Results concerning chain conditions in rings or, more generally, modules give information about the lattice of ideals of the ring or submodules of the module. By considering modular lattices we can not only generalize theorems from Ring Theory but also prove results which can be applied to Grothendieck categories. We shall try to justify this approach by showing that it is natural and does give additional information not least because result automatically dualize. As one might expect generalizing results to modular lattices presents difficulties and we shall illustrate some of these.

An algebra group is a group of the form G = 1 + J where J = J(A) is the Jacobson radical of a finite-dimensional associative algebra A (with identity). A theorem of Z. Halasi asserts that, in the case where A is defined over a finite field F, every irreducible complex representation of G is induced by a linear representation of a subgroup of the form H = 1 + J(B) for some subalgebra B of A. In this talk, we assume that F has odd characteristic p and (A,s) is an algebra with involution. Then, s naturally defines a group automorphism of G = 1 + J, and thus we may consider the fixed point subgroup G(s). In this situation, we may use Glauberman's correspondence to show that every irreducible complex representation of G(s) is induced by a linear representation of a subgroup of the form H(s) where H = 1 + J(B) for some s-invariant subalgebra B of A. A particular situation occurs for Sylow p-subgroups of the classical groups of Lie type (defined over F). If time permits, we will also introduce the notion of a supercharacter and discuss some applications to Combinatorics.

The idea that a noncommutative algebra which is a quantisation of a commutative Poisson algebra should have structure and representation theory which reflects the Poisson structure of the underlying commutative algebra (or variety) has a long history — for example, it lies at the heart of Kirillov's orbit method. I will review these concepts, describe a little of the history, and explain some recent results and some conjectures in this area, with specific reference to some or all of the following: enveloping algebras of Lie algebras; quantised function algebras of semisimple algebraic groups; and symplectic reflection algebras.

In this talk, some aspects of the theory of orthogonal polynomials will be in discussion. The elements of a classical polynomial sequence (Hermite, Laguerre, Bessel and Jacobi) are eigenfunctions of a second order linear differential operator with polynomial coeficientes L, known as the Bochner's operator. In an algebraic manner, a classical sequence is also characterised through the so-called Hahn's property, which states that an orthogonal polynomial sequence is classical if and only if the sequence of its (normalised) derivatives is also orthogonal.

To begin with, it is shown that an orthogonal polynomial sequence (OPS) is classical if and only if any of its polynomials fulfils a certain differential equation of order 2k, for some positive integer k. The structure of such differential equation is thoroughly revealed, permitting to explicitly present the corresponding 2k-order differential operator L_k. On the other hand, as a consequence of Bochner's result, any element of a classical sequence must be an eigenfunction of a polynomial with constant coefficients in powers of L. With the introduction of the so-called A-modified Stirling numbers (where A indicates a complex parameter), we are able to establish inverse relations between the powers of the Bochner operator L and L_k.

The second part of this talk is focused on a generalization on the Hahn's problem. Given certain lowering operators O (linear operators that decrease in one unit the degree of a polynomial), we will expound the search of all the O-classical sequences, in other words, all the orthogonal polynomial sequences {P_n}_n such that {OP_n}_n ≥ 0 is also orthogonal.

I will review the relationship between graded algebras and projective varieties, in particular recalling how more flexible gradings than the usual degree of polynomials allows one to model substantial parts of the classification of algebraic varieties (and maps, where lack of factoriality is often the key point) in low dimension. As well as familiar examples, I will show how new orbifold Riemann-Roch theorems can be applied to extend known partial classifications.

The representation theories of symmetric groups and of general linear groups are linked through Schur-Weyl duality. In 1937, Brauer asked the following question: which algebra has to replace the group algebra of the symmetric group in this situation if one replaces the general linear group by its orthogonal or symplectic subgroup? As an answer he defined the Brauer algebra. We will discuss this, and also see how a theory of Young modules leads to another Schur-Weyl duality for Brauer algebras. This is joint work with Robert Hartmann, Anne Henke and Steffen Koenig.

In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical grassmannian (Brown, Goodearl and Yakimov); and torus invariant prime ideals in the quantum grassmannian (Lenagan, Rigal and I). The aim of this talk is to present these results and explore the reasons for this coincidence.

The aim of the talk is to review and compare some old and recent results on PI Öre extensions. A special attention will be paid to iterated Öre extensions which appear naturally in the context of quantum algebras.

A fundamental problem in ring theory is to determine and describe ideal extensions of a given ring A by another ring B, i.e. all rings R with A an ideal of R and ^R⁄_A isomorphic to B. One meets such problems (more general or more specific) in many studies, especially when looking for some examples. In the talk we will discuss several such problems, present some constructions of ideal extensions and mention their applications.

There are different ways to define Schur functions. We take the combinatorial approach by means of tableaux. Thus the Bender-Knuth involution on tableaux will show that they are symmetric functions.

Jacobi-Trudi determinants provide another expression for Schur functions. Both tableaux and determinants can be seen as lattice paths. Using the Lindström-Gessel-Viennot method, the tableau and determinant formulas are unified.

Usando a noção de preunidade introduzida por Caenepeel and De Groot, apresentamos uma noção geral de produto cruzado num contexto débil, a qual generaliza as definidas por Blattner, Cohen e Montgomery, Doi e Takeuchi para álxebras de Hopf e a mais geral obtida por Brzezinski. Também, os produtos cruzados definidos recentemente por nós para álxebras de Hopf débis e extensões débis C-cleft associadas a estruturas entrelaçadas débis, são casos particulares desta teoria.

Using the notion of preunit introduced by Caenepeel and De Groot, we present a general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the more general obtained by Brzezinski. Also, the crossed products defined by us, for weak Hopf algebras and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory.

I consider nilpotent ideals in the Borel subalgebra of a simple Lie algebra. The question that I am going to address is: if the Lie algebra and its Borel subalgebra are fixed, how many such ideals are there which are annihilated by applying the Lie bracket K times? The first result concerning this question is due to Dale Peterson which says that the answer is 2ⁿ in the case of Abelian ideals (i.e., for K = 1), where n is the rank of the Lie algebra. I shall give a complete answer for any K and any Lie algebra. Interestingly, there are intimate connexions with the enumeration of lattice paths, which is not always well understood.

This is joint work with George Andrews, Luigi Orsina and Paolo Papi.

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G′| + 1, where |G′| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. In our talk we determine G for which upper (or lower) Lie nilpotency index are maximal or almost maximal, or the next highest possible value.

First, I will give a pedagogical introduction to the classical Robinson-Schensted correspondence between permutations and pair of Young tableaux (Schensted insertions, plactic monoid of Lascoux-Schützenberger, geometric interpretation with “light and shadow”, and Fomin's growth diagrams or “local rules”). This last presentation of the correspondence with “local rules” comes from an algebraic approach using the Heisenberg algebra defined by two operators U and D satisfying the commutation relation UD = DU + Id. I will finish the talk by giving some open combinatorial problems comming from quantum physics, with an algebraic approach extending the one given above with Young tableaux.

We introduce a mapping from orientations to spanning trees in graphs, from regions to hyperplane bases in real hyperplane arrangements, from reorientations to bases in oriented matroids (in order of increasing generality). We call it the active orientation-to-basis mapping, in reference to an extensive use of activities, a notion depending on a linear ordering, first introduced by W.T. Tutte for spanning trees in graphs. The active mapping, which preserves activities, can be considered as a bijective generalization of a polynomial identity relating two expressions of the Tutte polynomial of a graph, a hyperplane arrangement or an oriented matroid — one in terms of activities of reorientations, and the other in terms of activities of bases.

Specializations include bijective versions of well-known enumerative results related to the counting of acyclic orientations in graphs, or of regions in hyperplane arrangements. Another interesting feature of the active mapping is a tight relationship it establishes between linear programming and the Tutte polynomial.

(joint work with Emeric Gioan)

Os ideais equimúltiplos foram extensivamente estudados por causa das suas relações com a geometria, em particular no estudo das singularidades algébricas. Os módulos equimúltiplos foram introduzidos por A. Simis, B. Ulrich e W. Vasconcelos como uma classe particular de módulos ideais. Neste seminário usaremos a teoria das reduções de módulos para provar alguns critérios para que um módulo equimúltiplo seja de classe principal estendendo, para módulos, alguns dos resultados conhecidos para ideais.

A ringed space is a pair (X,R) of a topological space X and a topological ring R, R is a sheaf of rings over the base space X. Rings are related to sheaves of rings in such a way that every ring can be represented as a ring of global sections of a ringed space (X,R) (by a global section we mean a continuous function from the topological space X to R). Our aim in this talk will be to define sheaves of rings, the decomposition space of a ring and then to give the basic topological properties of these notions. Then we show that every ring is isomorphic to a ring of global sections of a ringed space. At the end, we will give a characterization of biregular rings in terms of stalks of sheaves of rings to demonstrate the use of this representation, and how local results can be lifted to global results in special ringed spaces.

Seja R um anel. Um R-módulo N diz-se c-injectivo se, para qualquer submódulo fechado L de qualquer R-módulo M, todo o homomorfismo de L para N se estende a M. Estudaremos estes módulos, em particular sobre domínios de Dedekind.

Calabi-Yau algebras are objects of great interest in Representation theory, Algebraic Geometry and Physics. It is known from the work of Ginzburg that, in dimension 3, these algebras are, in general, defined via a potential. In this talk, we will focus on the case where we have a 3-CY algebra presented by a quiver with potential. In this setting, we will prove that performing mutations (as introduced by Derksen, Weyman and Zelevinsky) induce derived equivalences between the original and the mutated algebras.

Throughout, all rings are commutative with non-zero identity and all modules are unitary. Let R be a ring and let M be an R-module. A proper submodule N of M is called p-prime (resp. p-primary) if rm ∈ N for r ∈ R and m ∈ M implies that either m ∈ M or r ∈ p = (N : M) (resp. m ∈ M or r ∈ p = √(N : M)). The radical of N in M, denoted by rad_M(N), is defined to be intersection of all prime submodules of M containing M, or rad_M(N) = M in case no prime submodule of M contains N. The envelope of N in M is the set of elements rm of M such that r ∈ R, m ∈ M and rkm ∈ N for some positive integer n. In general, E_M(N) is not a submodule of M, for a given submodule N. We denote by <E_M(N)> the submodule of M generated by the set E_M(N).

In this paper, we consider the problem of finding a generating set of the radical and envelope of a submodule N of R-module M from given generating set the submodule when R = k[x₁, …, x_n] is the polynomial ring over a field k and M is free module Rm for some positive integer m. Although a closely related problem that finding primary decomposition of a submodule in above setting has been extensively studied, it seems to be there is no method eveloped for finding a generating set of radical or envelope of a submodule. If q is a primary ideal of a ring, then it is well-known that rad(q) is a prime ideal. In module case, however, Q a primary submodule of M does not necessarily imply that rad_M(Q) is a prime submodule.

Moreover, it is not always true for submodules N and L of M that rad_M(N \ L) = rad_M(N) \ rad_M(L). Hence knowing a primary decomposition of a submodule N does not automatically give a generating set for radical of that submodule.

An R-module M is said to be a C1-module if every closed submodule of M is a direct summand. In this seminar, it is introduced and investigated the concept of the τ-C1 module where τ is a hereditary torsion theory on Mod-R. τ-C1 modules are a generalization of C1-modules.

We discuss the calculation of integral cohomology ring of LG/T and ΩG. First we describe the root system and Weyl group of LG, then we give some homotopy equivalences on the loop groups and homogeneous spaces, and calculate the cohomology ring structures of LG/T and ΩG for affine group A_n. We introduce Groebner-Shirshov basis and combinatorial integers (m,n_j) which play crucial roles in our calculations and give some interesting identities among these integers. Last we calculate generators for ideals and rank of each module of graded integral cohomology algebra in the local coefficient ring Z[½].

We consider R the ring of polynomials in x₁, x₂, …, x_n with coefficients from an infinite field k, i.e. R = k[x₁, x₂, …,  x_n] and the subsets R_d of all polynomials of degree d. The direct sum R = ⊕_d ∈ NR_d is called the degree grading. Let I ⊂ k[x₁, x₂, …,  x_n] be a homogeneous ideal. We define the vector space V_d(I) = R_d ∩ I. If {g₁, …, g_s} is a basis of I involving only homogeneous polynomials, then V_d(I) is generated by all monomial multiples x₁^α₁x₂^α₂…x_n^α_n g_i with α₁ + α₂ + ··· + α_n = d.

The known methods for finding a generating set for syzygy module of I involves a Gröbner basis computation. In this study our aim is to find a generating set for syzygy modules using only techniques of linear algebra. This will give us a method for finding H-basis of any polynomial ideal involving only techniques of linear algebra. It is well known that H-bases is more suitable than Gröbner basis in some applications such as solving polynomial systems and interpolation. Hence finding an H-basis without doing whole computation of Gröbner basis will be usefull.

Um hipermapa orientado regular é um terno ordenado composto por um grupo finito, designado grupo de monodromia, e dois geradores. O número de hiperfaces é o número de órbitas do grupo cíclico gerado pelo primeiro dos geradores considerados. Apesar desta caracterização, esta classificação não é meramente algébrica, de facto, um hipermapa regular orientado corresponde a um mergulho celular de um hipergrafo conexo numa superfície compacta. Um hipermapa orientado regular é reflexivo se admite uma reflexão, isto é, um automorfismo do hipergrafo subjacente invertendo a orientação global da superfície, caso contrário o hipermapa diz-se quiral.

Neste seminário será apresentada uma classificação dos hipermapas orientados regulares com um número primo de hiperfaces.

The determinantal formula gives an expression of Specht modules by permutational modules in the Grothendieck ring of modules over the symmetric group S_n. It was shown by Zelevinskii in “Resolutions, dual pairs and character formulas” (J. Funktsional. Anal. i Prilozhen. 21 p. 74–75) that for each Specht module over a field of zero characteristic there is a coresolution by permutational modules that realises this expression. We construct permutational coresolutions for Specht modules over a field of positive characteristic.

Roughly speaking, linear preserver problems consist on the characterization of linear maps between operator algebras that leave invariant certain quantities, properties or subsets.

In the last years, many mathematicians have studied linear preserver problems not only on operator algebras but on more general Banach algebras. In particular, a substantial attention has been paid to Kaplansky's problem concerned with the characterization of linear maps preserving invertibility, and also the related problem of spectrum preserving linear maps.

Based on some several partial positive results and some counterexamples, the Kaplansky's problem nowadays asks when a surjective unital invertibility preserving linear map between unital semisimple Banach algebras is a Jordan isomorphism. The problem is still open even for C*-algebras. Partial solutions are known for real rank zero C*-algebras, and semisimple Banach algebras with essential socle.

New important contributions to the study of linear preserver problems in the algebra L(H) of all bounded linear maps on an infinite dimensional complex Hilbert space, have been recently made by Mbekhta, Mbekhta, Rodman and Semrl, and Mebkhta and Semrl. They characterize unital surjective linear maps on L(H), preserving the set of Fredholm and semi-Fredholm elements in both directions.

In this talk we present some background on the Kaplansky's problem, and linear preserver problems, making emphasis in the two more favorable settings, that is, C*-algebras of real rank zero, and semisimple Banach algebras with essential socle.

We shall investigate some properties of Rad-supplemented modules and in general τ-supplemented modules where τ is a radical for R-MOD. One of the main questions we shall answer is when are all left R-modules Rad-supplemented. Whenever possible the related results are given in general for a radical τ for R-MOD.

The aim of this talk is to present some links between the combinatorics involved in Morita equivalence of two classes of algebras arising in noncommutative algebraic geometry: quantum tori (or MacConnell-Pettit algebras) and deformations of type A kleinian singularities (a particular case of generalized Weyl algebras).

Quantum tori are algebras of noncommutative Laurent polynomials, and we will review in the first part of the talk their definition and properties related to Morita equivalence. Then we will define generalized Weyl algebras and give results concerning their Morita equivalence in the particular case we are interested in here.

In the last part of the talk, we will focus on the links between these two studies.

This is joint work with Andrea Solotar.

Yang-Mills algebras have been defined and studied by A. Connes and M. Dubois-Violette. These algebras are interesting due to their applications in theoretical physics, specially in String Theory. In this talk I will presents results relating their representation theory to Weyl algebras. I will also describe their Hochschild homology.

We shall review some results in the theory of Hopf algebras and their tensor categories of representations, putting special emphasis in the classification problem in the semisimple case. We shall discuss the construction of extensions and semisimple Hopf algebras arising from matched pairs of finite groups as well as the more recent construction of group theoretical Hopf algebras. We shall also discuss normal Hopf subalgebras and simplicity and its relation with the twisting construction.

Contents:

1. (Semisimple) Hopf algebras. Tensor categories and the twisting construction. Character algebra and the Class Equation. Applications: Masuoka's pⁿ-Theorem.
2. Extensions. Normal Hopf subalgebras. Abelian extensions. The Kac exact sequence. Group-theoretical Hopf algebras.
3. Examples in low dimension. Simple Hopf algebras obtained by twisting.

References:

Part I.

H.-J. Schneider, Lectures on Hopf algebras, Trabajos de Matemática 31/95, FaMAF (1995)

Part II.

A. Masuoka, Extensions of Hopf algebras, Trabajos de Matemática 41/99, FaMAF (1999)
A. Masuoka, Hopf algebra extensions and cohomology, Math. Sci. Res. Inst. Publ. 43 (2002), 167–209.
P. Etingof, D. Nikshych and V. Ostrik, On fusion categories, Ann. Math. (2) 162, 581–642 (2005).

Part III.

C. Galindo and S. Natale, Simple Hopf algebras and deformations of finite groups, to appear in Math. Res. Lett. Preprint arXiv:math/0608734v2.
C. Galindo and S. Natale, Normal Hopf subalgebras in cocycle deformations of finite groups. Preprint arXiv:math/0708.3407.
S. Natale, Semisolvability of semisimple Hopf algebras of low dimension, Memoirs Amer. Math. Soc. 186, 123 pp. (2007).

Let a monoid S act on a ring R by injective endomorphisms. An over-ring A(R;S) of R is called the S-Cohn-Jordan extension of R if

1. the action of S on R extends to an action of S on A(R;S) by automorphisms and
2. for any a ∈ A(R;S), there exists s ∈ S such that s·a ∈ R.

A classical result of P.M. Cohn, which was originally formulated in much more general context of Ω-algebras (instead of rings), says that such an extension always exists provided the monoid S possesses a group S−1S = G of left quotients. The aim of the talk is to present a series of results relating various algebraic properties of R and that of A(R;S). For example primeness, Goldie conditions and other finiteness conditions will be considered. Some possible applications to the skew semigroup rings R # S and skew polynomial rings R[x;σ,δ] will be also discussed.

It is easy to check that the sum of any family of two-sided nil ideals of an associative ring is a nil ideal as well. Does the same hold for left nil ideals? Though this question looks very elementary and was raised more than seventy years ago (in 1930 by Köthe) it is still open. It is called Köthe's nil ideal problem and is one of the most famous open problems in ring theory. Attempts to solve it led to many interesting, deep and sometimes surprising results. There are also many related open problems. The aim of the talk is to present several such problems as well as some old and new results obtained in the area. In particular it will be shown that Köthe's problem is equivalent to a problem raised in 1969 by Andrunakievich.

We develop a (co)homology theory for the algebraic operad corresponding to algebras with bracket. We relate it with the Hochschild (co)homology of the underlying associative algebra. We use the homology theory with trivial coefficients to characterize universal central extensions of perfect algebras with bracket and to study the problem of lifting automorphisms or derivations in a covering.

Vamos definir acções parciais de grupos em conjuntos e a correspondente acção envolvente parcial. Toda acção parcial em conjuntos possui uma envolvente, daremos uma ideaia da prova desse fato. Também observaremos que acções parciais de grupos estão em correspondencia biunívoca com acções de certos semigrupos inversos associados ao grupo. Finalmente, consideraremos acções parciais e acções de semigrupos inversos em álgebras.

A relação entre códigos corretores e curvas algébricas tem determinado as linhas atuais de pesquisa sobre curvas. Nesta palestra pretendemos explicar rapidamente esta relação e fazer um resumo das linhas de pesquisa e dos resultados principais.

Consideraremos álgebras associativas livres sujeitas a acções lineares de álgebras de Hopf e discutiremos a estrutura das subálgebras de invariantes dessas acções.

Mais precisamente, seja R uma álgebra associativa livre sobre um corpo e seja H uma álgebra de Hopf pontual de dimensão finita que age linearmente em R. Mostraremos como construir uma correspondência de Galois entre o conjunto das subálgebras de H que são coideais à direita e o conjunto das subálgebras livres de R que contêm a subálgebra de invariantes R^H da acção de H em R. Uma consequência da existência dessa correspondência é o fato de R^H ser uma subálgebra livre de R. Além disso, indicaremos como a correspondência de Galois pode ser utilizada para mostrar que, no caso em que H é gerada por elementos group-like e skew-primitivos, R^H é uma álgebra finitamente gerada se, e somente se, R tiver posto finito e a acção de H em R for, de fato, escalar.

Esses resultados são extensões de resultados conhecidos de Kharchenko, Dicks e Formanek para acções lineares de grupos por automorfismos em álgebras livres.

Distributividade em anéis e módulos vem sendo estudados com certa intensidade desde os anos 70. A discussão de propriedades e da estrutura de co-álgebras a partir de uma abordagem da teoria dos módulos vem assumindo grande papel ultimamente. É neste espírito que pretendemos estudar a estrutura das co-álgebras cujo reticulado de co-ideais à direita é distributivo. Em particular, obtemos que uma tal co-álgebra é um co-produto de co-álgebras cujo reticulado de coideais à direita é uma cadeia (co-álgebras de cadeia). Estas últimas, por sua vez, são caracterizadas como duais finitos de anéis de cadeia noetherianos cujo corpo residual é uma álgebra de divisão finito-dimensional sobre seu corpo base. Dado uma álgebra de divisão D de dimensão finita sobre um corpo k, e uma estrutura de D-bimódulo sobre D, podemos construir uma co-álgebra de cadeia como uma co-álgebra tensorial. Apresentaremos tais construções. Esta palestra está baseada em um trabalho conjunto com Christian Lomp.

Referências Bibliográficas:

[1] Cuadra, J., Gómes-Torrecillas, J.; Serial coalgebras, J. Pure Appl. Algebra 189 (2004), 89–107.
[2] Green, J. A.; Locally finite representations, J. Algebra 41 (1976), 137–171.
[3] Lomp, C., Sant'Ana, A.; Chain coalgebras and distributivity, CMUP–2006–32, preprint (2006)
[4] Sthephenson, W.; Modules whose lattice of submodules is distributive, Proc. London Math. Soc. 28(3) (1974), 291–310.

The construction of the homogeneous coordinate ring of a toric variety, that I.M. Musson, D.A. Cox and others discovered in the early 1990s, takes a fan Δ and creates a torus action on an open subset of an affine space whose quotient is the toric variety of Δ. We reverse this process. Let k be an algebraically closed field of characteristic 0. Let H be an algebraic torus times a finite abelian group acting diagonally on the affine space

We create various fans whose toric varieties are the quotients under the action of an open subset of X. Let S be the Laurent polynomial ring

with n = r + s. Then S is multi-graded by the finitely generated group A = Hom(H,k^×). We prove the following statements to be equivalent:

• the fan is not contained in a half-space;
• S^H = k;
• every graded component S_a (a ∈ A) is finite dimensional.

When S_a is finite dimensional we can give a bounded polyhedron (polytope) whose number of lattice points equals the dimension of S_a. Let D(X) be the ring of differential operators on S and let D(X)^H be the subring of invariants under the action of H. We give results on the existence of finite dimensional representations of D(X)^H and we use the graded components of S to give families of finite dimensional D(X)^H-modules with “enough members”.

A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have aXb ≠ 0 whenever 0 ≠ a,b ∈ R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This talk is devoted to an investigation of uniform bounds of primeness in matrix rings over fields. The existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the n×n matrix ring over F.

Em Teoria dos Códigos, um código é um conjunto de palavras definidas sobre um dado alfabeto. O objectivo desta teoria é construir códigos que corrijam o máximo de erros na transmissão de informação por um dado canal, tendo em conta ainda a sua taxa de informação. O problema fundamental desta teoria é calcular o tamanho máximo que os códigos podem ter em função do comprimento das palavras e da sua distância mínima. Nesta palestra apresenta-se uma breve exposição sobre este problema, e mostra-se como este estudo pode ser usado em códigos de ADN.

In the eighties, Chatters and Jordan have developed the notion of noncommutative noetherian unique factorisation domain. In this talk, I will recall this notion and show that the algebra of generic quantum matrices which is a noncommuative deformation of the variety of matrices is actually a noncommutative noetherian UFD. To achieve this aim, we will generalise to the noncommutative world some results coming from the commutative world, and use the rich combinatorics of the algebra of generic quantum matrices. This is joint work with Tom Lenagan and Laurent Rigal.

By a normed algebra we mean a real or complex (possibly nonassociative) algebra A endowed with a norm ||·|| satisfying ||xy|| ≤ ||x||.||y|| for all x,y ∈ A. A complete normed associative algebra will be called a Banach algebra. A normed algebra is called norm-unital if it has a unit 1 such that ||1|| = 1. Unitary elements of a normunital normed associative algebra A are defined as those invertible elements u of A satisfying ||u|| = ||u⁻¹|| = 1. By a unitary normed associative algebra we mean a norm-unital associative normed algebra A such that the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. Relevant examples of unitary Banach algebras are all unital C^∗-algebras and the discrete group algebras l₁(G) for every group G. We are interested in the development of a general theory of unitary Banach algebras. For a real (respectively, complex) norm unital Banach algebra A, consider Property (S) which follows:

(S)  There exists a linear (respectively, conjugate-linear) algebra involution on A mapping each unitary element to its inverse.

It is known that Property (S) is fulfilled in the case that A is a unital C^∗-algebra, a discrete group algebra, or a finite-dimensional unitary Banach algebra. However, in general, unitary Banach algebras need not satisfy Property (S). We show that unitary semisimple commutative complex Banach algebras satisfy Property (S), and that, endowed with the involution given by such a property, they become hermitian ∗-algebras. From this theorem we get new characterization of unital C^∗-algebras. Later, we study Property (S) in the noncommutative case. To this end, we introduce “good” groups and prove that, if A is a real or complex unitary semisimple Banach algebra such that the group U_A is good, then A satisfies Property (S). It seems to be an open problem whether or not every group is good. We give some equivalent reformulations in terms of unitary normed algebras.

Historically, unitary Banach algebras have been considered only from an associative point of view, and the main topic of interest has been characterize C^∗-algebras among them. In this talk, we leave the associative scope in order to deal by the first time with nonassociative unitary normed algebras.

A more detailed abstract can be seen here.

Um R-módulo M é dito distributivo se seu reticulado de R-submódulos é distributivo, isto é, para todos R-submódulos A, B e C de M, vale

Um anel R é dito distributivo à direita, se o R-módulo R_R é distributivo. Um dos primeiros trabalhos importantes sobre anéis e módulos distributivos foi publicado por Stephenson no ano de 1974. A partir daí, muitos trabalhos sobre estas classes tem sido publicado. Por exemplo, em 1976, Brungs mostrou que se R é um domínio ou um anel noetheriano à direita, então R é distributivo se, e somente se, R é localmente um anel de cadeia à direita (reticulado de ideais à direita linearmente ordenado por inclusão). O resultado de Brugs permanece válido se R for um anel localizável. Recentemente Puninski, e depois Tuganbaev, construiram exemplos de anéis distributivos não localizáveis. A presente palestra se destina a apresentar uma série de resultados importantes sobre estes anéis e módulos e culminará com um resultado de Ferrero e Sant'Ana que caracteriza os anéis distributivos em função dos seus ideais saturados e que estende o resultado de Brungs, mesmo para anéis não localizáveis.

Na decada 50, Johnson e Utumi desenvolveram o conceito do anel de quocientes maximais. Em 1956, Findlay e Lambek apresentaram uma versão da teoria de módulos desta construção enquanto que Gabriel, no seu trabalho fundamental em 1961, unificou estas ideais com a localização classica de Ore e Goldie utilizando a noção de localização em categorias abelianas.

Neste processo da localização, os módulos de torsão resp. módulos livre da torsão foram designados módulos singulares resp. não singulares (ou poliformes). No nosso estudo procuramos conceitos duais com o objectivo de obter uma técnica de localização de co-álgebras. Vamos discutir estes novos conceitos e as suas relações com propriedades conhecidas de co-álgebras.

Seja G um grupo finito que age parcialmente sobre um anel com unidade R. Se R é Artiniano (Noetheriano), semiprimo, J-semisimples, semisimples, então em que condições o skew anel de grupo parcial e o subanel dos invariantes parciais preservarão estas propriedades? Faremos uma breve revisão do seja uma ação parcial e apresentaremos respostas a estas questões em casos em que existem envolvente para a ação parcial e apresentamos fórmulas que relacionam o radical de Jacobson e radical primo de R, com o radical do skew anel de grupo parcial e o subanel dos invariantes parciais, respectivamente.

Na teoria de anéis, a localização é uma técnica importante para a construção de vários anéis de quocientes. A dualização desta técnica na categoria de co-módulos vem sendo estudada por alguns autores. Nesse contexto, por exemplo, coberturas codensas dualizam extensões densas, etc.

Apresentamos a construção de cobertura maximal codensa de uma co-álgebra usando cobertura projetiva. Para uma co-álgebra C de dimensão finita é obtida uma cobertura maximal codensa D no sentido apresentado. No entanto, para uma co-álgebra coprima de dimensão infinita não podemos usar cobertura projetiva para construir sua cobertura maximal codensa isso se deve à ausência de co-módulos projetivos não-nulos na categoria dos co-módulos sobre tal co-álgebra. Vamos apresentar algumas definições básicas e após discutiremos tais idéias.

Morfismos irredutíveis (entre módulos sobre uma álgebra de dimensão finita) são morfismos que não admitem fatorizações não triviais. Na teoria de representações de álgebras de dimensão finita, tais morfismos são exatamente os que aparecem nas chamadas seqüências de Auslander-Reiten, essenciais no desenvolvimento atual da teoria. Nosso objetivo neste seminário será, após recordarmos o contexto acima, relatarmos recentes desenvolvimentos no estudo de compostas de morfismos irredutíveis.

Partial actions of groups on algebras have been studied and applied first in C*-algebras and then in several other areas of mathematics. In a pure algebraic context they were recently introduced and studied by M. Dokuchaev and R. Exel in Trans. Amer. Math. Soc. 357 (2005). In this lecture(s) we will present an introduction to the subject and will consider related questions like partial skew group rings, partial skew polynomial rings, etc.

In this talk we consider several primeness properties of comodules and corings, transferred from primeness properties of modules and rings. Making use of the so called internal coproduct of fully invariant subcomodules, we extend the notion of (pre-)coprime coalgebras over base fields to pre-coprime (pre-cosemiprime) comodules for corings over arbitrary ground rings. We study also coendo-prime (coendo-semiprime) comodules, i. e. comodules that are prime (semiprime) as canonical modules over their rings of colinear endomorphisms, and prime (semiprime) comodules, i. e. modules that are prime (semiprime) as rational modules over the dual rings of their ground corings. Moreover we clarify the relations between these different primeness properties and simplicity (semisimplicity) and irreducibility of the comodules under consideration. The results we get are applied then to present, study and characterize corings satisfying these different primeness conditions and clarify their relations with simple (semisimple) and irreducible corings.

Vai ser apresentada uma abordagem, baseada nos diagramas de Feynman, da teoria das representações de álgebras semi-simples. O objectivo, entre outros, é a decomposição, em componentes irredutíveis, de produtos tensoriais de uma álgebra de Lie simple de dimensão finita.

Vai ser mostrado como isso tem a ver com uma nova classificação de álgebras de Lie simples de dimensão finita (além da, bem conhecida, de Elie Cartan).

Generalized Weyl algebras, as defined by V. V. Bavula [St. Petersburgh Math. J., 1993], are a family of algebras containing both some classical objects (enveloping algebras and their prime quotients, Weyl algebras, invariant sub-algebras,…) and their quantum analogues. These algebras are generated by two generators over a k-algebra R, with relations given thanks to an automorphism and a central element of R.

We are interested in problems of classification for such algebras. In this talk based on a joint work with L. Richard, we will consider isomorphisms between generalized Weyl algebras, giving a complete answer to this problem in the quantum case for R = k[h]. We will give separation results too up to rational equivalence and Morita-equivalence for these algebras.

Em condições adequadas, o método das órbitas coadjuntas de Kirillov permite parametrizar as representações irredutíveis de um grupo linear unipotente. No caso geral, conjectura-se que uma parametrização deste tipo também seja possível. Neste seminário, definimos supercaracteres e superclasses de um grupo unipotente finito e salientamos o paralelismo com a teoria dos caracteres irredutíveis, classes de conjugação e órbitas coadjuntas. Discutimos também possíveis relações com algumas estruturas algébricas «próximas».

Em 1985 Drinfeld e Jimbo introduziram independentemente uma família de deformações a um parâmetro q da álgebra universal envolvente associada a uma álgebra de Lie complexa e semi-simples de dimensão finita g. Estas álgebras quânticas U_q(g) têm ligação a áreas diversas da matemática e física teórica, como sejam a teoria de nós, a mecânica estatística e a teoria de representação de álgebras de Kac-Moody.

A decomposição de U_q(g) como produto tensorial de U_q(g)⁻, U_q(g)⁰ e U_q(g)⁺, induzida pela decomposição triangular g = g⁻⊕h⊕g⁺, sugere U_q(g)⁺ como análogo quântico da álgebra universal envolvente da álgebra de Lie nilpotente g⁺. Neste seminário irei discutir vários resultados relativos às álgebras U_q(g)⁺ e às suas teorias de representação, parte dos quais foram obtidos no âmbito do meu programa de doutoramento.

Recordarei resultados clássicos sobre a estrutura e teoria de representação de álgebras de Lie nilpotentes, tendo por objectivo evidenciar as diferenças e semelhanças entre os panoramas clássico e quântico. Alguns conhecimentos básicos de álgebras de Lie e teoria de aneis serão assumidos, mas a palestra será acessível a não-especialistas destas áreas e a alunos de mestrado e do último ano da licenciatura.

Estruturas de Poisson em variedades diferenciáveis estão intimamente ligadas a estruturas de álgebras de Lie em algum espaço (possivelmente de dimensão infinita). Neste seminário vou tentar dar uma descrição de alguns problemas típicos da Geometria de Poisson do ponto de vista da teoria de Álgebras de Lie.

We will discuss the results on the computation of the Hochschild homology and cohomology of generalized Weyl algebras obtained in [1] together with M. A. Farinati and A. Solotar. We will describe the context and motivations that led us to the study of these algebras, we will explain the general ideas behind the method of proof, and connect this to related work that is still to be done.

[1] M. A. Farinati, A. Solotar, M. Suarez-Alvarez, Hochschild homology and cohomology of generalized Weyl algebras, Ann. Inst. Fourier (Grenoble) 53 no. 2, (2003) 465–488. arXiv:KT/0109025.

Dados números complexos α, β e γ, a álgebra «down-up» é a algebra A(α,β,γ) sobre os números complexos gerada por duas indeterminadas u e d com as relações

• d²u = αdud + βud² + γd;
• du² = αudu + βu²d + γu.

Nesta palestra determinamos o centro de A(α,β,γ) no caso β e α² + 4β não nulos.

Let R be a PID (principal ideal domain), possibly noncommutative, and A a matrix over R. One of the problems left over from the 1930s is: What happens to the Smith Normal Form of A when R is actually noncommutative? In other words: find a canonical form for A (under matrix equivalence) in this generality. More abstractly, this asks whether two such matrices A, B of the same size, say m×n, are equivalent if and only if the factor (right) module Rⁿ/(row-module of A) is isomorphic to the analogous factor module for B (the “only if” is trivial). Counter-examples of rank 1 were apparently well known. It turns out that there are no other counter-examples: The “if” implication holds in all ranks > 1; and, in those ranks, a possibly canonical form described by Nakayama in 1938 is actually a canonical form. The proof involves some ideas from the K-theory of noncommutative noetherian rings.

This talk will look at a recent application, due to Diracca and Facchini, of a well-known result in graph theory, namely, Hall's Marriage Theorem, to a Krull-Schmidt type theorem in the the theory of modules. Hall's Marriage Theorem gives a criterion for matchings in a bipartite graph, while a theorem of Krull-Schmidt type examines the uniqueness of decompositions of particular types of modules into direct sums of indecomposable modules.

This talk will look at (not-so-recent) work by H. Simmons on a lattice-theoretic approach to the study of modules of finite uniform dimension and modules for which every factor has finite uniform dimension. In particular, we will discuss an “escalator condition”, introduced by Simmons, in a complete upper continuous modular lattice, which he uses to describe modules for which every factor has finite uniform dimension.

This is joint work with Robert Guralnick and Charles Odenthal.

Corings C over an associative ring A were introduced by Sweedler as generalisation of coalgebras over commutative rings and have recently resurfaced in the theory of Hopf-type modules.

Coseparable corings are defined by dualising separable ring extensions. A relationship between coseparable A-corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is constructed for a coseparable coring with a grouplike element.

Regresso ao programa do seminário.