative rings (with V. Retakh) Advances
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
enveloping algebra of g. Our
of the current algebra has a striking resemblance to the commutator
of F used by M. Kapranov in his
approach to noncommutative
geometry. We also associate with each Lie algebra (g,A)(F) a "noncommutative
which naturally acts
(g,A)(F) by conjugations
conclude the paper with a number of examples of such groups.
symmetric and exterior algebras (with S. Zwicknagl)
Trans. Amer. Math. Soc., 360 (2008), 3429–3472.
notes on geometric crystals and their combinatorial analogues
(with D. Kazhdan)
aspect of integrable
of Japan, Tokyo, 2007.
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
Kashiwara crystal bases (presented by the first author in RIMS, August
and Unipotent Crystals II: From Unipotent Bicrystals to Crystal Bases
(with D. Kazhdan)
Contemp. Math., 433, Amer. Math. Soc., Providence,
For each reductive algebraic group G, we introduce and study unipotent
bicrystals which serve as a regular version of rational
and unipotent crystals introduced earlier by the authors. The
of unipotent bicrystals allows, on the one hand, to study
such varieties as Bruhat cells in G and their convolution
and, on the other hand, to give a new construction of many normal
crystals including those for Gv-modules,
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.
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
results are explicit positive matrix factorizations in the cells via
of matrices with noncommutative coefficents.
cluster algebras (with A.
Advances in Mathematics, vol.
2 (2005), 405–455.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky; their
continued in a series of papers including Cluster
III: Upper bounds and double Bruhat cells. This is a family of
rings designed to serve as an algebraic framework for the theory of
positivity and canonical bases in semisimple groups and their quantum
In this paper we introduce and study quantum deformations of cluster
algebras III: Upper bounds and double Bruhat cells (with S. Fomin
Duke Math. Journal, vol.
1 (2005), 1–52.
We continue the study of cluster algebras. We develop a new approach
on the notion of an upper cluster algebra, defined as an intersection
certain Laurent polynomial rings. Strengthening the Laurent phenomenon,
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
We prove that the coordinate ring of any double Bruhat cell in a
complex Lie group is naturally isomorphic to the upper cluster algebra
defined in terms of relevant combinatorial data.
product multiplicities, canonical bases and totally positive varieties
Zelevinsky) Invent. Math., vol. 143, 1 (2001), 77–128.
We obtain a family of explicit ``polyhedral" combinatorial expressions
multiplicities in the tensor product of two simple finite-dimensional
over a complex semisimple Lie algebra. Here ``polyhedral" means that
multiplicity in question is expressed as the number of lattice points
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
study combinatorial structure of Lusztig's canonical bases or,
of Kashiwara's global bases. Although Lusztig's and Kashiwara's
were shown by Lusztig to be equivalent to each other, they lead to
combinatorial parametrizations of the canonical bases. One of our main
is an explicit description of the relationship between these
Our approach to the above problems is based on a remarkable observation
G. Lusztig that combinatorics of the canonical basis is closely related
geometry of the totally positive varieties. We formulate this
in terms of two mutually inverse transformations: ``tropicalization"
and ``geometric lifting."
and unipotent crystals (with D. Kazhdan) Geom. Funct. Anal., Special
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
of rational actions of the multiplicative group Gm satisfying certain
relations. Such a structure induces a rational action of W
on X. Surprizingly many interesting rational actions of the
W come from geometric crystals. Also all the known
of the action of W which appear in the construction of
for the representations of the Langlands dual group Gv
in the recent work by A. Braverman and D. Kazhdan come from
crystals. There are many examples of positive geometric crystals on (Gm)l,
i.e., those geometric crystals
which the actions ei
and the morphism gamma are given by positive rational
One can associate to each positive geometric crystal X the
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
for any geometric SLn-crystal.
crystals are geometric analogues of normal Kashiwara crystals. They
a strict monoidal category. To any unipotent crystal built on a variety
we associate a certain gometric crystal.
orbits, moment polytopes, and the Hilbert-Mumford criterion
(with R. Sjamaar),
J. Amer. Math. Soc., 13
no. 2, 433–466.
In this paper we solve of the following problem: Given a reductive
G, and its reductive subgroup H, describe the momentum
This is a rational polyhedral cone spanned by all those dominant G-weights
which the simple G-module Vλ contains a non-trivial H-invariant.
Our result generalizes the result by Klyachko who has solved this
for G= GLn×GLn×GLn with the
embedded diagonally into G. We describe the facets of the cone Δo
in terms of the ``relative'' Schubert calculus of the flag varieties of
two groups. Another formulation of the result is the description of the
momentum cone Δ,
spanned by those pairs (λ,λ') for which the restriction
H of the simple G-module Vλ contains a simple H-module
tableaux, Schutzenberger involution and action of the symmetric group
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.
of weighted arithmetic means with applications (with Alex Vainshtein),
Math. (1997) 69, 120–126.
Schubert varieties (with A.
Math. Helv. 72 (1997), no. 1, 128–166.
In this paper we further develop the remarkable parallelism discovered
Lusztig between the canonical basis and the variety of totally positive
in the unipotent group.
of canonical bases and totally positive matrices (with S. Fomin and A.
122 (1996), 49–149.
We provide: (i) explicit formulas for Lusztig's transition maps related
the canonical basis of the quantum group of type A; (ii) formulas for the
of a square matrix into elementary Jacobi matrices; (iii) a family of
total positivity criteria.
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
The proof is based on a construction of some group-like elements
are q-analogs of elements in G).
Canonical bases for
quantum group of type A_r and piecewise-linear combinatorics
Math. J. 82 (1996), no. 3, 473–502.
We use the structure theory of the dual canonical basis B is to obtain a direct
proof of the Littlewood-Richardson rule (or rather, its
versions discussed above).
Another application of string technique is an explicit formula
the action of the longest element wo in
on the dual canonical basis in each simple slr+1-module.
Having been translated into the language of Gelfand-Tsetlin
and Young tableaux, this involution coincides with the Schützenberger involution.
String bases for
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,
We introduce and study a family of string bases for the guantum
of type Ar
(which includes the dual canonical basis). These bases are defined
and possess many interesting properties, e.g., they all are good
in the sense of Gelfand and Zelevinsky. For every string basis,
construct a family of combinatorial labelings by strings. These
in a different context appeared in more recent works by M. Kashiwara
by P. Littelmann. We expect that B has a nice multiplicative
Namely, we conjecture in  that B
all products of pairwise q-commuting elements of B. The conjecture was proved in
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
generated by involutions, Gel'fand-Tsetlin patterns, and combinatorics
of Young tableaux (with Anatol Kirillov),
i Analiz 7 (1995), no. 1, 92–152 (Russian). Translation in St.
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 Sn on Young
tableaux, discovered by
Lascoux and Schutzenberger. We introduced an action of Sn by
transformations on the space of Gelfand-Tsetlin patterns. In our
this group appears as a subgroup of the infinite group Gn, generated
quite simple piecewise-linear involutions (these involutions are
analogues of Bender-Knuth involutions acting on Young tableaux). The
not yet completely understood. Some relations were given in ; they
the famous Schützenberger involution
also belongs to Gn.
Another result of  is a conjectural description of Kashiwara's
operators for type A, in terms of Gn.
for sl(r+1) and the spectrum of the exterior algebra of the
representation (with A.Zelevinsky)
Algebraic Combin. 1 (1992), no. 1, 7–22.
When is the weight
equal to 1 (Russian) (with A.
Zelevinsky) Funkc. Anal.
Pril. 24 (1990), no. 4, 1–13; translation in Funct. Anal.
24 (1990), no. 4, 259–269.
and convex polytopes in partition space(with A.
Zelevinsky) J. Geom.
Phys. 5 (1988), no. 3, 453–472.
multiplicative analogue of the Bergstrom inequality for a matrix
product in the sense of
Hadamard (with Alex Vainshtein)
Mat. Nauk 42 (1987), no. 6(258), 181–182. Translation: Russian
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.
on Gelfand-Tsetlin patterns and multiplicities in skew GL(n)-modules(with
Zelevinsky) Soviet Math.
Dokl. 37 (1988), no. 3, 799–802.