RESEARCH STATEMENT
ALIMJON ESHMATOV
Introduction
My research interest centers around algebra, noncommutative algebraic geometry, symplectic geometry, and their interaction. More recently I have been working on description
of automorphism groups of algebras of geometric origin, such as: algebra of differential
operators on algebraic varieities, noncommutative algebraic torus, Cremona groups, etc.
To give a flavor of my research interest, I would like to explain one of basic principles
in noncommutative algebraic geometry. If k is a field, the set of all representations of an
associative k-algebra A in a finite-dimensional vector space V can be given the structure
of an affine k-scheme, called the representation scheme RepV (A). The group GLk (V) acts
naturally on RepV (A), with orbits corresponding to the isomorphism classes of representations. If k is algebraically closed and A is finitely generated, the equivariant geometry
of RepV (A) is closely related to the representation theory of A. This relation has been extensively studied (especially in the case of finite-dimensional algebras) since the late 70s,
and the schemes RepV (A) have become a standard tool in representation theory of algebras
(see, for example, [Ga], [Bo], [Ge] and references therein)
More recently, representation schemes have come to play an important role in noncommutative geometry. Let us recall that in classical (commutative) algebraic geometry,
there is a natural way to associate to a commutative algebra A a geometric object the
Grothendieck prime spectrum ”Spec(A)”. This defines a contravariant functor from commutative algebras to affine schemes, which is an (anti)equivalence of categories. Attempts
to extend this functor to the category of all associative algebras have been largely unsuccessful. M. Kontsevich and A. Rosenberg proposed a heuristic principle according to which
the family of schemes {RepV (A)} for a given algebra A should be thought of as a substitute
(or ”approximation”) for ”Spec(A)”. The idea is that every property or noncommutative
geometric structure on A should naturally induce a corresponding geometric property or
structure on RepV (A) for all V . This viewpoint provides a litmus test for proposed definitions of noncommutative analogues of classical geometric notions. In recent years, many
interesting structures in noncommutative geometry have originated from this idea: NC
smooth spaces, formal structures and noncommutative thickenings of schemes (Kapranov,
LeBruyn-Weyer), noncommutative symplectic and bisymplectic geometry, double Poisson
brackets and noncommutative quasi-Hamiltonian spaces.
On the other hand by functoriality of the construction, the group of automorphisms
Autk (A) acts in a canonical way on RepV (A) and RepV (A)GLk (V) . One can ask whether this
action is ”algebraic”, transitive. What are the orbits of this action? As it turned out, this action yields very interesting invariants of an algebra. In particular, for A0 = C[x, y] the representation variety can be identified with commuting variety Mn = {(X, Y) ∈ Mn (C) | XY =
Y X} for n ≥ 0. The action of Aut(A0 ) on Mn is given by evaluation : (σ−1 (X), σ−1 (Y))
for σ ∈ Aut(A0 ). This induces the action on Mn //GLn (C) and its a natural subvariety
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ALIMJON ESHMATOV
Hilbn (C2 ) - Hilbert scheme of n points in C2 which consists of isomorphism classes of
n-dimensional cyclic representations.
If we ”quantize” the affine plane C2 replacing the commutative polynomial ring C[x, y]
by the first complex Weyl algebra A1 := Chx, yi/(xy − yx − 1), one would expect the
representation variety of A1 is a certain deformation of the reprsentation variety of A0 .
However A1 has no finite dimensional representation. Thus, a priori, there seems no reason
to expect that there should be results for A1 similar to the ones indicated above for A0 :
nevertheless, we shall see that this is the case. To obtain such results, we have to relax
somewhat the notion of a representation, as follows. For each n ≥ 0 , let Cn be the space
of equivalence classes (modulo simultaneous conjugation) of pairs (X, Y) of n × n matrices
satisfying the condition
[X, Y] − I has rank at most one
We might think of Cn as the space of ”approximate n-dimensional representations” of A1 .
Let
G
Cn
C=
n≥0
be the disjoint union of the spaces Cn . In fact H. Nakajima has found that the Hilbert
scheme Hilbn (C2 ) is a hyperkahler manifold, and that after changing the complex structure
to a different one in the hyperkahler family, we obtain the space Cn . One can think of
it that this deformation of complex structure from Hilbn (C2 ) to Cn is resulted from the
deformation of algebras from A0 to A1 .
One can define a natural action of G = Aut(A1 ) on Cn similar to that of Hilbn (C2 ). Let
R denote the set of isomorphism classes of non-zero right ideals of A1 . In [BW1], Berest
and Wilson constructed an G-equivariant bijection ω : R → C, such that the action of G
on each Cn is transitive. This was somewhat surprising since the action of group Aut(A0 )
on Hilbn (C2 ) is not transitive, because any automorphism preserves the multiplicities of a
collection of n points of C2 .
The problem of describing the orbits of G in R goes back at least to Staffords paper [S]:
it is known to be equivalent to the problem of classifying rings of differential operators on
affine curves; or again, to classifying the algebras Morita equivalent to A1 . We will discuss
this more extensively in the next section and where we will also discuss solution to some
question of Stafford in [S].
In the first section I discuss some results in automorphism groups rings of differential operators on curves and resolution of some problems in the subject. The second and
third sections focus on my research in symplectic geometry. In last section, I list some of
research projects I am currently working on or planning to work in the near future.
1. Dixmier groups
1.1. Trees, amalgams and Calogero-Moser spaces. The classical result due to Dixmier
and Makar-Limanov gives explicit description of automorphism group of G = Aut(A1 ).
Specifically, G = A ∗U B is an amalgamated product of its subgroups A and B over U
where
A : (x, y) 7→ (ax + by + e, cx + dy + f ) ,
a, ..., f ∈ C ,
B : (x, y) 7→ (ax + q(y), a y + h) ,
∗
−1
U : (x, y) 7→ (ax + by + e, a y + h) ,
−1
a ∈ C , h ∈ C,
ad − bc = 1 ,
q(y) ∈ C[y] ,
a ∈ C , b, e, h ∈ C ,
∗
The aim of the [BEE1] is to generalize the above results to the case when A1 is replaced
by a noncommutative domain D, Morita equivalent to A1 as a C-algebra. We recall that the
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algebras D are classified, up to isomorphism, by a single integer n ≥ 0; the corresponding
isomorphism classes are represented by the endomorphism rings Dn := EndA1 (Mn ) of
certain distinguished right ideals of A1 and can be realized geometrically as algebras of
global differential operators on rational singular curves (see [BW1] and [BW2]). Thus
the Dixmier group Aut(A1 ) = Aut(D0 ) appears naturally as the first member in the family
{Aut(Dn ) : n ≥ 0} . Our goal is to give a presentation of Aut(Dn ) for arbitrary n > 0 in terms
of amalgamated products. To this end, we use G-equivariant bijection between Cn and the
space of isomorphism classes of right ideals of A1 and the fact that the G-action is transitive
to identify Aut(Dn ) with Gn , the isotropy group of G on Cn . Next, by the Bass-Serre theory
we can associate to Gn a graph Γn in terms of orbit spaces of the Calogero-Moser varieties
Cn :
G
G
(1)
Vert(Γn ) := (A\Cn )
(B\Cn ) , Edges(Γn ) = (U\Cn )
(U\Cn )∗ .
Two vertices are adjacent if and only if they correspond to intersecting A and B-orbits:
the set of all edges connecting an A-orbit OA and a B-orbit OB consists of the U-orbits in
the intersection OA ∩ OB . Now, to each vertex and each edge we can assign an isotropy
group of the corresponding orbit. Such an assignment will define a graph of groups Γn for
which Bass-Serre theory introduced π1 (Γn , T ), the fundamental group of Γn with respect
to a maximal subtree T of Γn . For n = 0, one can easily show that π1 (Γn , T ) A ∗U B. Our
main result in [BEE1] states
Theorem 1. For each n ≥ 0, the group Gn is isomorphic to π1 (Γn , T ).
This theorem answers the first question of Stafford. However, it is not clear to us how
to see from this description whether Aut is an invariant to distinguish Morita equivalent
algebras. In the next section, we describe a different approach to Aut(Dn ) which will
answer the second question of Stafford.
1.2. Dixmier groups as ind-groups. The notion of an ind-group goes back to Shafarevich
who called such objects simply infinite-dimensional groups (see [Sh]). The fundamental
example, which motivated the theory from the very beginning and still is one of its most
important applications, is the group Aut(Cd ) of polynomial automorphism of the affine dspace. It is known that Aut(A1 ) can be naturally identified (as a discrete group) either with
Autω (C2 ), a normal subgroup of Aut(C2 ) or with the subgroup of symplectic automorphims
of Chx, yi, the free algebra on two generators ([Cz, ML]). In [BEE2] we choose the indgroup structure on the later one. This structure can be defined in a simpler and somewhat
more natural way than that of Autω (C2 ) or Aut(A1 ). The reason for this is the remarkable
fact that the analogue of the Jacobian Conjecture for the free associative algebras is known
to be true in the rank two case (see [D1]).
We have studied the properties of Gn := Aut(Dn ) as an (infinite dimensional) algebraic
group. Our main result is a classification of Borel subgroups of Gn . Recall that a subgroup
H of a topological group G is called Borel if H is connected, solvable and maximal among
all connected solvable subgroups in G. Our first main result in [BEE2] gives an abstract
group-theoretic characterization of Borel subgroups of Gn .
Theorem 2. A discrete subgroup H of Gn is a Borel subgroup if and only if
(B1) H is a maximal solvable subgroup of G.
(B2) H contains no proper subgroups of finite index.
Theorem 2 is an infinite-dimensional generalization of a classical theorem of R. Steinberg [St] that asserts that conditions (B1) and (B2) characterize Borel subgroups in reductive affine algebraic groups over algebraically closed fields. We show that the groups
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ALIMJON ESHMATOV
Gn are pairwise non-isomorphic. The invariants that distinguish these groups are Borel
subgroups which we define in purely group-theoretic terms by analogy with Steinberg’s
abstract characterization of such subgroups in finite-dimensional algebraic groups.
Our second main result classifies the non-abelian Borel subgroups of Gn .
Theorem 3. For each n, the group Gn has exactly p(n) conjugacy classes of non- abelian
Borel subgroups, where p(n) is the number of partitions of n.
As a consequence, we get
Corollary 1. The groups Gn are pairwise non-isomorphic (as abstract groups).
The statement of Theorem 3 seems to be new even for n = 0, in which case it reads
exactly as in the classical case: all Borel subgroups of G are conugate to a single subgroup,
namely B. The Borel subgroups of Gn have geometric origin: they arise from the locus
of ‘nilpotent’ points in Cn , which can be characterised as the fixed point set of a natural
C∗ -action However, the proof is not entirely geometric nor algebraic; the crucial ingredient
is Friedland-Milnor’s classification of polynomial automorphisms of C2 according to their
dynamical properties (see [FM]).
1.3. Higher transitivity. The main theorem of [BW1] that states that the natural action
of G on the varieties Cn is transitive for all n. We extended this last transitivity result in
two ways [BEE2].
Theorem 4. For each n ≥ 1 , the action of G on Cn is doubly transitive.
Theorem 5. For any pairwise distinct natural numbers (n1 , n2 , . . . , nm ) ∈ Nm , the diagonal
action of G on Cn1 × Cn2 × . . . × Cnm is transitive.
One important consequence of the double transitivity is that the stabilizer of each point
P ∈ Cn is a maximal subgroup of G. Theorem 1 thus strengthens the main results of
[KT] and [W1], where it is shown that the subgroups S tabG (P) Gn coincide with their
normalizers in G.
Conjecture 1. The action of G on Cn is infinitely transitive.
2. On sum of coadjoint orbits
Also known as Horn problem is a problem about description of sum of spectra of two
selfadjoint operators with fixed spectra. It was posed by H. Weyl about 100 years ago
and was settled quite recently by efforts of many mathematician from different fields (see
more in [F]). During this period it was found to be related to many important problems in
areas such as: symplectic geometry, representation theory, combinatorics and cohomology
theory. From symplectic geometry point of view problem can be stated rather easily. It
is known selfadjoint matrices with fixed spectra form conjugacy class of diagonal matrix
diag(λ1 , . . . , λn ). In other words it is an orbit with respect to coadjoint action of SU(n)
which we call Oλ . It is basic example of a symplectic manifold with natural Hamiltonian
structure. Now the problem can be stated as given two coadjoint orbits Oλ and Oµ describe
the image of moment map
Φ : Oλ × Oµ → su(n)
n−1
(ξ, η) 7→ ξ + η
in a set of all diagonal matrices Diag R . One can show that this image in Rn−1 is a
convex polytope given by a collection of inequalities.
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There is a complete analogous solution of this problem when SU(n) is replaced with an
arbitrary compact Lie group with proper changes in the above data. However, the noncompact case seems to remain untouched, for a reason that the noncompact coadjoint orbits
do not possess the necessary nice properties and the sum of two non-compact orbits can in
general cover pretty much arbitrary spectra.
In our joint work with P. Foth [EF], we studied the problem for wide class of non-compact
Lie groups G. More precisely when Lie algebra g := Lie(G) is a quasi-Hermitian and
semisimple, there is a class of coadjoint orbits called admissible which share certain properties of the compact case. We have showed: given two admissible coadjoint orbits OA , OB
with spectra A and B respectively, the possible spectra of sum is given by explicitly defined
convex polyhedral set. As an application we used this result to describe sum of spectra
for elements in su(p, q). It would be desirable to extend this result probably with some
modification for other non-compact Lie groups.
3. Quasi-Hamiltonian spaces
Let G be a compact Lie group with an invariant inner product on its Lie algebra g. Let
ξ M a vector field generated by ξ ∈ g. Let θ, θ¯ and χ be canonical differential forms on G
defined by
1
[θ, θ] · θ in Ω3 (G)
θ = g−1 dg θ¯ = dgg−1 in Ω1 (G, g) , χ =
12
Definition. A quasi-Hamiltonian G-manifold is a smooth G-manifold M together with
ω ∈ Ω2 (M)G and a moment map Φ ∈ C ∞ (M, G)G such that
(1) dω = −Φ∗ χ
¯ ·ξ
(2) ι(ξ M )ω = 12 Φ∗ (θ + θ)
(3) ker ω ∩ ker (dΦ) = 0
Quasi-Hamiltonian spaces and their moment maps share many properties of Hamiltonian ones, such as convexity theorem and reduction. In fact there is a one-to-one correspondence between compact quasi-Hamiltonian G-spaces and infinite-dimensional Hamiltonian
LG-spaces with a proper moment map, where LG is the loop group of G
One of the motivations for developing the theory of group-valued moment maps comes
from one particularly important result. Namely, it has been shown that the moduli space
M(Σ) of flat G-bundles on a punctured Riemann surface Σ of given genus can be obtained
by a reduction of a finite dimensional quasi-Hamiltonian manifold. Hence it is a symplectic
manifold, result earlier obtained by Atiyah and Bott [AB].
The quasi-Hamiltonian implosion is certain operation on manifolds which from a given
quasi-Hamiltonian G-space M produces a quasi-Hamiltonian T -space Mimpl [HJS]. It is
somewhat similar to a reduction procedure, but instead of taking the quotient by the entire
stabilizer subgroup one quotients only by a maximal non-abelian subgroup. Although
it is usually a singular space, similar to a singular reduction an imploded space has a
natural stratification into symplectic manifolds. Hence one can think of implosion as an
“abelinization functor”
Impl : {q-Hamiltonian G-spaces} → {q-Hamiltonian T -spaces}
which assigns to each q-Hamiltonian space its “abelinization”.
It is known a “double” D(G) := G ×G has a canonical q-Hamiltonian G ×G-space structure (see ). In fact it can thought of as an analog of a cotangent bundle for q-Hamiltonian
spaces. If we implode this space with respect an action of G on the right, we obtain a
q-Hamiltonian G × T -space D(G)impl . It is observed in [HJS] that there are certain strata
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ALIMJON ESHMATOV
of D(G)impl where it is singular, but whose closure are smooth. This observation led them
to construct a new class of examples of quasi-Hamiltonian manifolds. In particular when
G is A-type i.e. G = SU(n), there is a stratum whose has a smooth closure diffeomorphic
to S 2n . As a result, they showed that S 2n is a q-Hamiltonian U(n)-space. Motivated by this
example, we studied the implosion for other type of Lie groups. In particular for, type C,
i.e. for Sp(n) the unitary quaternionic group we proved
Theorem 6. [E] There is a certain stratum of X = DSp(n)impl whose closure is a smooth
quasi-Hamiltonian Sp(n) × U(1)-manifold diffeomorphic to HPn .
It is natural to ask whether there are other examples of quasi-Hamiltonian spaces appearing in this context. Surprisingly, this question was answered negatively in [E1]. To be
precise, let G be any connected and simply-connected Lie group. We showed
Theorem 7. [E1] A stratum of DGimpl has a smooth closure only in above mentioned
examples.
That is, S 2n and HPn are the only examples where a stratum has a smooth closure. Our
proof based on the argument in [Po] and counting dimensions of strata.
Original definition of q-Hamiltonian spaces were defined only for compact Lie groups.
It certainly would be very interesting examples related to application in representation theory, algebraic geometry generally dealing with non-compact Lie groups such as : GLn (C), Sp2n (C)
etc.
4. Research plans
4.1. Dixmier groups. There are some important question which would help a lot to understand nature of Dixmier groups that we discussed in the first section.
Problem 1. Computing homology of groups H∗ (Gn ) for n ≥ 2.
One can use for this I. M. Chiswell result to compute homologies for graph of groups.
It is known group G is isomorphic to unimodular automorphism of plane. The following
question was proposed by E. Zelmanov
Problem 2. Whether groups Gn can described as automorphism groups of some structures/objects in C2 ?
It is easy to see that G1 can described as augmentation preserving automorphisms.
While we do not know for n ≥ 2. Note groups Gn were defined as automorphism of noncommutative algebras and understanding geometrically would be especially useful. The
following problem which J. Alev asked us
Problem 3. Describe all finite subgroups of Gn .
We have partial results on this problem. In particular we know complete description for
n = 0, 1, 2, 3 (case 0 and 1 were known before). We have some general statements about
this problem. This problems is quite attractive from point of view studying fixed subrings
of algebras Dn .
Recall we have a natural ind-variety structure on Gn . We proved in [BEE1] that G is
connected with respect to this topology. We expect this true for all Dixmier groups.
Conjecture. Groups Gn are connected for all n ≥ 0.
4.2. Hochschild and cyclic homologies of algebra of (pseudo)differential operators.
This is joint project with M. Khalkali and F. Fathizadeh. Singular homology is often
used in Algebraic Topology to obtain invariants of topological spaces. In the same spirit,
Hochschild and cyclic homology often provide interesting invariants of algebras. Hochschild
homology is usually thought of as a generalization of the notion of differential forms and
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(periodic) cyclic homology as a generalization of (de Rham) cohomology, from smooth
compact manifolds to essentially arbitrary algebras.
In [BG], J-L. Brylinski and E. Getzler computed the Hochschild and cyclic homology
groups of the algebra of pseudo-differential symbols on smooth compact manifold. While
M. Wodzicki computed the Hochschild and cyclic homology groups of the algebra of differential operators on smooth (algebraic, holomorphic, C ∞ ) manifold [Wo].
We compute the Hochschild and cyclic homology groups of the algebra of pseudodifferential symbols and differential operators on non-commutative tori. To compute Hochschild
homologies, we use a natural filtration on these algebras to reduce it to computing a spectral sequence. Then using Connes long exact sequence we should be able to compute cyclic
homologies of these algebras.
4.3. Automorphisms of noncommutative surfaces. In [GD] the group of automorphisms
of a certain class of quasihomogeneous surfaces were studied. Let X be a such surface.
They formed a tree 4X from standard completions of X on which Aut(X) acts. Then they
used the Bass-Serre theory to describe Aut(X).
From this point of view A1 is known as basic example of noncommutative affine surfaces ([SV]. It seems once we think of completions of surface in Danilov-Gizatullin as
certain filtration on ring of functions we should be able to extend this construction for
wide class of noncommutative algebras. We have natural action of automorphism group
on filtrations hence we have an action of a group on a graph (possibly even a tree) which
is a natural framework for applying Bass-Serre theory. It would be interesting to carry
out this construction for other algebras similar to A1 , such as: Oτ algebras which are family of noncommutative deformation of Kleinian singularities; certain class of deformed
preprojective algebras introduced by Crawley-Boevey; rational Cherednik algebras.
4.4. Cremona group of affine Weyl algebra. Also intimately related the problem stated
in last section is description of a birational automorphism group of (non)commutative surfaces. The problem have been completely solved for Cremona group, a group of birational
automorphisms of P2 . The complete set of generators were already known to NoetherCastelnuovo. However the complete set of relation have been obtained by V. Iskovskikh
in 1980’s. Shortly after D. Wright gave a very nice presentation of Cremona group in
terms of amalgamated product of its subgroups. For this he constructed a 1-connected
two-dimensional simplicitial complex where this group naturally acts and obtains desired
presentation using a version of Bass-Serre theory.
Natural problem is to study similar group, namely automorphisms of Q(A1 ) quotient
field of the first Weyl algebra which can be thought of as birational automorphism of noncommutative P2 . There are Iskovskikh type generators and also inner automorphism. One
can ask whether these form complete set of generators.
Problem 8. Describe set of all generators and relation for Aut(Q(A1 )).
Problem 9. Find amalgamated product type presentation of group Aut(Q(A1 )) similar
to to that in [W].
We believe the solution of problem 2 should involve some noncommutative projective
geometry (such as [SV]).
4.5. Generalized Dixmier Conjecture. Finally, we would like to propose an extension of
the well-known Dixmier Conjecture for A1 (see [D], Probl`eme 11.1) to the class of Morita
equivalent algebras. We recall that if D is a domain Morita equivalent to A1 , then there
is a unique integer n ≥ 0 such that D Dn , where Dn is the endomorphism ring of the
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ALIMJON ESHMATOV
right ideal Mn = xn A1 + (y + nx−1 ) A1 . For two unital C-algebras A and B, we denote by
Hom (A, B) the set of all unital C-algebra homomorphisms A → B .
Conjecture 2. For all n, m ≥ 0 , we have
∅
Hom (Dn , Dm ) =
Aut Dn
if
n,m
if
n=m
Formally, Conjecture 2 is a strengthening of the Dixmier Conjecture for A1 : in fact, in our
notation, the latter says that Hom (D0 , D0 ) = Aut D0 . We can show Dixmier Conjecture
partially implies that Conjecture 2. More precisely, Hom (Dn , Dn ) = Aut Dn is holds if
only if Dixmier Conjecture holds. However we do not know whether Dixmier Conjecture
imply stronger Conjecture 2?
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