We study a problem of birational equivalence for polynomial Poisson algebras over a field of arbitrary characteristic. More precisely, the quadratic Gel'fand-Kirillov problem asks whether the field of fractions of a given polynomial Poisson algebra is isomorphic (as Poisson algebra) to a Poisson affine field, that is the field of fractions of a polynomial algebra (in several variables) where the bracket of two generators is equal to their product (up to a scalar). We answer positively this question for a large class of polynomial Poisson algebras and their Poisson prime quotients. For instance, this class includes Poisson determinantal varieties.
A non-abelian tensor product of Hom-Lie algebras is constructed and studied. This tensor product is used to describe universal (α-)central extensions of Hom-Lie algebras and to establish a relation between cyclic and Milnor cyclic homologies of Hom-associative algebras satisfying certain additional condition.
In a previous paper, joint with Franois Dumas, we had studied a family of skew fields called “Mixed Weyl skew fields”. Recently we realized that for certain particular values of the parameter these appear as enveloping skew fields of certain super Lie algebras, opening the way to a super analog of the Gelfand-Kirilllov Hypothesis.
We present a purely combinatorial proof based on the explicit design of a bijective map, of the exact number of dominant regions having as a separating wall the hyperplane associated to the longest root in the m-extended Shi hyperplane arrangement of type A and dimension n − 1.
This is a joint work with Ricardo Mamede (CMUC, Portugal) and Eleni Tzanaki (University of Crete, Greece).
The main goal of this talk is to present the generalization of classical results that characterize universal central extensions of Leibniz (Lie) algebras to the framework of Hom-Leinbiz (Hom-Lie) algebras.
In the category of Hom-Leibniz (Hom-Lie) algebras we introduce the notion of Hom-co-representation (Hom-L-module) as adequate coefficients to construct the chain complex from which we compute the Leibniz (Lie) homology of Hom-Leibniz (Hom-Lie) algebras.
Nevertheless, in this generalization fails the key result that claims the composition of central extensions is central as well. This singularity motivates the introduction of new concepts as α-perfect Hom-Leibniz (Hom-Lie) algebra and α-central extension. Then the corresponding characterizations are given (see [1] and [2]). We also provide the recognition criteria for these kinds of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them (see [2]).
We introduce Hom-actions, semidirect product and establish the equivalence between split extensions and the semi-direct product extension of Hom-Leibniz algebras. We analyze the functorial properties of the universal (α)-central extensions of (α)-perfect Hom-Leibniz algebras. We establish under what conditions an automorphism or a derivation can be lifted in an α-cover and we analyze the universal α-central extension of the semi-direct product of two α-perfect Hom-Leibniz algebras (see [4]).
References:
[1] Casas, J. M.; Insua, M. A.; Pacheco Rego, N. On universal central extensions of Hom-Lie algebras, arXiv: 1209.5887 (2012).
[2] Casas, J. M.; Insua, M. A.; Pacheco Rego, N. On universal central extensions of Hom-Leibniz algebras, arXiv: 1209.6266 (2012).
[3] A. Makhlouf, A.; Silvestrov, S. Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008), no. 2, 51–64.
[4] Casas, J. M.; Insua, M. A.; Pacheco Rego, N. On the universal α-central extension of the semi-direct product of Hom-Leibniz algebras, arXiv: 1309.5204 (2013).
For a given module M, the intersection graph G(M) of M has been defined in [1, 2] as the simple graph whose set of vertices consists of non-trivial submodules of M and two distinct submodules U, V are adjacent if and only if U ∩ V is not zero. For a given ring R the intersection graph of R is defined as G(_RR), where _RR denotes the left R-module R. One of the first remarkable results in the area was obtained in [3], where an isomorphism problem on the intersection graph of finite abelian groups (regarded as modules over the ring of integers) was solved. From this result it follows in particular that if A, B are finite abelian p-groups for a prime p, then A ≃ B if and only if G(A) ≃ G(B). This result does not extend to all modules or even to all finite abelian groups but it shows that properties of the intersection graph of modules carry quite important information on modules and can be applied to study their structure. The aim of the talk is to present some results on the subject obtained in [1, 2, 4].
References:
[1] S. Akbari, R. Nikandish and M. J. Nikmehr: Intersection graph of sub-modules of a module. J. Algebra Appl. 11 (2012) 1250019.
[2] S. Akbari, H. A. Tavallaee and S. Khalashi Ghezelahmad: Some results on the intersection graph of rings. J. Algebra Appl. 12 (2013) 1250200.
[3] D. Bertholf, G. Walls: Graphs of finite abelian groups. Czechoslovak Math. J. 28 (103) (1978), 365–368.
[4] M. Nowakowska and E. R. Puczylowski: On the intersection graph of modules and rings. Preprint.
The Eulerian and Narayana polynomials are the generating functions of many combinatorial objects. They appear in several areas of mathematics and share a list of common properties. In this talk I will present a family of polynomials, most of them unknown, that have the same properties as the Eulerian and Narayana polynomials. In particular, a novel and simple proof of real-rooted-ness for the Narayana polynomials is obtained as a direct consequence of our approach. Likewise, we provide an explicit formula to compute the gamma numbers for our family of polynomials. We will say a few words on the geometric and combinatorial significance for these numbers.
The aim of the talk is to present some results on idempotents and clean elements of ring extensions R < S, where S stands for one of the rings S = R[x1,x2, … ,xn] (polynomial ring), S = R[x1±1,x2±1, … ,xn±1] (Laurent polynomial ring), S = R[[x1,x2, … ,xn]] (formal power series ring). In particular, criterions for an idempotent of S to be conjugate to an idempotent of R will be presented. Some applications will be given.
It is known that the polynomial ring is never a clean ring. A description of the set of all clean elements in the polynomial R[x] over a 2-primal ring (i.e. R/B(R) is a reduced ring, where B(R) is the prime radical of R) will be presented. It appears that, in general, the description of the set of all clean elements of R[x] is related to the Koethe's conjecture.
Ring epimorphisms are important from a representation-theoretical point of view as they provide a way to compare categories of modules. In this talk we will use the theory of localisations of rings and modules with respect to a torsion class (set up by Gabriel in his thesis) to construct ring epimorphisms. Moreover, we will describe precisely which ring epimorphisms can be obtained through this technique.
We consider the multiplication and differentiation operators x and ∂/∂x, which generate the Weyl algebra. Fix a nonzero polynomial h = h(x) and let y be the operator h.∂/∂x, so that x and y satisfy the commutation relation [y,x] = h. The algebra generated by x and y is denoted by Ah and is a subalgebra of the Weyl algebra. Noteworthy algebras in this family are the Weyl algebra A1, the enveloping algebra of the two-dimensional non-abelian Lie algebra Ax, and the Jordan plane Ax².
We will discuss the representations of the algebras Ah over a field of arbitrary characteristic, including all the irreducible representations. When the base field has prime characteristic some interesting combinatorics emerge, which we will discuss and phrase in the language of partitions.
This is joint work in progress with Georgia Benkart and Matt Ondrus.
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 long 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 this problem led to many interesting, deep and sometimes surprising results. There are also many related open problems. The aim of the talk is to present some new results obtained in the area.
We consider rings over which the injective hulls of simple modules are locally Artinian. We will give an outline of the problem and talk about recent developments. In particular, we will see when the injective hulls of simple modules over an Ore extension R = K[x][y;d] are locally Artinian. Specifically we will show that the injective hulls of simple modules over R are locally Artinian if and only if R is isomorphic to the first Weyl algebra or to the polynomial ring K[x,y].
This is a joint work with Paula A.A.B. Carvalho and Christian Lomp.
The depth of a subalgebra B in an algebra A is a number computed by considering tensor powers of modules. The notion touches on various areas of algebra with some nice classification arguments. There are a few different approaches to working with depth: this talk will reveal them.
A left R-module M is called morphic if M/im(f) is isomorphic to ker(f), for every endomorphism f of M, that is if the dual of the Nöther isomorphism theorem holds. Having as a starting point the categorial definition of morphic object given by Professor Grigore Calugareanu in the article “Morphic abelian groups” we recover most of its proprieties under suitable conditions.
In 1976 Erlich proved that an endomorphism f of a module M is unit regular if and only if it is regular and M/im(f) is isomorphic to ker(f), i.e. if the dual of the Nöther isomorphism theorem holds. After nearly 30 years, the interest for this dual property florished in 2003–2004 when Nicholson and Sánchez Campos publised a series of papers and nowadays the subject continues to be investigated. In this paper, some results are generalized in so called Puppe-exact categories (exact by Mitchell and Herrlich, Strecker) and most of them are recovered in abelian categories. We also discuss how the Mitchell Embedding Theorem can be used in order to reduce proofs from abelian categories to the categories of modules. Connection with unit regular and regular objects is made and some special examples are given together with some applications.
This is joint work with Grigore Calugareanu.
In this talk I will present Galois correspondence between subalgebras of a Hopf Galois extension and quotients of the structure Hopf algebra. I will also show some characterisations of its closed elements. I will also show some examples and a computation of lattices of generalised quotients of the universal enveloping algebra of a Lie algebra.
Ore localisation is a classical way to obtain ring epimorphisms with a flatness condition. Universal localisations of rings are generalisations of Ore localisations that, although not in general flat, still have interesting homological properties. In this talk we will discuss the notion of universal localisation and give sufficient conditions of a homological nature for a ring epimorphism to be a universal localisation.
This is joint work with Frederik Marks.
The ring of Hurwitz integers, being both a left and a right PID, could be thought of as being arithmetically fairly simple. However, the fact that it is not commutative entails some complications, but also some surprises, as well as some interesting open problems. In this talk we will describe Conway and Smith metacommutation problem, and some almost forgotten results on the related ring of Lipschitz integers. We will then present some results, obtained in joint work with Luis Roçadas, on some relationships between the arithmetic and the geometry of Lipschitz integers, namely certain divisibility relations between a given Lipschitz integer and some other integers built from it using the vector product. Some speculations on a possible integer factorization method involving quaternion arithmetic will also be presented.
We introduce a new notion of exact sequences in arbitrary pointed (non-exact) categories. We apply this notion to provide restricted versions of the Short Five Lemma and the Snake Lemma in the category of cancellative right semimodules over a semirings.
We prove a generalized version of a conjecture of A. Tyszka on the relative magnitude of solutions of certain linear systems with integer coefficients. The proof uses combinatorial and linear algebra techniques.
Proper classes were introduced by Buchsbaum to axiomatize conditions under which a class of short exact sequences of modules can be computed as Ext groups corresponding to a certain relative homology. In this talk, we shall present some typical ways of generating a proper class from a given class of modules or from a class of short exact sequences. As an application, the proper classes generated by simple modules and the least proper class determined by weak supplements will be discussed.
Last year, Izakhian and Rhodes developed a theory of representation of matroids by boolean matrices where all matroids become representable, unlike the case of classical representations over fields. In a joint work with Rhodes, we explore a similar approach in the context of finite graphs, leading to new notions of rank and independence of vertices and establishing new connections to matroid theory.
A ring R is called right McCoy if whenever non-zero polynomials f(x) and g(x) in R[x] satisfy f(x)g(x) = 0, then f(x)r = 0 for some non-zero r ∈ R. A ring R is Armendariz if f(x)g(x) = 0 implies that all pairwise products of coefficients of f(x) and g(x) are zero. In my talk I am going to present some new results concerning above mentioned classes of rings.
(This talk is based on joint work with Ryszard Mazurek.)
Recently, Brualdi et al. defined a partial order on an interesting class of binary matrices which generalizes the classical Bruhat order on the symmetric group, seen as the set of permutation matrices. We study the structure of this poset, and we give some enumerative results, e.g. about the maximal length of a chain and the largest size of an antichain.
This is a joint work with Carlos Martins da Fonseca and Ricardo Mamede.
Athanasiadis introduced separating walls for a region in the m-extended Catalan arrangement and used them to generalize the Narayana numbers.
In this paper, we fix a hyperplane in the m-extended Catalan arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.
A hypermaps is, in its topological form, a cellular embedding of a connected hypergraph on a compact connected surface. This definition can be translated to an algebraic form, allowing us to use group theory to prove some results about hypermaps. We will give some examples of how this can be done and we will also show that the topological version of a hypermap is sometimes an important tool to prove some algebraic results.
Given a group G grading a k-linear category C, a Galois covering of C has been associated to this data in [CM] using the “smash product” construction, which is provided of a G-action. Since the category of Galois coverings of C is equivalent to the full subcategory whose objects are the coverings obtained in this way, the computation of the fundamental group of C may be restricted to this subcategory. It is worth to notice that having a G-grading is the same as having a kG-coaction, or, whenever G is finite, a kG-action. The theory developed in [CS] suggests that one should extend the notion of grading of a k-linear category to the notion of H-module category, where H is a Hopf algebra, as it has been successfully done for algebras. Another option is to consider not only actions of groups on C but partial actions, generalizing in this way the theory of partial actions on k-algebras (see [AB] for more details and for bibliographical references). The guiding idea is that this may allow us to obtain more “coverings”. Our aim in this article is to develope both points of view, considering thus what we call partial H-module linear categories, with a particular interest on kG-module categories.
This talk is based on joint work with Edson R. Alvares, Marcelo M. S. Alves and Eliezer Batista.
References:
[AB] M. M. S. Alves and E. Batista; Enveloping Actions for Partial Hopf Actions, Comm. Algebra 38 (2010), 2872–2902.
[CM] C. Cibils and E. Marcos; Skew category, Galois covering and smash product of a category over a ring, Proc. Amer. Math. Soc. 134 (2006), no. 1, 39–50.
[CS] C. Cibils and A. Solotar; Galois coverings, Morita equivalence and smash extensions of categories over a field, Doc. Math. 11 (2006), 143–159.
Let R be a ring with identity. By a central ideal of R we mean an ideal of R which can be generated by central elements of R. An ideal I of R is called hypercentral provided there exists a transfinite chain of ideals of R
where for each ordinal 0 ≤ α < ρ the ideal Iα + 1/Iα is a central ideal of the ring R/Iα and Iα = ∪0 ≤ β < α Iβ for every limit ordinal 0 < α ≤ ρ. If I is a hypercentral ideal of R then the least such ordinal ρ from a chain of ideals of the above type is called the height of I. If M is a unitary right R-module and J an ideal of R then we set annM(J) = {m ∈ M : mJ = 0}.
Our starting point is a theorem of Robinson proved in 1974 that states that if M is a Noetherian right R-module and I a hypercentral ideal of R of height ω then there exists a positive integer n such that
There is a dual result for Artinian modules proved by Newell in 1976. Robinson showed that his theorem does not extend to hypercentral ideals of height ω + 1 and Dark (1976) showed that the same was true for Newell's theorem. A good source of examples of hypercentral ideals is to be found in group rings. Roseblade (1971) proved that every ideal of the integral group ring ZG is hypercentral if and only if the group G is hypercentral. This leads to a consideration of the Artin Rees Property and to certain chain conditions in groups and in rings.
We consider rings whose injective hulls of simple modules are locally Artinian. After a brief discussion of this ring theoretic property and after a list of examples, we consider this property for super Lie algebras and classify all finite dimensional complex nilpotent super Lie algebras whose enveloping algebras satisfy this property.
This is a joint work with C. Lomp.
I will discuss primitive ideals of the algebra of quantum matrices. In particular, I will explain how, roughly speaking, the problem reduces to the computation of the rank of certain integral matrices. Then I will explain combinatorial tools to compute the rank of these integral matrices.
For a comodule algebra over a Hopf algebra we construct a Galois correspondence between the complete lattices of of subalgebras and the complete lattice of generalised quotients of the structure Hopf algebra. The construction involves techniques of lattice theory and of Galois connections. Such a ‘Galois Theory’ generalises the classical Galois Theory for field extensions, and someimportant results of S. Chase and M. Sweedler, F. van Oystaeyen, P. Zhang and P. Schauenburg. Using the developed theory we positively answer the question raised by S. Montgomery: ‘is there a bijective correspondence between generalised subobjects of a Hopf algebra and its generalised quotients?’ for finite dimensional Hopf algebras.
The aim of the talk is to introduce the notion and present results on ring endomorphisms having large images. In particular, we will show that:
Examples showing that the assumptions imposed on R can not be weakened to R being a prime Goldie ring will be presented and some problems will be formulated.
This talk is concerned with the existence of a Dixmier map for nilpotent super Lie algebras and its applications to the representation theory of super Yang-Mills algebras. More precisely, we shall state results concerning the Kirillov orbit method a la Dixmier for nilpotent super Lie algebras, i.e. that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras. Moreover, our construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach. One key fact in the construction is the existence of polarizations for (solvable) super Lie algebras, generalizing the concept in the nonsuper case. As a corollary of the previous description, we obtain that the quotient of the enveloping algebra of a nilpotent super Lie algebra by a maximal ideal is isomorphic to the tensor product of a Clifford algebra and a Weyl algebra, and we determine explicitely the indices of both of them, we get a direct construction of the maximal ideals of the underlying algebra of enveloping algebra and also some properties of the stabilizers of the primitive ideals. All of these results can be used to study the representation theory of super algebras related to the super Yang-Mills theory of interest in physics.
In the first part of the talk we outline the basic ideas of synthetic approach to differential geometry. The main idea of this approach, which originates from considerations of Sophus Lie is very simple: All geometric constructions are performed within a suitable base category in which space forms are objects. In the second part we indicate how a synthetic method could be employed in the context of Noncommutative Differential Geometry.
This talk is addressed to mathematicians who have some very basic familiarity with general category theory culture and are familiar with elementary concepts of geometry and algebra. The aim is to explain synthetic approach to commutative and noncommutative geometry on two examples of geometric notions. First we explain all categorical ingredients that enter the synthetic definition of a principal bundle (in classical geometry) and then we show that noncommutative generalisation of this definition yields in particular principal comodule algebras or faithfully flat Hopf-Galois extensions.
I will report on recent joint work with I. Heckenberger. We are studying systematically the Nichols algebra (or quantum symmetric algebra) of a Yetter-Drinfeld module over any Hopf algebra (with bijective antipode) which is a finite direct sum of finite-dimensional irreducible Yetter-Drinfeld modules. In this general context we define reflection maps in joint work with N. Andruskiewistch. In the special case of the quantum groups in Lusztig's book these maps are essentially the restriction of the Lusztig automorphisms to the plus part of the quantum group. Under mild assumptions we associate a generalized root system (in the sense of Heckenberger and Yamane) and a Weyl groupoid to the Nichols algebra. Using these invariants it is possible to decide when the Nichols algebra is finite-dimensional. We obtain a coproduct formula which seems to be new even for the classical quantum groups. Then we describe the right coideal subalgebras of the Nichols algebra by words in the Weyl groupoid. As a special case we obtain a proof of a recent conjecture of Kharchenko which says that the number of right coideal subalgebras of the plus part of the quantum group of a semisimple Lie algebra is the order of the Weyl group.
The Goldie dimension of a module M is defined as the supremum of all cardinalities λ such that M contains the direct sum of λ nonzero submodules. This definition can be easily extended to modular lattices with 0 and it extends the notion of the linear dimension of linear spaces to modules or, further, to modular lattices. It is natural to ask how far the fundamental properties of the linear dimension can be extended to the Goldie dimension. Problems of that sort were studied in many papers. The aim of the talk is to present some old and new results concerning that topic.
There are two natural notions of deformation applicable to N-Koszul algebras.
First, such an algebra can be presented by generators and relations using homogeneous relators of degree N, and it is natural to consider those algebras which are obtained by changing the relators by adding to them terms of lower degree. Among these, there is a subclass of algebras characterized by a certain Poincaré-Birkhoff-Witt propery, which is particularly interesting in view of their applications. These deformations have studied by R. Berger and V. Ginzburg [BG] under the name of PBW-deformations, generalizing previous work of A. Braverman and D. Gaitsgory for the quadratic case. In particular, they were able to exhibit a sequence of conditions on non-homogeneous deformations, generalizing the Jacobi condition of Lie algebras, which are equivalent to the PBW property in the general situation.
On the other hand, we have the general theory of formal deformations of algebras as initialted by M. Gerstenhaber [G] in terms of Hochschild cohomology.
The purpose of the talk is to present a result showing the equivalence of these two notions in the case of N-Koszul algebras.
I will recall the general definition of N-Koszul algebras, introduce with some detail the two notions of deformations which are involved in the result, discuss examples and explain some of the details of the proof.
This is joint work with A. Solotar and E. Herscovich.
[BG] Berger, R.; Ginzburg, V. Higher symplectic reflection algebras and non-homogeneous N-Koszul property. J. of Algebra 304 (2006), 577—601.
[G] Gerstenhaber, M. On the deformation of rings and algebras. Ann. of Math. (2) 79 (1964), 59—103.
[BG] Braverman, A.; Gaitsgory, D. The Poincare-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. of Algebra 181 (1996), 315—328.
I shall talk about some new developments in the theory of coalgebra. In particular I will talk about quasi-co-Frobenius coalgebras and the splitting problem for certain chain and serial coalgebras as well as for semiartinian profinite algebras. I will also discuss some questions of Nastasescu, Van Oystaeyen & Torrecillas and their relations to the previously mentioned topics.
Depth two is a nice little theory for subrings (reminiscent to and generalizing H-separable ring extensions) that combines properties of being normal in the sense of Hopf subalgebras and normal in the sense of Galois field extensions. We will discuss why depth two when applied to Hopf subalgebras is precisely normality; giving a new view of normality in left and right versions (based on a paper by Boltje and Kuelshammer).
When depth two is applied to subalgebra pairs of split semisimple artinian algebras, it is identical with a more experimental notion of normality (Rieffel, 1979). A subalgebra of depth n greater than two is the distance out along a tower of right endomorphism algebras and their left regular representations iterated n − 2 times before it becomes normal in this sense. When applied to subgroups and their representations over C, depth n is a condition on the matrix M of the induction-restriction table of the subgroup, its product with its transpose matrix and itself alternatingly. We will discuss this and why Young diagrams show that Sn in the permutation group of n + 1 letters is depth 2n − 1 (based on a joint paper with Burciu and Kuelshammer). For certain subgroups that are products of permutations groups, the entries of the matrix M are the Littlewood-Richardson numbers in combinatorics.
In the first part of the talk we will mention the definition and general informations on the Gelfand-Kirillov dimension, and its connections with noncommutative ring theory, noncommutative projective algebraic geometry and Lie Algebras. We will mention open questions and results on the Gelfand-Kirillov dimension of domains, primitive algebras, algebraic algebras and Golod-Shafarevich algebras, and modules and rings satysfying some special relations.
Let the finite group G act linearly on the vector space V over the field k of arbitrary characteristic, and let H ≤ G be a subgroup. The have an induced action on the ring k[V] of polynomial functions on V. We write k[V]G for the subalgebra of G-invariant polynomials.
The extension of invariant rings k[V]G⊂k[V]H is studied using so called modules of covariants. An example of our results is the following. Let W be the subgroup of G generated by the reflections on V contained in G. A classical theorem due to Serre says that if k[V] is a free k[V]G-module then G = W. We generalize this result as follows. If k[V]H is a free k[V]G-module, then G is generated by H and W. Furthermore, in that case the invariant ring k[V]H∩W is free over k[V]W and is generated as an algebra by H-invariants and W-invariants.
This is joint work with Jianjun Chuai.
A list of generators and relations offers a succinct presentation for an algebra over a field, but what can we deduce when looking at this presentation? If two algebras have similar presentations, they may also share other characteristics. I will illustrate several ways in which deformations of presentations can preserve ring theoretic properties. Unfortunately, this may require an infinite amount of additional information. I will discuss work with Brad Shelton (U. of Oregon) in which the problem is made finite via a homological constant attached to an algebra. I will demonstrate how this finite calculation works in several examples, and lay out the analogy to the enveloping algebras of Lie algebras.
Starting with a multiparameter quantum polynomial algebra, we can associate to it a noncommutative projective space in the sense of Artin and Zhang. The aim of this talk is to present current work towards a classification of such spaces, up to several types of equivalence, in terms of the parameters of the algebra.
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 Lk. 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 Lk.
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 {Pn}n such that {OPn}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 radM(N), is defined to be intersection of all prime submodules of M containing M, or radM(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, EM(N) is not a submodule of M, for a given submodule N. We denote by <EM(N)> the submodule of M generated by the set EM(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, …, xn] 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 radM(Q) is a prime submodule.
Moreover, it is not always true for submodules N and L of M that radM(N \ L) = radM(N) \ radM(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 An. We introduce Groebner-Shirshov basis and combinatorial integers (m,nj) 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, …, xn with coefficients from an infinite field k, i.e. R = k[x1, x2, …, xn] and the subsets Rd of all polynomials of degree d. The direct sum R = ⊕d ∈ NRd is called the degree grading. Let I ⊂ k[x1, x2, …, xn] be a homogeneous ideal. We define the vector space Vd(I) = Rd ∩ I. If {g1, …, gs} is a basis of I involving only homogeneous polynomials, then Vd(I) is generated by all monomial multiples x1α1x2α2…xnαn gi with α1 + α2 + ··· + αn = 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 Sn. 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:
References:
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
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 RH da acção de H em R. Uma consequência da existência dessa correspondência é o fato de RH 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, RH 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:
When Sa is finite dimensional we can give a bounded polyhedron (polytope) whose number of lattice points equals the dimension of Sa. 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 UA 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 RR 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 Uq(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 Uq(g) como produto tensorial de Uq(g)−, Uq(g)0 e Uq(g)+, induzida pela decomposição triangular g = g−⊕h⊕g+, sugere Uq(g)+ como análogo quântico da álgebra universal envolvente da álgebra de Lie nilpotente g+. Neste seminário irei discutir vários resultados relativos às álgebras Uq(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
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.