XVIII Latin American Algebra Colloquium S˜ao Pedro, SP, Brazil, August 3rd – 8th, 2009 Held at Hotel Fonte Colina Verde, S˜ao Pedro, SP, Brazil. http://www.ime.usp.br/

(1)

XVIII Latin American Algebra Colloquium

S˜ao Pedro, SP, Brazil, August 3rd – 8th, 2009

Held at Hotel Fonte Colina Verde, S˜ao Pedro, SP, Brazil.

http://www.ime.usp.br/∼xviiicla xviiicla@ime.usp.br

(2)

(3)

Organized by:

• Universidade de S˜ao Paulo

• Instituto Nacional de Ciˆencia e Tecnologia de Matem´atica (INCTMat)

Abstracts

Self-dual codes over F_q[u]/(u^t) Ricardo Alfaro

University of Michigan-Flint, USA

Two characterizations of self-dual codes over Fq[u]/(u^t) are determined in terms of linear codes overF_q.An algorithm to produce such self-dual codes is also established.

On Moore’s conjecture: overview and some new results Eli Aljadeff

Technion-Israel Institute of Technology, Israel

A theorem of J.P. Serre (1969) says that if Γ is a torsion free group and H is a subgroup of finite index then they have the same cohomological dimension.

In 1976 J. Moore posed a conjecture which is a far reaching generalization of Serre’s theorem:

If Γ is torsion free and H is a subgroup of finite index then the projective dimension of any ZΓ-moduleM coincides with the projective dimension ofM as a module overZH. In particular (and in fact equivalently) anyZΓ-moduleM which is projective overZH is projective also over ZΓ.

The conjecture is known for large families of groups, as solvable groups, linear groups and some special groups (as the Thompson’s group). In the lecture I will present some old results (joint work with Cornick, Ginosar and Kropholler) together with some recent results (joint work with Udi Meir).

From trisections in module categories to quasi-directed components Edson Ribeiro Alvares

UFPR, Brazil

Joint with Assem, I., Coelho, F.U., Pe˜na, M.I, Trepode, S.

We define a special type of trisections in a module category, namely the compact trisections which characterise quasi-directed components. We apply this notion to the study of laura algebras and we use it to define a class of algebras with predictable Auslander-Reiten components.

(22)

On partial actions of Hopf algebras Marcelo Muniz Silva Alves

UFPR, Brazil

Partial group actions on algebras were first studied in the context of operator algebras, but soon enough became an independent topic of interest in ring theory. Partial actions of Hopf algebras were introduced as a natural generalization of partial actions of groups (which correspond, in a certain way, to partial actions of the Hopf algebra kG). In this talk it will be shown that important results regarding partial group actions also hold in the Hopf case, as the existence of an envelopingH-module algebraB for a partial H-module algebraA, and of a strict Morita context between the partial smash product A#H and B#H. Results concerning partial coactions of a Hopf algebraH and will also be presented, as well as some results on the extensionA^H ⊂A, whereA^H is the subalgebra of (partial) invariants ofA.

On pointed Hopf algebras with non-abelian group Nicol´as Andruskiewitsch

Universidad Nacional de C´ordoba, Argentina

I will survey recent developments on the classification of finite-dimensional pointed Hopf algebras with non-abelian group. Concretely, I will describe two main topics:

i) How to attach a generalized root system, or equivalently a Weyl groupoid, to a completely reducible Yetter-Drinfeld module W. Reference: [AHS]. This is done by looking at the Nichols algebraB(W); I will try to avoid technicalities and concentrate on some applications.

ii) How to show that for a given finite non-abelian group G, there are no finite-dimensional pointed Hopf algebras with group G. So far, we know that this happens for the alternating groups An,n >6 orn= 5; for some sporadic groups (some of the Mathieu, some of the Janko, the Suzuki group and the Held group); and for a few finite groups of Lie type. References: [AF, AFZ, AFGV1, AFGV2, F, FGV]. This is done by a variety of techniques concerning special subracks of the different conjugacy classes ofG.

References

[AF] N. Andruskiewitsch and F. Fantino. New techniques for pointed Hopf algebras. arXiv:0803.3486v1

<http://arxiv.org/abs/0803.3486>. 29 pages. To appear in Proceedings of the Sixth Workshop in Lie Theory and Geometry, Contemp. Math.

[AFGV1] N. Andruskiewitsch, F. Fantino, M. Gra˜na and L. Vendramin, Finite-dimensional pointed Hopf algebras with alternating groups are trivial, submitted.

[AFGV2] N. Andruskiewitsch, F. Fantino, M. Gra˜na and L. Vendramin, On pointed Hopf algebras associated to sporadic groups, in preparation.

[AFZ] N. Andruskiewitsch, F. Fantino and S. Zhang. On pointed Hopf algebras associated with the symmetric groups. arXiv:0807.2406v1<http://arxiv.org/abs/0807.2406>. 14 pages. Manuscripta Math., accepted.

[AHS] N. Andruskiewitsch, I. Heckenberger and H.-J. Schneider, The Nichols algebra of a semisimple Yetter- Drinfeld module, arXiv:0803.2430v1. Amer. J. Math., to appear.

[F] F. Fantino, On pointed Hopf algebras associated with Mathieu groups, arXiv: 0711.3142v2 [math.QA].

[FGV] S. Freyre, M. Gra˜na and L. Vendramin, On Nichols algebras over GL(2, Fq) and SL(2, Fq), J. Math. Phys. 48 (2007), 123513-1 – 123513-11.

22

(23)

C´odigos de controle da paridade e c´odigos de Goppa Jaime Edmundo Apaza Rodriguez

UNESP, Brazil

Os códigos controle da paridade surgiram no século passado como exemplo de uma fam´ılia de códigos detectores de um erro simples. Seu nome provém de um código binário que acrescentava um s´ımbolo extra para que o número de uns fosse par. Inicialmente, os códigos detectores e corretores de erros foram criados usando unicamente conceitos de álgebra e teoria dos números.

Posteriormente, em 1977, V. D. Goppa introduziu uma nova forma de construir códigos lineares usando curvas algébricas definidas sobre corpos finitos. Esses códigos são conhecidos hoje como os códigos geométricos de Goppa. Neste trabalho usamos algumas das ideias de Goppa para construir a classe de códigos de controle da paridade e exibimos alguns exemplos. Para isso usaremos a técnica de restri¸cão de um código linear, concretamente, a restri¸cão de um código de Goppa racional.

Spinor class fields for lattices in function fields Luis Arenas-Carmona

Universidad de Chile, Chile

The theory of spinor class fields allows the study of the set of maximal orders in a central simple algebra over a number field, or the set of quadratic lattices that are isometric at every completion of the number field. We extend this theory to the function field case through the use os schemes to study how the spinor class field depends on the choice of an affine subset of a projective curve.

(24)

On Cox rings of K3 surfaces Michela Artebani

Universidad de Concepci´on, Chile

LetX be a smooth projective surfaceXoverCwith finitely generated Picard group Pic(X).

The Cox ring ofX is the graded ring

R(X) = M

D∈Pic(X)

H⁰(X,O_X(D)).

It is known thatR(X) is a polynomial ring if and only ifX is a toric surface [2] and an explicit description of the Cox ring is available for Del Pezzo surfaces [3]. In general, it is even difficult to decide if the Cox ring of a surface is finitely generated.

In [1] we investigate Cox rings ofK3 surfaces i.e. simply connected compact complex surfaces with trivial canonical bundle. We prove that the Cox ring of a K3 surface is finitely generated if and only if its effective cone is polyhedral. Moreover, we compute the Cox ring of some K3 surfaces which either have Picard number two or are double covers of rational surfaces. An example of both types is the generic K3 surface with Picard lattice isometric to

U(2) =

0 2 2 0

.

In this case theZ²-graded Cox ring is:

R(X)∼= C[a₁, a₂, b₁, b₂, c]

(c²−f4,4(a, b)) , where deg(a_i) = (1,0), deg(b_i) = (0,1) and deg(c) = (2,2).

References

[1] M. Artebani, J. Hausen, A. Laface,On Cox rings of K3-surfaces, arXiv:0901.0369v2.

[2] D. Cox,The homogeneous coordinate ring of a toric variety.J. Algebraic Geom. 4(1), 17–50, (1995).

[3] A. Laface, M. Velasco,Picard-graded Betti numbers and the defining ideals of Cox rings, (2007).

Cluster-tilted algebras without clusters Ibrahim Assem

Universit´e de Sherbrooke, Canada

This is a report on a work with progress with Thomas Br¨ustle and Ralf Schiffler. We give an algorithm allowing to start from a tilted algebra and construct the transjective component of the corresponding cluster-tilted. In the Dynkin case, this yields the whole Auslander-Reiten quiver.

We also introduce a notion of reflection allowing to obtain all tilted algebras corresponding to a given cluster-tilted.

24

(25)

Partial actions of ordered groupoids on rings Dirceu Baggio

UFSM, Brazil

In this joint work with A. Paques and D. Flores we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring.

We present sufficient conditions under which the partial skew groupoid ring is either associative or unital. Also, we show that there is a one-to-one correspondence between partial actions of an ordered groupoid G on a ring R, in which the domain of each partial bijection is an ideal, and meet-preserving global actions of the Birget-Rhodes expansionG^BR ofGonR. Using this correspondence we prove that the partial skew groupoid ring is a homomorphic image of the skew groupoid ring constructed through the Birget-Rhodes expansion.

Questions on towers of function fields motivated by applications in multi-party computation and secret sharing

Alp Bassa

CWI & Leiden University, The Netherlands

In a series of papers, Cascudo, Chen, Cramer and Xing have shown how towers of algebraic function fields find applications in the construction of “multi-party computation friendly” secret sharing schemes. In this talk, I will try to outline this connection and discuss some of the questions about towers, which emerge naturally from it.

Weight of codewords from evaluation codes Peter Beelen

Technical University of Denmark, Denmark

Let F be a finite field and R a finite dimensional F-algebra. Further let Ev : R → Fⁿ be an isomorphism of F-algebras. Error-correcting codes can be obtained from this setup by evaluating all elements in a linear subspace L of R. Many interesting codes can be described in this way, such as algebraic geometry codes, generalized Reed-Muller codes and toric codes.

In this talk we will investigate the weight of codewords from codes obtained in this way by studying certain linear operators on R. Also we will show the connection with several other known techniques to estimate the weight of a codeword.

(26)

Train algebras which satisfy the additional identity (x²−ω(x)x)² = 0 Antonio Behn

Universidad de Chile, Chile

(Joint work with Irvin Roy Hentzel, Department of Mathematics, Iowa State University.) In the following we assume that A is a commutative algebra over a field F of characteristic not 2. The principal powers of x ∈A are defined as followsx¹ =x and xⁿ⁺¹ = xⁿx ∀ n≥1.

The plenary powers of an elementx ∈A are defined as x^[1] =x and x^[n+1]=x^[n]x^[n] ∀n≥1.

We observe thatx^[2] =x².

IfA has a non-zero algebra homomorphism ω:A→F, then the pair (A, ω) is called abaric algebra. A baric algebra (A, ω) is called aplenary train algebra, if there are scalarsγ1, . . . , γr−1

such that every x∈A satisfies the followingplenary identity of degree 2^r−1: (1) x^[r]+γ1ω(x)ⁿ¹x^[r−1]+γ2ω(x)ⁿ²x^[r−2]· · ·+γr−1ω(x)ⁿ^r−1x= 0

wherer ≥2 andni = 2^r−i−1(2ⁱ−1) for alli∈ {1, . . . , r−1}.

A baric algebra (A, w) is called a principal train algebra if there existγ₁,· · · , γt−1 ∈K such that every elementx∈A satisfies the equation

(2) xⁿ+γ1ω(x)xⁿ⁻¹+· · ·+γn−1ω(x)ⁿ⁻¹x= 0

In recent work with Prof. I.R. Hentzel we consider plenary train algebras of arbitrary rank.

Given the identity 1

we consider the rootsλ1, λ2, . . . , λr of the associative polynomial x^r+γ1x^r−1+γ2x^r−2· · ·+ γr−1x. A sufficient condition for the algebra to satisfy the additional identity (x²−w(x)x)²= 0 is that the pairwise products of these roots are all distinct, i.e. λiλj 6=λ_lλ_k ∀(i, j)6= (l, k).

We also show that a train algebra with train identity (1) which satisfies the additional identity (x²−w(x)x)²= 0 has an idempotent as long as

r−1

X

i=1

(r−i)γi6= 0.

Another interesting fact is that ifAis a commutative algebra which satisfies (x²−w(x)x)² = 0, thenA is principal train if and only if it is plenary train.

If x ∈ A is such that ω(x) = 1, then the subalgebra of A generated by x has a unique idempotent, and we can fully characterize its structure.

References

[1] I. M. H. Etherington. Commutative train algebras of ranks 2 and 3.J. London Math. Soc., 15:136–149, 1940.

[2] J. Carlos Guti´errez Fern´andez. Principal and plenary train algebras.Comm. Algebra, 28(2):653–667, 2000.

[3] Alicia Labra and Avelino Suazo. On plenary train algebras of rank 4.Comm. Algebra, 35(9):2744–2752, 2007.

Derived tame local and two-point algebras Viktor Bekkert

UFMG, Brazil

We determine representation type of the bounded derived category of finitely generated modules over finitely generated complete local and two-point algebras.

This is a joint work with Yuriy Drozd (Institute of Mathematics, Ukraine) and Vyacheslav Futorny (Universidade de S˜ao Paulo, Brazil).

26

(27)

Almost split sequences for Hopf Algebras Fernando Araujo Borges

UFPR, Brazil

In this poster we will present results regarding almost-split sequences in the context of group algebraskG(following Auslander and Carlson), such as the behaviour of almost-split sequences under tensoring by indecomposable modules. We will also present extensions of some of these results for Hopf algebras (finite-dimensional, with involutive antipode); for example, if M is a indecomposableH-module, then the trivial modulekis a summand ofEndk(M) iff tensoring by M the almost-split sequence that ends at the trivial module kgives a non-split sequence. Fol- lowing work of Green, Marcos and Solberg, we present sufficient conditions for these equivalent properties to hold.

The multi-Frobenius non-classical curves Herivelto Borges

The University of Texas at Austin, USA

An irreducible curveF defined overF_q is calledq-Frobenius non-classical if the imageF r(P) of each simple pointP of F under the Frobenius map lies on the tangent line at P.

Based on [2], Hefez and Voloch extended the study of theq-Frobenius non-classical curves in [1], where some interesting arithmetic and geometric properties of such curves were first pointed out.

In this talk, I will present and characterize all irreducible plane curves defined overFq which are simultaneously Frobenius non-classical for different powers ofq. Such characterization gives rise to many previously unknown curves which turn out to have some interesting properties.

For instance, for n≥3 a plane curve which is both q- and qⁿ-Frobenius non-classical will have its number ofF_qⁿ-rational points attaining the St¨ohr-Voloch bound.

References

[1] A. Hefez and J.F. Voloch, Frobenius non classical curves, Arch. Math.54, (1990) 263–273.

[2] St¨ohr, K-O. and Voloch, J.F., Weierstrass Points and Curves over Finite Fields, Proc. London Math. Soc.(3) 52(1986)1–19.

On the classification of irreducible modules over finite simple Lie conformal superalgebras Carina Boyallian

We construct all finite irreducilble modules over Lie conformal superalgebras of type W, S and K. We also give a complete description of them in terms of differential forms.

(28)

On numerical semigroups and algebraic codes Maria Bras-Amoros

Universitat Rovira i Virgili, Spain

Given a rational pointP of a curve with Weierstrass semigroup Λ = {λ₀ < λ₁ < λ₂ < . . .} one can find an infinite basisz₀, z₁, z₂, . . . of the ring of functions having poles only atP such thatvP(zi) =−λ_i. Consider a set of rational points P1, . . . , Pn different fromP. To each finite subset W ⊆ N0 we associate the one-point code CW =< (zi(P1), . . . , zi(Pn)) : i∈ W >^⊥. By extension, we say that W is the set of parity checks ofC_W.

The ν sequenceof Λ is defined byνi = #{j ∈N0 :λi−λj ∈Λ}.The minimum set of parity checks that are needed to correctterrors is given byR(t) =e {i∈N0 :νi <2t+ 1} [10, 12]. The codes determined byR(t) are called Feng–Rao improved codes. Thee ν sequence is also used to define the order bound on the minimum distance for one-point codes [9, 13, 12].

We say that the points P_i₁, . . . , P_i_t (P_i_j 6=P) are generically distributed if no function generated by z0, . . . , zt−1 vanishes in all of them. Generic errors are those errors whose non-zero positions correspond to generically distributed points. Generic errors of weightt can be a very large portion of all possible errors of weightt [11]. By restricting the errors to be corrected to generic errors the decoding requirements become weaker and we are still able to correct almost all errors. The minimum set of parity checks that are needed to correct generic errors of weight tis Re^∗(t) ={i∈N0:λi6=λj+λ_k for anyj, k≥t} [7].

We can define a new sequence τ [5] that allows us to describe Re^∗(t) in terms of τ in a similar way as how R(t) is described in terms ofe ν. Then, by studying the increasingness of τ we can compare the codes determined by Re^∗(t) with standard codes and by studying the relation between the sequences ν and τ we can compare the codes determined by Re^∗(t) with the Feng–Rao improved codes.

We will present some results related to classification [2, 15, 16] and characterization [2, 5]

of numerical semigroups from the perspective of algebraic codes and their decoding and some results and open questions related to counting of numerical semigroups [14, 3, 4, 6, 8, 1].

References

[1] Victor Blanco and Justo Puerto. Computing the number of numerical semigroups using generating functions.

arXiv:0901.1228v1.

[2] Maria Bras-Amor´os. Acute semigroups, the order bound on the minimum distance, and the Feng-Rao improvements.IEEE Trans. Inform. Theory, 50(6):1282–1289, 2004.

[3] Maria Bras-Amor´os. Fibonacci-like behavior of the number of numerical semigroups of a given genus.Semi- group Forum, 76(2):379–384, 2008.

[4] Maria Bras-Amor´os. Bounds on the number of numerical semigroups of a given genus.J. Pure Appl. Algebra, 213(6):997–1001, 2009.

[5] Maria Bras-Amor´os. On numerical semigroups and the redundancy of improved codes correcting generic errors.Designs, Codes and Cryptography, Accepted. 2009.

[6] Maria Bras-Amor´os and Stanislav Bulygin. Towards a better understanding of the semigroup tree.Semigroup Forum, Accepted. 2009.

[7] Maria Bras-Amor´os and Michael E. O’Sullivan. The correction capability of the Berlekamp-Massey-Sakata algorithm with majority voting.Appl. Algebra Engrg. Comm. Comput., 17(5):315–335, 2006.

[8] Sergi Elizalde. Improved bounds on the number of numerical semigroups of a given genus. arXiv:0905.0489v1.

[9] Gui Liang Feng and T. R. N. Rao. A simple approach for construction of algebraic-geometric codes from affine plane curves.IEEE Trans. Inform. Theory, 40(4):1003–1012, 1994.

[10] Gui-Liang Feng and T. R. N. Rao. Improved geometric Goppa codes. I. Basic theory.IEEE Trans. Inform.

Theory, 41(6, part 1):1678–1693, 1995. Special issue on algebraic geometry codes.

[11] Johan P. Hansen. Dependent rational points on curves over finite fields—Lefschetz theorems and exponential sums. In International Workshop on Coding and Cryptography (Paris, 2001), volume 6 ofElectron. Notes Discrete Math., page 13 pp. (electronic). Elsevier, Amsterdam, 2001.

[12] Tom Høholdt, Jacobus H. van Lint, and Ruud Pellikaan. Algebraic geometry of codes. InHandbook of coding theory, Vol. I, II, pages 871–961. North-Holland, Amsterdam, 1998.

[13] Christoph Kirfel and Ruud Pellikaan. The minimum distance of codes in an array coming from telescopic semigroups.IEEE Trans. Inform. Theory, 41(6, part 1):1720–1732, 1995. Special issue on algebraic geometry codes.

28

(29)

[14] Jiryo Komeda. Non-Weierstrass numerical semigroups.Semigroup Forum, 57(2):157–185, 1998.

[15] Carlos Munuera and Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups.Adv. Math. Commun., 2(2):175–181, 2008.

[16] Anna Oneto and Grazia Tamone. On numerical semigroups and the order bound. J. Pure Appl. Algebra, 212(10):2271–2283, 2008.

Incidence algebras Rosali Brusamarello UEM, Brazil

LetX ={x₁, x2, . . . , xn}be a finite partially ordered set (poset) andKa field. Theincidence algebra I(X, K) is the K-algebra which has basis elements labelled E_ij for each i, j for which x_i ≤x_j and has multiplication defined via

E_ijE_kl=

(E_il if j=k 0 if j6=k

(i.e., theK-vector space with basis{E_ij}withx_i≤x_j and the multiplication defined as above).

SinceX is finite, we can always label the elements of X in such way thatxi ≤xj impliesi≤j, thenI(X, K) can be identify with a subalgebra of the upper triangular matrices U T_n(K).

These algebras were introduced in the 1960s by G.-C. Rota and R.P. Stanley, two famous combinatorialists. From then on there have been many articles on these kinds of algebras, sometimes called structural matrix algebras, and the results are somewhat scattered in the literature.

In this talk we will collect some of this results in order to answer the following natural questions.

(I) To what extent doesI(X, K) determines the setX?

(II) When doesZ(I(X, K)) =K? That is, when is I(X, K) a centralK-algebra?

(III) Can we describe the group Aut(I(X, K))? Are there cases when every automorphism of I(X, K) is inner?

(IV) When doesI(X, K) admit anti-automorphisms and involutions?

We will also present some new material on involutions. This talk is based on a joint work with Prof. David W. Lewis.

Free field realizations of the elliptic affine Lie algebras Andre Bueno

Let E be an affine complex elliptic curve with two points removed. Here, the elliptic affine

(30)

Classification of irreducible non-dense modules for A⁽²⁾₂ Thomas Bunke

IME-USP, Brazil

Letgbe an affine Kac-Moody algebra with Cartan subalgebrah, root system ∆ and centerCc.

A g-module V is called a weight if V =L

λ∈h^∗V_λ, V_λ = {v∈V |hv =λ(h)v for all h∈h^∗}.

If V is an irreducible weight g-module then c acts on V as a scalar, called level of V. For a weight g-module V, the support is the set supp(V) ={λ∈h^∗|Vλ 6= 0}. The root lattice Q is the free abelian group over ∆. If V is irreducible then supp(V) ⊂λ+Qfor some λ∈h^∗. An irreducible weight g-moduleV is called non-dense, ifsupp(V)(λ+Q,

This work contains the classification of irreducible non-dense modules for the Kac-Moody algebra A⁽²⁾₂ with at least one finite-dimensional weight subspace. The classification for non- dense irreducible A⁽¹⁾₁ -modules with a finite-dimensional weight subspace has been done by V.

Futorny [1]. The classification problem is also solved for all affine Kac-Moody algebras for non-zero level modules with all finite-dimensional weight subspaces (V. Futorny and A. Tsylke [2]). In these cases an irreducible module is either a quotient of a classical Verma module, or of a generalized Verma module, or of a loop module (induced from a Heisenberg subalgebra).

That this will hold for all irreducible non-dense modules of affine Kac-Moody algebras has been conjectured by V. Futorny [6]. With this work we confirm the conjecture for non-dense irreducibleA⁽²⁾₂ -modules with a finite-dimensional weight subspace.

We also obtain a classification of all possible supports for irreducibleA⁽²⁾₂ -modules. The proof is elementary and involves only the combinatorics of the root system.

References

[1] Futorny, V.,Irreducible non-denseA⁽¹⁾₁ -modules, Pacific J. of Math. Vol. 172, No. 1, 83-97, 1996

[2] Futorny, V., Tsylke, .A.,Irreducible non-zero level modules with finite- dimensional weight spaces for Affine Lie algebras, J. of Algebra 238, 426-441, 2001

[6] Futorny, V.,Representations of Affine Lie algebras, Queen’s Papers in Pure and Applied Mathematics 106, v. 1, Kingston, 1997

[7] Chari, V., Pressley, A.,Integrable Representations of Twisted Affine Lie Algebras, J. of Algebra 113, 438-46, 1986

Interaction of Koszul and Ringel dualities Paula Andrea Cadavid Salazar

IME-USP, Brazil

Joint work with Eduardo do Nascimento Marcos.

In the theory of quasi-hereditary algebras there are two classical dualities: the Ringel duality, associated with the characteristic tilting module, and the Koszul duality, associated with the category of linear complexes of projective modules. In [1, 2] it is shown that a certain class of Koszul quasi-hereditary algebras is stable with respect to taking of both Koszul and Ringel duals and that on this class of algebras the Koszul and Ringel dualities commute. In this poster we present and discuss these results.

References

[1] I. ´Agoston, V. Dlab, E. Luk´as, Quasi-hereditary extension algebras. Algebras and Representation Theory 6 (2003), no. 1, 97-117.

[2] V. Mazorchuck, S. Ovsienko, A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra. With an appendinx by Catharina Stroppel. J. Math Kyoto Univ. 45 (2005), no. 4, 711-741.

30

(31)

C´odigos geom´etricos de Goppa Joyce dos Santos Caetano

UFABC, Brazil

Este trabalho tem o apoio financeiro da Funda¸c˜ao Universidade Federal do ABC.

A teoria dos códigos é hoje uma área com uma enorme atividade de pesquisa, juntando es- for¸cos de matemáticos e engenheiros. Seus aspectos teóricos têm motivado ainda mais as já intensas investiga¸cões sobre corpos de fun¸cões em uma variável (ou equivalentemente, investiga¸cões sobre curvas algébricas) definidos sobre corpos finitos. Historicamente, os códigos de Goppa foram introduzidos por meio de rela¸cões polinomiais, onde V. D. Goppa abriu cam- inho para uma constru¸cão mais geral dos códigos algébricos-geométricos, nosso objetivo será apresentar a constru¸cão de tais códigos sobre curvas algébricas definidas sobre um corpo finito.

On the cohomology ring of truncated quiver algebras Leandro Cagliero

FaMAF - CONICET, Argentina

By results of Cibils and Locateli it is well known that the even cohomology ring of truncated quiver algebras over a field of characteristic zero (TQA’s) is described by certain special class of parallel paths called medals.

It is also known that, for TQA’s, the cup product of odd-degree cohomology classes is zero.

Therefore the medals play a central role in the study of the structure of the cohomology ring of a TQA.

In this talk we present a joint work with Paulo Tirao in which we give a complete description of the set of medals of a given quiver. This description allows us to determine the structure of the cohomology ring for a large family of TQA’s whose cup product is not zero. We think that we will be able to obtain a condition on the quiver Q and on the integer N in order to determine whether the TQA associated to Qand N has a non-trivial cohomology ring.

Una introducción a la teor´ıa de representaciones de álgebras Yohny Ferney Calderón Henao

Universidad de Antioquia, Colombia

En este poster se pretende describir dos de las técnicas que se volvieron esenciales a lo largo de los últimos años en el estudio de la teor´ıa de representaciones de álgebras, las cuales son:

Por un lado, los m´etodos diagram´aticos usados por Gabriel [ ver 1] y por otro lado, las secuencias que casi se dividen y morfismos irreducibles dadas por Auslander-Reiten en conjunto con la aljaba de Auslander-Raiten introducida por Ringel [ver 2].

Bibliograf´ıa.

(32)

Jacobians with complex multiplication Angel Carocca

PUC-Chile, Chile

Joint work with Herbert Lange and Rub´ı E. Rodr´ıguez.

We construct and study two series of curves whose Jacobians admit complex multiplication.

The curves arise as quotients of Galois coverings of the projective line with Galois group meta- cyclic groups G_q,3 of order 3q with q ≡ 1 mod 3 an odd prime, and G_m of order 2^m+1. The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the CM-types and show that the Jacobians are simple abelian varieties.

Weierstrass semigroups at several points and generalized Hamming weights Cicero Carvalho

UFU, Brazil

The Weierstrass semigroup at several points is a natural generalization of the well-known Weierstrass semigroup at a point. Its systematic study started with papers by Homma and Kim in mid 90’s, and since then it has been studied by many authors. In this talk we intend to present several results on this semigroup and also some results from an ongoing work on an application of this semigroup to the construction of Goppa codes which have lower bounds for their generalized Hamming weights better than the usual bound.

Freely connected algebra Diane Castonguay

UFG, Brazil

In this talk, we consider A to be a triangular algebra. To each bound quiver (Q, I) of A, one can define an homotopy relation which lead to the definition of the fundamental group of the bound quiver. This group also can be related to a Galois covering of the bound quiver and thus of the algebra. In this talk, we present the quiver of homotopy of an algebra as defined by P. Le Meur in [M06] together with some of the results of this work. Using this technique, we show that a monomial algebra (that is an algebra who admits a bound quiver where the ideal is generated by paths) without double arrows is freely connected. An algebra is freely connected if all its associated fundamental groups are free groups. Some known examples of freely connected algebras are the simply connected algebras and the finite representation type algebra.

[M06] Le Meur, P., Revêtements galoisiens et groupe fondamental d’algèbres de dimension finie, thesis de doutorado, Université Montpellier II - Sciences et Techniques du Languedoc - (2006-02-10), Claude Cibils (Dir.)

32

(33)

Divisibility of the Number of Solutions of Polynomials with Prescribed Leaders Monomials Francis Castro

University of Puerto Rico, Puerto Rico

In this presentation we consider the following problem over finite fields: To compute the exact divisibility of exponential sums of the type

S= X

x1,...,xn∈Fp

ψ(x₁· · ·x_d+x^d₁+· · ·+x^d_d+G(x₁, . . . , x_d)),

whereψis an additive character and the degree ofGis less thand. Also we compute the exact divisibility of the number of solutions of the polynomial equation

X₁· · ·X_d+X₁^d+· · ·+X_d^d+G(X₁, . . . , X_d) = 0,

overFp, where the degree ofGis less thand. This is a jointly work with I. Rubio and R. Figueroa.

Conjugacy separability and commensurability Sheila Campos Chagas

UFAM, Brazil

A groupGis called conjugacy separable if wheneverxandyare non-conjugate elements ofG, there exists some finite quotient ofGin which the images ofxandyare non-conjugate. We shall discuss criterions when conjugacy separability is preserved by commensurability. Several results (obtained jointly with Pavel Zalesskii) when it holds and examples of groups not satisfying this property will be presented.

Grados de morfismos irreducibles y tipo de representacion de un algebra Claudia Chaio

Universidad Nacional de Mar del Plata, Argentina

El objetivo de esta charla es mostrar como el grado de ciertos morfismos irreducibles puede darnos información sobre el tipo de representación de un álgebra. La noción de grado de un morfismo irreducible fue introducido por S. Liu en 1992. En este trabajo consideraremos álgebras de dimensión finita sobre un cuerpo algebraicamente cerrado. Daremos condiciones necesarias y suficientes para que el grado de un morfismo irreducible sea finito. Como aplicación de estos resultados, obtendremos una caracterización de cuando un álgebra es de tipo de representación finito.

Trabajo conjunto con P. Le Meur y S. Trepode.

(34)

Idempotents for some cyclic codes Gladys Chalom

IME-USP, Brazil

Explicit expressions for idempotents of minimal codes of the ring F_lC_pⁿ_q ( where l, p, andq are odd primes with some additional hypothesis) were obtained by Bakshi and Raka in (*) in terms of generating polynomials of the corresponding ideals. In this work (that is a joint work with Ferraz, Guerreiro and Polcino) we present this idempotents (and some code parameters) in terms of the generators of the cyclic groups, instead.

(*) Minimal Cyclic Codes of length pⁿq Gurmeet K. Bashi and Madhu Raka Finite Fields and theirs Aplications 9 (2003) 432-448.

The Weyr form for matrices and various varieties of commuting matrices John Clark

University of Otago, New Zealand

The Weyr form is little known but a useful alternative to the Jordan canonical form for matrices. In this talk we describe the Weyr form and indicate how it is useful in problems concerning approximately simultaneously diagonalisable matrices, using algebraic geometry techniques.

Abel maps for curves of compact type Juliana Coelho

UFF, Brazil

Fix a stable curve C of compact type. For each integer d ≥ 1 we contruct two degree-d Abel maps for C having different target spaces and we compare the fibers of both maps. As an application we get an characterization of stable hypereliptic curves of compact type having only two components.

This is a joint work with Marco Pacini (UFF).

34

(35)

Further examples of maximal curves Luciane Quoos Conte

UFRJ, Brazil

(Joint work with Miriam Abdon and Juscelino Bezerra.)

We show that the non-singular model of the curve X(q;n) defined over Fq²ⁿ, with n≥3 an odd integer, by the affine equation:

y^q²−y =x

qn+1 q+1 ,

is maximal. These curves have an interesting feature: the curveX(3; 3) is not Galois-covered by the Hermitian curve, in contrast to the curveX(2;n) which is Galois-covered by the Hermitian curve ([1], [2], [3]). We also explore the family of maximal curves X_b given over Fq²ⁿ by the affine equation:

y^N =−x^b(x+ 1), 1≤b≤N −1,

whereN is an odd divisor of qⁿ+ 1, and gcd(N;b) = gcd(N;b+ 1) = 1.Explicit isomorphisms between them are found.

References:

[1] Abd´on, M., Bezerra, J., Quoos, L.: Further examples of maximal curves. Journal of Pure and Applied Algebra, v. 213, pp. 1192-1196, 2009.

[2] Garcia, A., Torres, F.: On unramified coverings of maximal curves. In: Arithmetic, Ge- ometry and Coding Theory (AGCT-10), 2009, Luminy-Marseille. SminairesCongrs. Marseille : Societe Mathematique de France, vol. 21, pp. 35-42, 2009.

[3] Garcia, A., Stichtenoth, H.: A maximal curve which is not a Galois subcover of the Hermitian curve, Bulletin of the Brazilian Mathematical Society, vol.37, pp. 139-152, 2006.

Classifying indecomposable RA Loops Mariana Cornelissen

UFSJ, Brazil

In this work, we will provide a complete classification of finitely-generated indecomposable groups which quotient by its center is isomorphic to the direct product of two cyclic groups of prime orderp. Applying this result for the case p= 2, we will classify all the finitely-generated indecomposable RA loops, further generalizing the classification done in 1995, by Eric Jespers, Guilherme Leal and C´esar Polcino Milies for the finite case.

(36)

On partial skew Armendariz rings Wagner Cortes

UFRGS, Brazil

In this work we consider ringsRwith a partial actionαof an infinite cyclic groupGonR. We will introduce the concept of partial skew Armendariz rings and partial α-rigid rings. We will obtain that partial α-rigid rings implies partial skew Armendariz rings, the bijection between the set of right annihilator in R and the set of right annihilators in R[x;α] is equivalent to R be partial skew Armendariz ring. We study the relationship between Baerness, ascending chain condition on right annihilators property, right p.p.-property and right zip property betweenR and R[x;α] using the concept of a partial skew Armendariz ring. It will be provided examples where partial skew Armendariz, Baerness, quasi-Baerness, p.q.-Baerness and p.p-property do not imply the associativity of partial skew polynomial rings. Moreover, when (R, α) has enveloping action (T, σ), whereσis an automorphism ofT we show if the partial action is of finite type, thenR is either Baer or quasi-Baer or p.q-Baer or p.p. ring if and only ifT is either Baer or quasi-Baer or p.q.-Baer or p.p ring.

Tba

Severino Collier Coutinho UFRJ, Brazil

Fast finite field arithmetic for cryptographic applications Ricardo Dahab

UNICAMP, Brazil

Finite field arithmetic is pivotal in the efficiency of cryptographic methods based on elliptic curves. In this talk I will review some of the relevant finite field algorithms and discuss recent advances.

Modular invariants for group-theoretical modular data Alexei Davydov

University of Sydney, Australia

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.

36

(37)

Tba

Jos´e Antonio de la Pe˜na UNAM, Mexico

On the loop space of a 2-category Matias L. del Hoyo

UBA, Argentina

The construction of nerves and classifying spaces associate geometric objects to categorical structures. The relation between categories and the homotopy types of their classifying spaces has been widely studied, with important applications. Examples of this are Quillen’s development of higher algebraic K-theory, Thomason’s theorems on homotopy colimits, and Segal’s work on delooping topological spaces that arise from monoidal categories.

In this talk we shall discuss the notions of nerves and classifying spaces for 2-categories. We also state sufficient conditions under which the loop space ΩBC of the classifying space of a 2-categoryCcan be recover (up to homotopy) as the spaceB(Hom(x, x)), wherexis an object of C and Hom(x, x) is the category of maps x→x inC.

This theorem says that taking loops can be done algebraically, and extends in one direction the results of Segal about monoidal categories.

(38)

Invariantes de ´algebras de grupo parciais M. Dokuchaev

IME-USP, Brazil

Asrepresenta¸cões parciaisde grupos foram introduzidas em teoria de álgebras de operadores independentemente por R. Exel [4], e J. C. Quigg e I. Raeburn [5] como uma ferramenta rele- vante no estudo dessas álgebras. O estudo algébrico das representa¸cões parciais foi iniciado em [1]. De modo análogo ao caso das representa¸cões usuais existe uma álgebra, denominada álgebra de grupo parcial, que controla as representa¸cões parciais de grupos. Um resultado estrutural sobre essa álgebra foi obtido em [1] no caso de grupos finitos. Isso permite derivar conseqüências sobre a estrutura das representa¸cões parciais de grupos finitos por um lado e, por outro, estudar uma questão natural no contexto de álgebras de grupo parciais, a saber, a doProblema do Iso- morfismo: o que podemos dizer sobre os grupos GeH seK_parcG∼=K_parcH comoK-álgebras?

Com respeito a esse problema foi provado em [1] que para um dom´ınio K de caracter´ıstica zero, dois grupos abelianos finitos G e H s˜ao isomorfos se e somente se K_parcG ∼= K_parcH.

Um resultado análogo com char K > 0 foi obtido em [2]. Esses resultados mostram eviden- temente a diferen¸ca surpreendente entre a álgebra de grupo parcial e a usual, já que para um grupo abeliano finitoGa álgebra de grupoCGsobre os complexosC“lembra” só a ordem deG.

Em [3] obtemos uma lista completa de invariantes, em termos de subgrupos de G e suas K-representa¸c˜oes, para ´algebras de grupo parciais de um p-grupo finito G sobre um corpo K algebricamente fechado com charK 6=p.

References

[1] M. Dokuchaev, R. Exel, P. Piccione, Partial representations and partial group algebras,J. Algebra,226(1), (2000), 505-532.

[2] M. Dokuchaev, C. Polcino Milies, Isomorphisms of partial group rings,Glasgow Math. J., 46, (1), (2004), 161-168.

[3] M. Dokuchaev, J. J. Sim´on, Invariants of partial group rings ofp-groups,Preprint.

[4] Exel, R., Parcial Actions of Groups and Actions of Semigroups,Proc. Amer. Math. Soc. 126, no. 12 (1998), 143-151.

[5] Quigg, J. C., Raeburn, I., Characterizations of Crossed Products by Parcial Actions,J. Operator Theory37 (1997), 311-340.

38

(39)

Parametrization of representations of braid groups Claudia Egea

In these works we give a method to produce representations of the braid group Bn of n−1 generators τ₁, . . . , τn−1, (n ≤ ∞). These representations satisfy certain conditions and every representations which verify these conditions can be obtained by this method.

More specifically, the principal results are the following.

Given a 5-tuple (π, X, µ, ν, U), where (1) X is a measurable space, (2) π :Bn→Aut(X) is an action, (3) µ is aπ-quasi-invariant measure,

(4) ν :X →N∪ {∞} is aπ-invariant measurable function, (5) U :Bn×X → B(H) is a cocycle, where H = R

XH_xdµ(x) is a direct integral Hilbert space associated to (X, µ, ν), (that is ν(x) = dim(H_x) for almost allx∈X)

then we can construct a non unitary representation φ=φ(π,X,µ,ν,U) of Bn.

Moreover, if the cocycle U verifies that the operator U(τ_k, x) of H_x is diagonal, for almost all x∈X and for allk, 1≤k≤n−1, thenφ_{(π,X,µ,ν,U}₎ satisfies the following properties, for all k, j, 1≤j, k≤n−1,

(a) ψ(τ_k)ψ(τ_k)^∗ has discrete spectral decomposition;

(b) [ψ(τk)ψ(τk)^∗, ψ(τj)ψ(τj)^∗] = 0;

(c) ψ(τk)P ψ(τk)⁻¹ ∈ N, for all projection P ∈ N, where N is the von Neumann algebra generated by

F ={ψ(τ_k)ψ(τ_k)^∗:k= 1, . . . , n−1}

Conversely, given a representationsρ of Bn which satisfies (a), (b) and (c), then there exists a 5-tuple (π, X, µ, ν, U) such that ρ is equivalent toφ_{(π,X,µ,ν,U}₎.

Furthermore, we obtain conditions for these representations to be irreducible. Ifφ(π,X,µ,ν,U)

is a self-adjoint, µis ergodic, ν(x) = 1 for allx∈X, and U is not constant, thenφ_{(π,X,µ,ν,U}₎ is an irreducible representation.

References

[1] C.M. Egea, E. Galina, Some Irreducible Representations of the Braid GroupBn of dimension greater that n, to appear in Journal of Knot Theory and Its Ramifications, ArXiv math.RT/0809.4173v2 (2008).

[2] C.M. Egea, E. Galina, Parametrization of representations of Braid Groups, ArXiv math.RT/0904.0491 (2009).

(40)

Solvable Lie bialgebras Marco Farinati

UBA, Argentina

We propose a definition of solvability for Lie bialgebras. For these classe of Lie bialgebras, the extension problem can be “solved”, and we discuss applications to the classification problem in low dimensions.

Units in U(ZCp) Raul Ferraz IME-USP, Brazil

LetZ be the ring of rational integers, p be a prime,C_p be the (cyclic) group of order p and U(ZCp) be the group of units of the integral group ring of Cp. It is known that U(ZCp) can be writen as the direct product of ±C_p and a free abelian subgroup of finite rank of U(ZC_p).

In this work we give a multiplicatively independent subset ofU(ZCp), which generates a direct factor to ±C_p inU(ZCp), forp less than or equal to 67.

Relative invariant theory Walter Ferrer

Universidad de La Republica, Uruguay

We continue with the development of a relative viewpoint in geometric invariant theory. In particular we prove that if an algebra group acts in a linearly reductive way on an affine variety, then the variety can be split as a product of the unipotent radical of the group with an affine variety where the action is given by the action by a reductive group. The study of the action of the original group on the unipotent radical, is given by a twisting of the conjugation action.

The study of this twisting seems to be crucial in order to understand the new features provided by a relative reductive action.

Partial actions on semiprime rings: Globalization Miguel Ferrero

UFRGS, Brazil

In this talk we find conditions under which a partial action of a group on a semiprime ring, which does not necessarily have an identity element, has an enveloping action. In the case this enveloping action does exist it is also semiprime and is unique unless equivalence. In particular, if the semiprime ring has an identity element we recover the well-known result on the existence of an enveloping action formerly obtained by M. Dokuchaev and R. Exel.

40