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introduce the symplectic group Sp2(A,σ) over a
noncommutative algebra A with
an anti-involution σ.
We realize several classical Lie groups as Sp2 over various
noncommutative algebras, which provides new insights into their structure theory.We construct several
geometric spaces, on which the groups Sp2(A,σ) act.
We introduce the space of isotropic A-lines,
which generalizes the projective line. We describe the action of Sp2(A,σ) on
generalize the Kashiwara-Maslov index of triples and the cross ratio of
quadruples of isotropic A-lines as
invariants of this action. When the algebra A is
Hermitian or the complexification of a Hermitian algebra, we introduce the
symmetric space XSp2(A,σ),
and construct different models of this space. Applying this to classical
Hermitian Lie groups of tube type (realized as Sp2(A,σ)) and their complexifications,
we obtain different models of the symmetric space as noncommutative
generalizations of models of the hyperbolic plane and of the three-dimensional
hyperbolic space. We also provide a partial classification of Hermitian
algebras in Appendix A.
multiplicities (with Y. Li),
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 Gv-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 X is a monoid,
i.e., the corresponding Gv-module is an
algebra, we expect that in many cases, the spectrum of this algebra is an
affine Gv-variety Xv, and thus the correspondence XàXv has a flavor of both the Langlands duality and mirror symmetry.
Duality and Poisson-Lie Duality via Cluster Theory and Tropicalization
(with A. Alekseev,
B. Hoffman, Y. Li) Selecta
Mathematica, 27 (4) DOI:
be a connected semisimple Lie group. There are two natural duality
constructions that assign to it the Langlands dual group Gv and the
Poisson-Lie dual group G*. The main result of this paper is the
following relation between these two objects: the integral cone defined by the
cluster structure and the Berenstein-Kazhdan potential on the double Bruhat cell Gv;wo,e.
is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure
on the partial tropicalization of K*⊂ G* (the Poisson-Lie dual of
the compact form K ⊂ G). The first cone parametrizes the
canonical bases of irreducible G-modules.
The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by
the highest weight of the representation. As a by-product of our construction,
we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of K*
are equal to symplectic volumes of the corresponding
coadjoint orbits in k*. To achieve these goals,
we use (Langlands dual) double cluster varieties defined by Fock
and Goncharov. These are pairs of cluster varieties whose
seed matrices are transpose to each other. There is a naturally defined
isomorphism between their tropicalizations. The isomorphism between the cones
described above is a particular instance of such an isomorphism associated to
the double Bruhat cells G wo,e ⊂ G and Gv;wo,e.
On cacti and crystals (with J. Greenstein and Jian-Rong Li) Representations and Nilpotent Orbits of Lie
Algebraic Systems: in honor of the 75th Birthday of Tony Joseph, Progress in Mathematics, 330,
In the present
work we study actions of various groups generated by involutions on the
category Oqint(g) of integrable highest
and their crystal bases for any symmetrizable Kac-Moody algebra g. The most notable of them are the cactus group
and (yet conjectural) Weyl group action on any highest weight integrable module
and its lower and upper crystal bases. Surprisingly, some generators of cactus
groups are anti-involutions of the Gelfand-Kirillov model for Oqint(g) closely related to the remarkable
quantum twists discovered by Kimura and Oya.
structures and potentials (with A. Alekseev, B. Hoffman, Y. Li) Lie Groups, Geometry, and Representation
Theory: A Tribute to the Life and Work of Bertram Kostant,
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).
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 Cn which belong to the
free Laurent polynomial algebra in n generators. Our noncommutative
numbers admit interesting (commutative and noncommutative) specializations, one
of them related to Garsia-Haiman (q,t)-versions, another -- to solving
noncommutative quadratic equations. We also establish total positivity of the
corresponding (noncommutative) Hankel matrices Hn
and introduce accompanying noncommutative binomial coefficients.
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 G,
their parabolic subgroups, basic affine spaces and many others. It turns out
that tensor products of factorizable algebras are also factorizable and it is
easy to create a factorizable algebra out of virtually any g-module algebra. We also have quantum
versions of all these constructions in the category of Uq(g)-module algebras. Quite surprisingly,
our quantum factorizable algebras are naturally acted on by the quantized
enveloping algebra Uq(g*) of
the dual Lie bialgebra g* of
Hecke-Hopf algebras (with D. Kazhdan) Advances in
Mathematics, Vol. 353 (2019), 312–395.
Let W be a Coxeter group. The goal of the paper is to construct new Hopf algebras contain Hecke algebras Hq(W) as (left) coideal subalgebras. Our Hecke-Hopf
algebras H(W) have a number of applications. In
particular they provide new solutions of quantum Yang-Baxter equation and lead to
a construction of a new family of endo-functors of the category of Hq(W)-modules. Hecke-Hopf algebras for the
symmetric group are related to Fomin-Kirillov
algebras; for an arbitrary Coxeter group W the
“Demazure” part of H(W) is being acted upon by generalized
braided derivatives which generate the corresponding (generalized) Nichols
Canonical bases of quantum
Schubert cells and their symmetries (with J. Greenstein),
J. Sel. Math. New
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), 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 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
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 exye-x=Σk≥0
1/k! (ad x)k(y). We also obtain
combinatorial applications to q-exponentials, q-binomials, and
decompositions (with J.
Greenstein), Comptes Rendus Mathematique, Doi :
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.
clusters (with J.
Greenstein, D. Kazhdan), Comptes Rendus Mathematique,Vol 353, 5 (2015), 387–390.
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.
bases (with J.
Greenstein), Advances in Mathematics,Vol. 316 (2017),
We introduce a new class of bases for quantized universal
enveloping algebras Uq(g)
and other doubles attached to semisimple and Kac-Moody Lie algebras. These
bases contain dual canonical bases of upper and lower halves of Uq(g)
and 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)
spans its center.
Mystic reflection groups (with Y. Bazlov), SIGMA 10 (2014), 040, 11
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
characters of Hall algebras
(with D. Rupel), Selecta Mathematica, DOI:
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 VàXV,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=(io,io), where io 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  and . 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  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 .
Cocycle twists and extenstions of braided doubles (with Y. Bazlov),
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 sln+1.
Hall algebras (with J. Greenstein),
Pacific Journal of Mathematics, Vol. 281, No. 2, 2016.
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 Σn≥0 anbn
for non-commuting a and b, Catalan numbers and non-commutative
quadratic equations (with V. Retakh, C. Reutenauer, D. Zeilberger ), Contemp.
Math. 592 (2013), 103–109.
The 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 formal series in a and b. We
show that the inversion satisfies a non-commutative quadratic equation and that
the number of certain monomials in its homogeneous components equals to a
Catalan number. We also study general solutions of similar quadratic equations.
Quantum Chevalley groups (with J. Greenstein), Contemp.
Math., 592 (2013), 71–102.
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.
for reflection groups (with E. Richmond), Advances in
Mathematics, Vol 284 (2015), 54–111.
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*T(G/P).
A short proof of
Kontsevich cluster conjecture (with V.
Retakh), C. R. Math. Acad. Sci., Paris 349
(2011), no. 3–4, 119–122.
We give an elementary proof
of the Kontsevich conjecture that asserts that the iterations of the
noncommutative rational map Kr:(x,y)→(xyx-1,(1+y-r)x-1)
are given by noncommutative
Stability inequalities and
universal Schubert calculus of rank 2 (with M. Kapovich),
16, Issue 4 (2011), 955–1007.
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
Quantum folding (with J. Greenstein), Int. Math. Res. Not. 2011, no. 21, 4821–4883.
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 gσ
along a Dynkin diagram automorphism σ of g. For each quantum folding we
by its Langlands dual (gσ)v
and construct a nilpotent Lie algebra n
which interpolates between the nilpotent parts of g and (gσ)v,
together with its quantized enveloping algebra Uq(n) and a Poisson structure on S(n). Remarkably, for the pair
the algebra Uq(n) admits an action of the Artin
braid group Brn and contains a new
algebra of quantum n x n matrices with an adjoint action of Uq(sln),
which generalizes the algebras constructed by K. Goodearl
and M. Yakimov. The hardest case of quantum folding is, quite expectably, the
pair (so8,G2) for which the PBW presentation of Uq(n) and the corresponding Poisson bracket on S(n) contain more than 700
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.
Operators and Canonical Invariants of Reflection Groups (with Yu. Burman), SIGMA 5 (2009), 057, 18 pages.
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
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 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.
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
algebras and Lie groups over noncommutative rings (with V. Retakh), Advances in
Mathematics, Vol. 218, 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 universal enveloping algebra
of g. Our description of the current algebra has a striking
resemblance to the commutator expansions of F used by M. Kapranov in his
approach to noncommutative geometry. We also associate with each Lie algebra (g,A)(F) a “noncommutative algebraic” group G
which naturally acts on (g,A)(F) by
conjugations and 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.
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
notes on geometric crystals and their combinatorial analogues (with D. Kazhdan), Combinatorial
aspect of integrable systems, MSJ Memoirs,
17, Mathematical Society of Japan, Tokyo, 2007.
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).
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 rational 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 Gv-modules,
where Gv 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.
Noncommutative Double Bruhat cells and their factorizations (with V. Retakh), Int. Math. Res. Not., 8
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
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 III: Upper
bounds and double Bruhat cells
(with S. Fomin
and A. Zelevinsky) Duke Math. Journal,
vol. 126, 1 (2005), 1–52.
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.
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),
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:XàT and a
compatible family ei:Gm×XàX, i in I of rational actions of
the multiplicative group Gm
satisfying certain braid-like relations. Such a structure induces a
rational action of W on X. Surprisingly many interesting rational
actions of the group W come from
geometric crystals. Also all the known examples of the action of W
which appear in the construction of Gamma-functions for the
representations of the Langlands dual group Gv
in the recent work by A. Braverman and D. Kazhdan come from
geometric crystals. There are many examples of positive geometric
crystals on (Gm)l, i.e., those
geometric crystals for which the actions ei
and the morphism gamma are given by positive rational expressions.
One can associate to each positive geometric crystal X the Kashiwara’s 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 isomorphism Xà>γ-1(e)×T
for any geometric SLn-crystal.
Unipotent crystals are geometric analogues of normal Kashiwara crystals. They form
a strict monoidal category. To any unipotent crystal built on a variety X
we associate a certain geometric crystal.
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 Δo.
This is a rational polyhedral cone spanned by all those dominant G-weights
λ for which the simple G-module Vλ
contains a non-trivial H-invariant. Our result generalizes the result by
Klyachko who has solved this problem for G= GLn×GLn×GLn with the subgroup H=GLn embedded diagonally into G.
We describe the facets of the cone Δo in terms of the “relative”
Schubert calculus of the flag varieties of the two groups. Another formulation
of the result is the description of the relative momentum cone Δ,
which is spanned by those pairs (λ,λ') for which
the restriction to H of the simple G-module Vλ contains a simple H-module
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,
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 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
ring. The proof is based on a construction of some group-like elements in Gq (which are q-analogs of elements in G).
Canonical bases for the
quantum group of type Ar and piecewise-linear
combinatorics (with A. Zelevinsky), Duke Math. J. 82
(1996), no. 3, 473–502.
We use the structure theory of the dual canonical basis B is
to obtain a direct representation-theoretic proof of the Littlewood-Richardson
rule (or rather, its piecewise-linear versions discussed above).
Another application of string technique is an explicit formula for the action of the
longest element wo in Sr+1 on the dual
canonical basis in each simple slr+1-module.
Having been translated into the language of Gelfand-Tsetlin patterns and Young tableaux, this involution
coincides with the Schützenberger involution.
bases for quantum groups of type Ar (with A. Zelevinsky) I. M.
Gelfand Seminar, 51–89, Adv. Soviet Math.,
16, Part 1, Amer. Math. Soc., Providence, RI, 1993.
We introduce and study a family of string
bases for the quantum groups of type Ar
(which includes the dual canonical basis). These bases are defined
axiomatically and possess many interesting properties,
e.g., they all are good in the sense of Gelfand and Zelevinsky. For every string basis, we construct a family
of combinatorial labelings by strings. These labelings in a different context appeared in more recent
works by M. Kashiwara and by P. Littelmann.
We expect that B has a nice
multiplicative structure. Namely, we conjecture in  that B
contains all products of pairwise q-commuting
elements of B. The conjecture was proved
in  for A2 and A3. 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 quantum minors).
generated by involutions, Gel’fand-Tsetlin patterns,
and combinatorics of Young tableaux (with Anatol
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 Sn
on Young tableaux, discovered by Lascoux and Schutzenberger. We introduced an action of Sn
by piecewise-linear transformations on the space of Gelfand-Tsetlin
patterns. In our approach, this group appears as a subgroup of the infinite
group Gn, generated by quite simple
piecewise-linear involutions (these involutions are continuous analogues of
Bender-Knuth involutions acting on Young tableaux). The structure of Gn is not yet completely understood. Some
relations were given in ; they involve the famous Schützenberger
involution which also belongs to Gn. Another result of
 is a conjectural description of Kashiwara’s
crystal operators for type A, in terms of Gn.
Triple multiplicities for sl(r+1) and the spectrum of the exterior
algebra of the adjoint representation
(with A. Zelevinsky), J. Algebraic Combin.
1 (1992), no. 1, 7–22.
When is the weight
multiplicity equal to 1 (Russian) (with A. Zelevinsky) Funkc.
Anal. Pril. 24 (1990), no. 4, 1–13; translation: Funct.
Anal. Appl. 24 (1990), no. 4, 259–269.
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.
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.
on Gelfand-Tsetlin patterns and multiplicities in
skew GL(n)-modules (with A. Zelevinsky) Soviet Math. Dokl.
37 (1988), no. 3, 799–802 592 (2013), 71–102.