MAT5201 - Seminários de Álgebra I
1° Semestre de 2025
Schedule:
| 10/03 |
Ana Carolina de Oliveira Silva
|
Braided Hopf Algebras Arising from Matched Pairs of Groups
|
| 17/03 |
Micael Said Garcia
|
Graded polynomial identities of UT over an arbitrary field
|
| 31/03 |
Zaqueu Cristiano Moreira |
Graded maximal rings of quotients of groupoid graded rings
|
| 07/04 |
Italo Bauer Rodrigues da Silva
|
The image of multilinear polynomials on upper triangular matrices
|
| 28/04 |
Victor Kioshi Higa
|
On the existence of fixed elements in a ring with no \(|G|\)-torsion
|
| 05/05 |
Rosiele Trindade Barbosa
|
Upper triangular matrices: gradings and polynomial identities
|
| 26/05 |
Micael Said Garcia
|
Classifying gradings on full matrix algebras
|
| 02/06 |
Victor Kioshi Higa
|
From the subring of fixed elements and the skew group ring to the subalgebra of invariants and the smash product
|
| 09/06 |
Rosiele Trindade Barbosa
| Graded polynomial identities of block-triangular matrices
|
| 16/06 |
Zaqueu Cristiano Moreira
|
Rosiele Trindade Barbosa
Title: Graded polynomial identities of block-triangular matrices
Abstract: here.
Date and time: 09/06/2025, 14:00
Room: A267
Victor Kioshi Higa
Title: From the subring of fixed elements and the skew group ring to the subalgebra of invariants and the smash product
Abstract: In this work, the Bergman-Isaacs theorem will be presented, concerning the existence of fixed elements under the action of a finite group \(G\) on a ring \(R\) without \(|G|\)-torsion, along with some of its most important consequences. Three Morita
contexts will be constructed: the first one associated with the rings \(R^G\) and \(R*G\), the second associated with the rings \(A_1\) and \(A\# (kG)^*\), where \(A_1\) is the component of index \(1\) of a \(G\)-graded algebra \(A\), and the third associated with the
rings \(A^H\) and \(A\# H\), where \(H\) is a finite-dimensional Hopf algebra, \(A\) is an \(H\)-module algebra, and \(A^H\) is the subalgebra of invariants of \(A\). Additionally, the Blattner-Montgomery duality theorem will be demonstrated. Finally, the semiprimality
of the mentioned rings will be investigated, especially the smash product \(A\# H\).
Date and time: 02/06/2025, 14:00
Room: A267
Micael Said Garcia
Title: Classifying gradings on full matrix algebras
Abstract: Graded algebras are important in many areas of mathematics, for instance in theory of PI-algebras and in the classification of simple Lie Algebras. In this talk, we will present some important concepts of the theory of graded algebras and to
show the classification for the gradings on the full matrix algebras over an algebraically closed field.
Date and time: 26/05/2025, 14:00
Room: A267
Rosiele Trindade Barbosa
Title: Upper triangular matrices: gradings and polynomial identities
Abstract: here
Date and time: 05/05/2025, 14:00
Room: A267
Victor Kioshi Higa
Title: On the existence of fixed elements in a ring with no \(|G|\)-torsion
Abstract: In [1], there is an ingenious proof that if a ring \(R\), on which a finite group \(G\) acts by automorphisms, is not nilpotent and has no \(|G|\)-torsion, then its subring of fixed elements \(R^G\) is also not nilpotent. This theorem motivates
the study of this fixed subring, especially at a time when the theory concerning the structure of rings alongside the structure of finite groups is undergoing significant development. In this seminar, we will present a sketch of the proof of the theorem
given by G. M. Bergman and I. M. Isaacs in [1], as well as some immediate applications of the Theorem.
[1] G. M. Bergman and I. M. Isaacs. "Rings with fixed-points-free group actions". Journal of Algebra 95 (1985), pp. 153-172
Date and time: 28/04/2025, 14:00
Room: A267
Italo Bauer Rodrigues da Silva
Title: The image of multilinear polynomials on upper triangular matrices
Abstract: The L'vov-Kaplansky conjecture states that the image of a multilinear polynomial evaluated on the algebra of \(n\times n\) matrices over a field is a vector space. In this talk, I will present a variant of this conjecture, which
asserts that the image of a multilinear polynomial on the algebra \(UT_n(\mathbb{K})\) of \(n\times n\) upper triangular matrices over an infinite field \(\mathbb{K}\) equals \(J^r\), a power of its Jacobson ideal \(J=J(UT_n(\mathbb{K}))\).
Date and time: 07/04/2025, 14:00
Room: A267
Zaqueu Cristiano Moreira
Title: Graded maximal rings of quotients of groupoid graded rings
Abstract: In this work, we study the graded maximal right (left, symmetric) ring of quotients of groupoid graded rings. In order to define and prove properties of these graded rings of quotients, we generalized several concepts and results from Ring
Theory and Group Graded Ring Theory to the groupoid graded context, some of which did not exist in the literature yet. We characterize when the graded maximal right ring of quotients is a von Neumann gr-regular ring and when it is a gr-semisimple ring.
This work is part of the Master's thesis supervised by Prof. Javier Sánchez Serdà, with financial support from FAPESP (grant #2021/14132-2), and was defended at IME-USP in February 2024.
Date and time: 31/03/2025, 14:00
Room: A267
Micael Said Garcia
Title: Graded polynomial identities of UT over an arbitrary field
Abstract: We study the graded polynomial identities of the infinite dimensional upper triangular matrix algebra over an arbitrary field. Since this algebra is non-PI, we find conditions under which a grading on such an algebra satisfies a nontrivial graded
polynomial identity. Under the assumption that the grading group is finite, we obtain that the ideal of graded polynomial identities admits a finite basis. Finally, in contrast to the finite-dimensional case we provide examples showing that two nonisomorphic
gradings can have the same set of graded polynomial identities.
Date and time: 17/03/2025, 14:00
Sala: A267
Ana Carolina de Oliveira Silva
Title: Braided Hopf Algebras Arising from Matched Pairs of Groups
Abstract: here
Date and time: 10/03/2025, 14:00
Room: A267