Meeting Lyon − São PauloAlgebra, Group and Logic
Lyon, April 22–26, 2013

Abstracts
Monday, April 22, 2013:
1. Weight Representations of Kac-Moody Algebras, by V. Futorny (USP).
Abstract: Classification problem of weight irreducible representations is open even for finite-dimensional simple Lie algebras (exceptsl2 ). S. Fernando and O. Mathieu classified such representations
with finite-dimensional weight spaces. This is the largest subcategory of modules where classification of irreducible objects is known.
On the other hand, Y. Drozd, V. Futorny and S. Ovsienko introduced a larger category of so-called Gelfand-Tsetlin modules whereclassification is known up to a certain finiteness. Recently, a classification of irrreducible Gelfand-Tsetlin modules was completed forsl3 by D. Grantcharov, V. Futorny and E. Ramires. We will discussthese results and also comment on the current situation with thesame problem for affine Kac-Moody algebras.
2. Groups Definable in Orthogonal Sets, by F. Wagner (UdL, Lyon 1).
Abstract: Two sets X and Y are orthogonal if every definable subsetof X ×Y is a finite union of rectangles A×B with A ⊆ X and B ⊆ Y .
I shall study groups definable in the product X × Y . Examples showthat they need not be isomorphic to the product of a group definable
in X and one definable in Y — in fact there need not be any groups
definable in X or Y separately. I shall give a structure theorem underan additional hypothesis on X, and make a conjecture for the generalcase involving local and approximate groups.
3. Noncommutative Algebras and Poisson Algebras, by A. Pichereau(UdL, St. Étienne).
Abstract: This is a joint work with R. Berger (St-Étienne) Following
the idea that classical mechanics should be a limit case of quantum1mechanics, P.A.M. Dirac explained that the commutator of dynamical variables in quantum mechanics should be the analogue of thesymplectic Poisson bracket of R2r in classical mechanics. Working
in a mathematical setting, we consider a non-commutative algebra
B, which can be seen as a deformation of a Poisson algebra T . Thisalgebra B belongs to a family of 3-Calabi-Yau algebras defined bypotentials and depending on a natural integer n and the algebra B
is for us the most interesting example in the case n = 2. We give
cohomological links between B and T , as we obtain the Poisson cohomology of T and prove that the Hochschild cohomology of B isisomorphic to the Poisson cohomology of T .
4. Problems and Conjectures in Division Rings: new Advances, by J.Z.
Gonçalves (USP).
Abstract: Let D be a division ring with center k and multiplicative
group D∗ = D\{0}. We will discuss the history and progress obtained trying to prove the following conjectures:
Conjecture(G) (Lichtman, 1977): D∗ contains a free (noncyclic) subgroup.
Conjecture(A) (Makar-Limanov, 1982): If D is finitely generated andinfinite-dimensional over k then D contains a free (noncyclic) algebra.
Conjecture(GA): Under the same conditions as in (A), D contains
the k-group algebra of a free group of rank 2.
We also address the problem of constructing free groups and freealgebras in D.
5. Cubic Forms over F2 and the Hurwitz-Radon Function, by V. Ovsienko(UdL, Lyon 1).
Abstract: We consider cubic forms over the field with 2 elements(sometimes called ”boolean cubic functions”). The classification problem of such forms is widely open. Our goal is to define an invariant
that can be understood as an analog of the classical Arf invariant of
quadratic forms, and show its relation to the Hurwitz-Radon numbers. We will show a combinatorial consequence of this relation.
Tuesday, April 23, 2013:
1. On Idempotents in Algebraic Semigroups, by M. Brion (Grenoble).
Abstract: The objects of the talk are algebraic semigroups, thatis, algebraic varieties equipped with an associative composition law
which is a morphism of varieties. We shall show that every such
semigroup S admits idempotent elements; moreover, the set E(S)of idempotents has an unexpectedly rich structure under additional
assumptions on S. For example, if S is commutative then E(S) isfinite; if in addition S is irreducible (as a variety), then E(S) is theset of faces of some rational polyhedral convex cone.

22. On Some Pro-p Groups from Infinite-Dimensional Lie Theory,(joint work with Bertrand Rémy), by I. Capdeboscq (Warwick).
Abstract: We initiate the study of some pro-p-groups arising frominfinite-dimensional Lie theory. These groups are completions ofsome subgroups of incomplete Kac-Moody groups over finite fields,with respect to various completions of algebraic or geometric origin.
We show topological finite generation for the pro-p Sylow subgroups
in many complete Kac-Moody groups. This implies abstract simplicity of the latter groups. We also discuss with the question of(non-)linearity of these pro-p groups.
3. Finitely Presented Groups with Infinitely Many non-Homeomorphic
Asymptotic Cones, by A. Ould Houcine (UdL, Lyon 1).
Abstract: We construct a finitely presented group with infinitely
many non-homeomorphic asymptotic cones. We also show that theexistence of cut points in asymptotic cones of finitely presented groups
does, in general, depend on the choice of scaling constants and ultrafilters. This is a joint work with D. Osin.
4. Satake Isomorphism for Kac-Moody groups, by S. Gaussent (UdL,St-Étienne).
Abstract: We use a Kac-Moody version of the Bruhat-Tits building
to define the spherical Hecke algebra H for a Kac-Moody group overa local non-archimedean field. Then, we prove the Satake isomorphism between H and the algebra of Weyl-invariant elements of theLooijenga’s coweight algebra. The proof involves a parabolic retraction of the hovel onto an extended tree inside the generalization ofthe building. This is a joint work with G. Rousseau.
5. Trisections in Module Categories and Quasi-Directed Components,by F.U. Coelho (USP).
Abstract: The purpose of this talk is to discuss new results concerning the existence of trisections in the module categories in relation tothe components of the Auslander-Reiten quiver called quasi-directed.
Such components appear naturally in several classes of algebras defined by homological properties.
Wednesday, April 24, 2013:
1. Definable Subgroups of Minimal Simple Groups of Finite Morley Rank,by T. Altınel (UdL, Lyon 1).
Abstract: Morley rank is an ordinal dimension that generalizes theZariski dimension in algebraic geometry over algebraically closed
fields. Simple groups that admit a finite rank are conjectured to belinear algebraic. This conjecture has frequently been attacked using
inductive methods inspired by finite group theory. Such a strategy
necessitates a better understanding of the minimal simple groups offinite Morley rank.

3In a joint work with Jeffrey Burdges and Olivier Frecon, we haveobtained a fairly precise structural description of the definable subgroups of most minimal simple groups of finite Morley rank. The
proof establishes an abstract Jordan decomposition and yields ananswer to an open question about the splitting of Borel subgroups in
one of the known configurations.
2. A Banachic Version of a Theorem of Delorme and Applications, byR. Tessera (UdL, ENS Lyon).
Abstract: Delorme proved in 1977 that a unitary representation with
non-trivial first cohomology of a connected solvable Lie group hasa dimension 1 subrepresentation. In 2000, Shalom proved that aslightly weaker property passes to lattices. With Cornulier, we give a
new geometric proof of Delome’s result, which allows us to extend it
to isometric actions on any reflexive Banach spaces. We also enlarge
the class of locally compact groups satisfying the conclusion.
3. The Profinite Topology on Groups, by P. Zalesskii (Brasilia).
Abstract: We shall start with a discussion of residual properties ofgroups and their interpretation in connection with the profinite completion. According to J-P. Serre a group G is good if the cohomology
b are naturally isomorphic
groups of G and its profinite completion G
on finite coefficients. The talk will be about residual properties andgoodness for arithmetic groups and groups of geometric nature.
Thursday, April 25, 2013:
1. Nonassociative Lie Theory, by I. Shestakov (USP).
Abstract: Non-associative Lie theory, that is, Lie theory for nonassociative products, appeared as a subject of its own in the works ofA.I. Malcev who constructed the tangent structures corresponding toMoufang loops. Its general development has been slow; nevertheless,by now many of the basic features of the theory have been understood, with much of the progress happening in the last ten years.
In the present talk we outline the non-associative Lie theory in general and review the recent developments, related with the notions ofSabinin algebras and non-associative Hopf algebras.
2. On the Classification of ZN -Graded Lie Algebras (joint work with K.
Iohara), by O. Mathieu (UdL, Lyon 1).
Abstract: In this talk, we will explain recent advances on the classification of simple ZN -graded Lie algebras. Here we consider Liealgebras whose homogenous components are one-dimensional.
Although this looks like a very abstract problem of algebra, the result
involves concrete Lie algebras, namely some affine lie algebras andsome sub-algebras of symbols of twisted PDO on the circle. Someparts of the proof are based on classical topics, like Jordan algebra
theory or the question of non local equivariant binary maps between
tensor densities modules over the circle.
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3. Algebraic Methods in Parameter Estimation, by R. Ushirobira (INRIA, Lille).
Abstract: In this talk, we will review some basic concepts of thetheory of automatic control and the algebraic structures related toit. We will show how the problem of parameter estimation can betreated by algebraic methods in an efficient manner.
4. A Global Version of Grozman’s Theorem (joint work with O. Mathieu), by K. Iohara (UdL, Lyon 1).
Abstract: The Witt algebra W is the Lie algebra of algebraic vector
fields on C∗ whose central extension is known as the Virasoro algebra. The classification of the Z-graded indecomposable W- modulewith multiplicity 1 has been achieved by I. Kaplansky and L. J. Santharoubane in 1984. In this talk, we explain the classification of
the W-equivariant bilinear maps M × N → P where M, N, P are
Z-graded indecomposable W-modules with multiplicity 1.
5. Free Symmetric Algebras in Skew Fields Generated by Orderable, byV.O. Ferreira (USP).
Abstract: The canonical involution on a non-abelian ordered groupcan be extended to an involution on the division ring of fractions of
the group algebra. With respect to this involution, a pair of symmetric elements generating a free subalgebra is shown to exist. (This isa joint work with J. Goncalves and J. Sanchez at IME-USP, Brazil.)Friday, April 26, 2013:
1. Jordan Algebras and Their One-sided Representations, by I. Kashuba(USP).
Abstract: This talk is a survey on the joint results with S. Ovsienko,
V. Serganova and I. Shestakov. It is devoted to the problem of
the classification of indecomposable Jordan bimodules over finitedimensional Jordan algebras when squared radical is zero.
Recall, that for a Jordan algebra J the category J -bimod of kfinite dimensional J -bimodules is equivalent to the category Umod of (left) finitely dimensional modules over an associative algebra
U = U(J ), which is called the universal multiplication envelope of
J . If J has finite dimension the algebra U is finite dimensional aswell. In particular, in accordance with the representation type of thealgebra U one can define Jordan algebras of the finite, tame and wild
representation types.
From the other side to each Jordan algebra corresponds a Lie algebra TKK (J ). Moreover there is a correspondence between thefinite dimensional Jordan modules over J and finite dimensional
Lie modules over TKK (J ).
This allows us to apply to the category J -bimod all the machinery
developed in the representation theory of finite dimensional algebras
as well as the representation theory of Lie algebras.
52. Full Groups of Minimal Homeomorphisms of the Cantor Space, byJ. Melleray (UdL, Lyon 1).
Abstract: Full groups of minimal homeomorphisms of the Cantor
space are useful objects, which remember the structure of the orbit decomposition of the space under the action of the homeomorphism. These groups have counterparts in ergodic theory which canbe endowed with a Polish group topology, enabling one to use Bairecategory methods to study them. I will explain why such a topology cannot exist in the topological context, and discuss some relatedproblems.
3. A Reduction Theorem for the Computation of the Yoneda Algebra,
by E. N. Marcos (USP).
Abstract: (This is a joint work in progress with Ed Green and DagMadsen.) We take in an Artin Algebra Λ an idempotents e and f withe+f = 1. Let Λ∗ = Lf Λf , we show that,with some strong hypothesis,
the Yoneda algebra i≥0 Λ/(ri Λ) is eventually isomorphic, as ringsL without unit, to i≥0 Λ∗ / ri (Λ∗ ) . We also show that this can begeneralized for the graded case.
4. Unimodularity and Calabi-Yau Duality for the Algebras of Differential Operators, by P. Le Meur (Clermont-Ferrand).
Abstract: The algebra of differential operators, or enveloping algebra, associated with a compatible action by derivations of a Liealgebra L on a commutative algebra S was defined by Rinehart(1963). The most classical examples arise from Lie algebras, fromPoisson structures and from Lie algebroids. Such a pair (S, L) is now
called a Lie-Rinehart algebra. In his study of Lie-Rinehart algebras,
Huebschmann established (1999) some Poincaré duality involving the(co)homology groups associated with (S, L). This duality is related
to the existence of a suitable modular class which he defines, andwhich is tightly related to the modular class introduced by Weinstein. On the other hand, in many examples, the enveloping algebraof (S, L) appears to be Calabi-Yau in the sense of Ginzburg. Thisis the case for Weyl algebras and for enveloping algebras of finitedimensional and unimodular Lie algebras. The talk shall present
some links between the duality and the modular class on (S, L), andthe Calabi-Yau duality on the corresponding algebra of differentialoperators.
5. Gradings and Fundamental Group of a Linear Category, by C. Cibils(CIMPA).
Abstract: The intrinsic fundamental group à la Grothendieck of a category over a field is obtained through its connected gradings, which
are related to its Galois coverings. In some cases there exists a universal (or a versal) grading. After considering the theoretical tools, I
will concentrate in the Schurian generated case, that is on linear cat-6egories which are generated by their spaces of morphisms which areone-dimensional : an universal grading exists. The results are basedon the joint paper on arXiv with Maria-Julia Redondo and Andrea
Solotar, ”On universal gradings, versal gradings and Schurian generated categories” to appear in J. Noncommut. Geom.

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