Noncommutative Algebra and Applications

Projeto Temático FAPESP No.2015/09162-9, coordenado por César Polcino Milies

Research group seminars

2015 Seminars:

Lecturer

Title

Date

Time

Room

Misha Dokuchaev (IME-USP)

On invariants of partial group rings

November 10, 2015

14:30-15:30

241-A

Abstract:

My talk will be based on a joint paper with Juan Jacobo Simon. We show that the partial group ring of a finite group over a commutative ring is determined by the subgroup lattice of the base group and the orders of the subgroups. For partial group rings of finite groups of odd orders a list of natural invariants is given. Some applications to the isomorphism problem will be given.



Lecturer

Title

Date

Time

Room

Makar Plakhotnyk (IME-USP)

On some properties of the generators of the
additive semigroup of non-negative exponent matrices

October 27, 2015

14:30-15:30

241-A

Abstract:

We will discuss algorithms to find the generators of the semigroup of the non-negative exponent matrices and some results on associated quivers.


Lecturer

Title

Date

Time

Room

Nicola Sambonet (IME-USP)

The unitary cover of a finite group

October 20, 2015

14:30-15:30

241-A

Abstract:

An explicit description of the Schur multiplier of a finite group is often a difficult task. We introduce the unitary cover, a very interesting tool to achieve information about the exponent of the multiplier. We discuss some open problems while describing part of the obtained bounds.


Lecturer

Title

Date

Time

Room

Pham Ngoc Anh

(Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary)

Peirce decompositions, Peirce idempotents and Peirce rings

October 13, 2015

14:30-15:30

241-A

Abstract:

It is well-known that idempotents determine essentially the structure of rings when there are a lots of them as in the case of matrix rings or in coordinating projective geometries. In this talk we introduce the notion of n-Peirce idempotents and rings by analysizing a two-sided Peirce decomposition of a ring by a given idempotent. Surprising enough, one can develop a structure theory of n-Peirce rings quite similarly to the one of semiperfect rings where the 1-Peirce idempotents take almost the same role as primitive idempotents played in the classical theory.


Lecturer

Title

Date

Time

Room

Raul Antonio Ferraz (IME-USP)

Abelian Groups with constant weight

October 6, 2015

14:30-15:30

241-A

Abstract:

Let G be a finite abelian group and F_q a finite field with q elements. An q-ary abelian code will be defined as a non trivial ideal of the group algebra F_q G. In this lecture we determine when an abelian code C has constant weight. That is, when the support of allnon trivial elements of C have the same cardinality. (Joint work with Prof. Ruth Nascimento (UTFPR-Guarapuava-PR).)


Lecturer

Title

Date

Time

Room

Javier Sánchez Serdà (IME-USP)

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras (III)

September 29, 2015

14:30-15:30

241-A

Abstract:

This is a joint work with Vitor O. Ferreira and Jairo Z. Gonçalves. For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring D(L) constructed by Cohn (and simplified later by Lichtman). If U(L) is an Ore domain, D(L) coincides with its ring of fractions. Consider now the principal involution of L, L---> L, x-->-x. It is well known that the principal involution of L can be extended to an involution of U(L). It was proved by Cimpric, that this involution can be extended to D(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that D(L) contains noncommutative free algebras generated by symmetric elements (with respect to the extension of the principal involution). This class contains all noncommutative Lie algebras over a field of  characteristic zero such that U(L) is an Ore domain.


Lecturer

Title

Date

Time

Room

Javier Sánchez Serdà (IME-USP)

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras (II)

September 22, 2015

14:30-15:30

241-A

Abstract:

This is a joint work with Vitor O. Ferreira and Jairo Z. Gonçalves. For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring D(L) constructed by Cohn (and simplified later by Lichtman). If U(L) is an Ore domain, D(L) coincides with its ring of fractions. Consider now the principal involution of L, L---> L, x-->-x. It is well known that the principal involution of L can be extended to an involution of U(L). It was proved by Cimpric, that this involution can be extended to D(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that D(L) contains noncommutative free algebras generated by symmetric elements (with respect to the extension of the principal involution). This class contains all noncommutative Lie algebras over a field of  characteristic zero such that U(L) is an Ore domain.



Lecturer

Title

Date

Time

Room

Javier Sánchez Serdà (IME-USP)

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

September 15, 2015

14:30-15:30

241-A

Abstract:

This is a joint work with Vitor O. Ferreira and Jairo Z. Gonçalves. For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring D(L) constructed by Cohn (and simplified later by Lichtman). If U(L) is an Ore domain, D(L) coincides with its ring of fractions. Consider now the principal involution of L, L---> L, x-->-x. It is well known that the principal involution of L can be extended to an involution of U(L). It was proved by Cimpric, that this involution can be extended to D(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that D(L) contains noncommutative free algebras generated by symmetric elements (with respect to the extension of the principal involution). This class contains all noncommutative Lie algebras over a field of  characteristic zero such that U(L) is an Ore domain.


Lecturer

Title

Date

Time

Room

Izabella Stuhl (University of Debrecen, Hungary)

Steiner Loops satisfying Moufang's Theorem

August 25, 2015

14:30-15:30

241-A

Abstract:

A loop satisfies Moufang's theorem whenever the subloop generated by three associating elements is a group. Moufang loops satisfy Moufang's theorem, but it is possible for a loop that is not Moufang to nevertheless satisfy Moufang's theorem. Steiner loops that are not Moufang loops are known to arise from Steiner triple systems in which some triangle does not generate a subsystem of order 7, while Steiner loops that do not satisfy Moufang's theorem are shown to arise from Steiner triple systems in which some Pasch configuration does not generate a subsystem of order 7.
Furthermore, examples are given of non-commutative loops that satisfy Moufang's theorem yet are not Moufang loops.


Lecturer

Title

Date

Time

Room

Jairo Gonçalves (IME-USP)

Free objects in Ore extensions and free groups in division rings II

August 18, 2015

14:30-15:30

241-A

Abstract:

Let K be a field, let \sigma be an automorphism of K with fixed field k, and let \delta be a \sigma-derivation of K. We show that if char(k)=0, then division ring D=K(X;\sigma,\delta) is either locally PI, or it contains a free algebra over its center. If k_0 is the prime field of K, and tr.deg(k/k_0) is infinite, then D-{0}, the multiplicative group of D, contains a free subgroup. Some applications of these results will be given. (Joint work with Jason P. Bell, from U. Waterloo.)


Lecturer

Title

Date

Time

Room

Jairo Gonçalves (IME-USP)

Free objects in Ore extensions and free groups in division rings

August 11, 2015

14:30-15:30

241-A

Abstract:

Let K be a field, let \sigma be an automorphism of K with fixed field k, and let \delta be a \sigma-derivation of K. We show that if char(k)=0, then division ring D=K(X;\sigma,\delta) is either locally PI, or it contains a free algebra over its center. If k_0 is the prime field of K, and tr.deg(k/k_0) is infinite, then D-{0}, the multiplicative group of D, contains a free subgroup. Some applications of these results will be given. (Joint work with Jason P. Bell, from U. Waterloo.)


Lecturer

Title

Date

Time

Room

Mykola Krypchenko (IME-USP)

Extensions of semilattices of groups by groups II

June 2, 2015

14:30-15:30

144-B

Abstract:

We introduce the concept of an extension of a semilattice of groups A by a group G and describe all the extensions of this type which are equivalent to the crossed products A*G by twisted partial actions [2] of G on A. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and Sieben's twisted modules (see [1,3,4]) over E-unitary inverse semigroups. This is a joint work with M. Dokuchaev.

[1] Buss, A., and Exel, R. Twisted actions and regular Fell bundles over inverse semigroups. Proc. Lond. Math.Soc. 103, 2 (2011), 235-270.

[2] Dokuchaev, M., Exel, R., and Simon, J. J. Crossed products by twisted partial actions and graded algebras. J. Algebra 320, 8 (2008), 3278-3310.

[3] Lausch, H. Cohomology of inverse semigroups. J. Algebra 35 (1975), 273-303.

[4] Sieben, N. C*-crossed products by twisted inverse semigroup actions. J. Operator Theory 39, 2 (1998), 361-393.



Lecturer

Title

Date

Time

Room

Mykola Krypchenko (IME-USP)

Extensions of semilattices of groups by groups I

May 26, 2015

14:30-15:30

144-B

Abstract:

We introduce the concept of an extension of a semilattice of groups A by a group G and describe all the extensions of this type which are equivalent to the crossed products A*G by twisted partial actions [2] of G on A. As a consequence, we establish a one-to-one correspondence, up to an isomorphism, between twisted partial actions of groups on semilattices of groups and Sieben's twisted modules (see [1,3,4]) over E-unitary inverse semigroups. This is a joint work with M. Dokuchaev.

[1] Buss, A., and Exel, R. Twisted actions and regular Fell bundles over inverse semigroups. Proc. Lond. Math.Soc. 103, 2 (2011), 235-270.

[2] Dokuchaev, M., Exel, R., and Simon, J. J. Crossed products by twisted partial actions and graded algebras. J. Algebra 320, 8 (2008), 3278-3310.

[3] Lausch, H. Cohomology of inverse semigroups. J. Algebra 35 (1975), 273-303.

[4] Sieben, N. C*-crossed products by twisted inverse semigroup actions. J. Operator Theory 39, 2 (1998), 361-393.