Noncommutative Algebra and Applications
Projeto Temático FAPESP No.2015/09162-9, coordenado por César Polcino Milies
Research group seminars
2016 Seminars:
Lecturer |
Title |
Date |
Time |
Room |
Arnaldo Mandel (IME-USP) |
Automorphisms of max-closed integral polyhedral cones II |
November 08, 2016 |
14:30-15:30 |
138-B |
Abstract:
Two
natural associative operations on the set of k-tuples of
non-negative integers are +, componentwise addition, and \oplus,
componentwise maximum. We consider subsets which are closed (thus
submonoids) under both operations. Those which are
finitely |
Lecturer |
Title |
Date |
Time |
Room |
Arnaldo Mandel (IME-USP) |
Automorphisms of max-closed integral polyhedral cones |
November 01, 2016 |
14:30-15:30 |
138-B |
Abstract:
Two
natural associative operations on the set of k-tuples of
non-negative integers are +, componentwise addition, and \oplus,
componentwise maximum. We consider subsets which are closed (thus
submonoids) under both operations. Those which are
finitely |
Lecturer |
Title |
Date |
Time |
Room |
Makar Plakhotnyk (IME-USP) |
Algebraic structures on non-negative exponent matrices (part III) |
October 25, 2016 |
14:30-15:30 |
138-B |
Abstract:
We
will consider a class of integer non-negative matrices, precisely
exponent matrices, which appeared in the theory of tiled orders.
The set of exponent matrices is the semigroup with respect to
entry-wise addition (called tropical product) and also is the
semigroup with respect to element-wise maximum (tropical sum).
The evident distributivity of tropical operations imply that our
matrices form a semiring. We will describe the generators of the
tropical semiring of exponent matrices and the automorphisms of
the mentioned algebraic structures. |
Lecturer |
Title |
Date |
Time |
Room |
Makar Plakhotnyk (IME-USP) |
Algebraic structures on non-negative exponent matrices (part II) |
October 11, 2016 |
14:30-15:30 |
138-B |
Abstract:
We
will consider a class of integer non-negative matrices, precisely
exponent matrices, which appeared in the theory of tiled orders.
The set of exponent matrices is the semigroup with respect to
entry-wise addition (called tropical product) and also is the
semigroup with respect to element-wise maximum (tropical sum).
The evident distributivity of tropical operations imply that our
matrices form a semiring. We will describe the generators of the
tropical semiring of exponent matrices and the automorphisms of
the mentioned algebraic structures. |
Lecturer |
Title |
Date |
Time |
Room |
Makar Plakhotnyk (IME-USP) |
Algebraic structures on non-negative exponent matrices (part I) |
October 04, 2016 |
14:30-15:30 |
138-B |
Abstract:
We
will consider a class of integer non-negative matrices, precisely
exponent matrices, which appeared in the theory of tiled orders.
The set of exponent matrices is the semigroup with respect to
entry-wise addition (called tropical product) and also is the
semigroup with respect to element-wise maximum (tropical sum).
The evident distributivity of tropical operations imply that our
matrices form a semiring. We will describe the generators of the
tropical semiring of exponent matrices and the automorphisms of
the mentioned algebraic structures. |
Lecturer |
Title |
Date |
Time |
Room |
Raul Antonio Ferraz (IME-USP) |
Unidades centrais em ZC_{p,q} |
September 27, 2016 |
14:30-15:30 |
138-B |
Abstract:
Sejam
p e q primos, e C_{p,q} o produto semi direto de um grupo de
ordem q por outro de ordem p. Denotemos por ZC_{p,q} o anel de
grupo integral do grupo C_{p,q} e por \mathcal{Z} o grupo de
unidades centrais normalizadas deste anel. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Free
groups in a normal subgroup of the field of |
September 13, 2016 |
14:30-15:30 |
138-B |
Abstract: Let k(t) be the field of rational functions over the field k, let \sigma be a k-automorphism of K=k(t), let D=K(X; \sigma) be the ring of fractions of the skew polynomial ring K[X; \sigma], and let D^{\bullet} be the multiplicative group of D. We show that if N is a noncentral normal subgroup of D^{\bullet}, then N contains a free subgroup. We also prove that when k is algebraically closed with char(k) \neq 2 and \sigma of infinite order, there exists a specialization from D to a quaternion algebra. This allow us to present explicitly free subgroups in D^{\bullet}. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Free
groups in a normal subgroup of the field of |
August 23, 2016 |
14:30-15:30 |
138-B |
Abstract: Let k(t) be the field of rational functions over the field k, let \sigma be a k-automorphism of K=k(t), let D=K(X; \sigma) be the ring of fractions of the skew polynomial ring K[X; \sigma], and let D^{\bullet} be the multiplicative group of D. We show that if N is a noncentral normal subgroup of D^{\bullet}, then N contains a free subgroup. We also prove that when k is algebraically closed with char(k) \neq 2 and \sigma of infinite order, there exists a specialization from D to a quaternion algebra. This allow us to present explicitly free subgroups in D^{\bullet}. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Free
groups in a normal subgroup of the field of |
August 16, 2016 |
14:30-15:30 |
138-B |
Abstract: Let k(t) be the field of rational functions over the field k, let \sigma be a k-automorphism of K=k(t), let D=K(X; \sigma) be the ring of fractions of the skew polynomial ring K[X; \sigma], and let D^{\bullet} be the multiplicative group of D. We show that if N is a noncentral normal subgroup of D^{\bullet}, then N contains a free subgroup. We also prove that when k is algebraically closed with char(k) \neq 2 and \sigma of infinite order, there exists a specialization from D to a quaternion algebra. This allow us to present explicitly free subgroups in D^{\bullet}. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
Free
groups in a normal subgroup of the field of |
August 09, 2016 |
14:30-15:30 |
138-B |
Abstract: Let k(t) be the field of rational functions over the field k, let \sigma be a k-automorphism of K=k(t), let D=K(X; \sigma) be the ring of fractions of the skew polynomial ring K[X; \sigma], and let D^{\bullet} be the multiplicative group of D. We show that if N is a noncentral normal subgroup of D^{\bullet}, then N contains a free subgroup. We also prove that when k is algebraically closed with char(k) \neq 2 and \sigma of infinite order, there exists a specialization from D to a quaternion algebra. This allow us to present explicitly free subgroups in D^{\bullet}. |
Lecturer |
Title |
Date |
Time |
Room |
Vitor O. Ferreira (IME-USP) |
Free algebras in division rings with an involution (part II) |
June 28, 2016 |
14:30-15:30 |
268-A |
Abstract: Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra. (This is a joint work with Erica Z. Fornaroli (UEM) and Jairo Z. Goncalves (IME-USP).) |
Lecturer |
Title |
Date |
Time |
Room |
Vitor O. Ferreira (IME-USP) |
Free algebras in division rings with an involution |
June 14, 2016 |
14:30-15:30 |
268-A |
Abstract: Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra. (This is a joint work with Erica Z. Fornaroli (UEM) and Jairo Z. Goncalves (IME-USP).) |
Lecturer |
Title |
Date |
Time |
Room |
Leonid Makar-Limanov (Wayne State University, USA) |
A
description of two-generated subalgebras |
June 7, 2016 |
14:30-15:30 |
268-A |
Abstract: The famous AMS (Abhyankar-Moh-Suzuki) theorem states that if two polynomials f and g in one variable with coefficients in a field F generate all algebra of polynomials, i.e. any polynomial h in one variable can be expressed as h = H(f,g) where H is a polynomial in two variables, then either the degree of f divides the degree of g, or the degree of g divides the degree of f, or the degree of f and the degree of g are divisible by the characteristic of the field F. There were several wrong published proofs of this theorem and there are many correct published proofs of this theorem but all of them are either long or not self-contained. Recently I found a (relatively) short and self-contained proof which will be discussed. The talk is accessible to undergraduate students knowing elementary linear algebra. |
Lecturer |
Title |
Date |
Time |
Room |
Javier Sánchez (IME-USP) |
A way of obtaining free group algebras from free algebras inside division rings (part III) |
May 31, 2016 |
14:30-15:30 |
268-A |
Abstract: In
the mid eighties, L. Makar-Limanov conjectured the
following: |
Lecturer |
Title |
Date |
Time |
Room |
Javier Sánchez (IME-USP) |
A way of obtaining free group algebras from free algebras inside division rings (part II) |
May 24, 2016 |
14:30-15:30 |
268-A |
Abstract: In
the mid eighties, L. Makar-Limanov conjectured the
following: |
Lecturer |
Title |
Date |
Time |
Room |
Javier Sánchez (IME-USP) |
A way of obtaining free group algebras from free algebras inside division rings |
May 17, 2016 |
14:30-15:30 |
268-A |
Abstract: In
the mid eighties, L. Makar-Limanov conjectured the
following: |
Lecturer |
Title |
Date |
Time |
Room |
Misha Doduchaev (IME-USP) |
Partial actions and subshifts (part IV) |
May 03, 2016 |
14:30-15:30 |
268-A |
Abstract: Given
an arbitrary subshift X over a finite alphabet with n letters,
let V be the vector space with base X over a field K of
characteristic 0, and let B be the tensor product over K of the
algebra of all linear operators of V with the group algebra KF,
where F is the free group of rank n. We construct a partial
representation u: F -> B, and define the shift algebra O as
the subalgebra of B generated by u(F). We prove that O is
isomorphic to the crossed product of a commutative subalgebra A
by a partial action of F on A. The algebra A is generated by
idempotents and can be seen as a subalgebra of linear operators
on V. This isomorphism gives a possibility to use facts on
partial crossed products to study O. For this purpose one needs a
way to deal with A, which is given by realizing A as the algebra
of all locally constant functions on a topological space, which
we call the spectrum of A. The spectrum is the set of all algebra
homomorphisms from A to K. Taking the discrete topology on K, we
naturally obtain a topology on the spectrum so that it becomes a
totally disconnected compact Hausdorff space. Borrowing an idea
from the theory of C*-algebras, we obtain from the partial action
of F on A a partial action of F on the spectrum, which we call
the spectral partial action. Our main effort is concentrated on
the study of the dynamical properties of the spectral partial
action such as the topological freeness and minimality. Then we
apply our results to discuss the simplicity of the shift algebra
O. The C* version of the construction is also elaborated, and it
is shown that the C*-algebra O is isomorphic to the so-called
Carlsen-Matsumoto C*-algebra. |
Lecturer |
Title |
Date |
Time |
Room |
Misha Doduchaev (IME-USP) |
Partial actions and subshifts (part III) |
April 26, 2016 |
14:30-15:30 |
268-A |
Abstract: Given
an arbitrary subshift X over a finite alphabet with n letters,
let V be the vector space with base X over a field K of
characteristic 0, and let B be the tensor product over K of the
algebra of all linear operators of V with the group algebra KF,
where F is the free group of rank n. We construct a partial
representation u: F -> B, and define the shift algebra O as
the subalgebra of B generated by u(F). We prove that O is
isomorphic to the crossed product of a commutative subalgebra A
by a partial action of F on A. The algebra A is generated by
idempotents and can be seen as a subalgebra of linear operators
on V. This isomorphism gives a possibility to use facts on
partial crossed products to study O. For this purpose one needs a
way to deal with A, which is given by realizing A as the algebra
of all locally constant functions on a topological space, which
we call the spectrum of A. The spectrum is the set of all algebra
homomorphisms from A to K. Taking the discrete topology on K, we
naturally obtain a topology on the spectrum so that it becomes a
totally disconnected compact Hausdorff space. Borrowing an idea
from the theory of C*-algebras, we obtain from the partial action
of F on A a partial action of F on the spectrum, which we call
the spectral partial action. Our main effort is concentrated on
the study of the dynamical properties of the spectral partial
action such as the topological freeness and minimality. Then we
apply our results to discuss the simplicity of the shift algebra
O. The C* version of the construction is also elaborated, and it
is shown that the C*-algebra O is isomorphic to the so-called
Carlsen-Matsumoto C*-algebra. |
Lecturer |
Title |
Date |
Time |
Room |
Misha Doduchaev (IME-USP) |
Partial actions and subshifts (part II) |
April 19, 2016 |
14:30-15:30 |
268-A |
Abstract: Given
an arbitrary subshift X over a finite alphabet with n letters,
let V be the vector space with base X over a field K of
characteristic 0, and let B be the tensor product over K of the
algebra of all linear operators of V with the group algebra KF,
where F is the free group of rank n. We construct a partial
representation u: F -> B, and define the shift algebra O as
the subalgebra of B generated by u(F). We prove that O is
isomorphic to the crossed product of a commutative subalgebra A
by a partial action of F on A. The algebra A is generated by
idempotents and can be seen as a subalgebra of linear operators
on V. This isomorphism gives a possibility to use facts on
partial crossed products to study O. For this purpose one needs a
way to deal with A, which is given by realizing A as the algebra
of all locally constant functions on a topological space, which
we call the spectrum of A. The spectrum is the set of all algebra
homomorphisms from A to K. Taking the discrete topology on K, we
naturally obtain a topology on the spectrum so that it becomes a
totally disconnected compact Hausdorff space. Borrowing an idea
from the theory of C*-algebras, we obtain from the partial action
of F on A a partial action of F on the spectrum, which we call
the spectral partial action. Our main effort is concentrated on
the study of the dynamical properties of the spectral partial
action such as the topological freeness and minimality. Then we
apply our results to discuss the simplicity of the shift algebra
O. The C* version of the construction is also elaborated, and it
is shown that the C*-algebra O is isomorphic to the so-called
Carlsen-Matsumoto C*-algebra. |
Lecturer |
Title |
Date |
Time |
Room |
Misha Doduchaev (IME-USP) |
Partial actions and subshifts |
April 5, 2016 |
14:30-15:30 |
268-A |
Abstract: Given
an arbitrary subshift X over a finite alphabet with n letters,
let V be the vector space with base X over a field K of
characteristic 0, and let B be the tensor product over K of the
algebra of all linear operators of V with the group algebra KF,
where F is the free group of rank n. We construct a partial
representation u: F -> B, and define the shift algebra O as
the subalgebra of B generated by u(F). We prove that O is
isomorphic to the crossed product of a commutative subalgebra A
by a partial action of F on A. The algebra A is generated by
idempotents and can be seen as a subalgebra of linear operators
on V. This isomorphism gives a possibility to use facts on
partial crossed products to study O. For this purpose one needs a
way to deal with A, which is given by realizing A as the algebra
of all locally constant functions on a topological space, which
we call the spectrum of A. The spectrum is the set of all algebra
homomorphisms from A to K. Taking the discrete topology on K, we
naturally obtain a topology on the spectrum so that it becomes a
totally disconnected compact Hausdorff space. Borrowing an idea
from the theory of C*-algebras, we obtain from the partial action
of F on A a partial action of F on the spectrum, which we call
the spectral partial action. Our main effort is concentrated on
the study of the dynamical properties of the spectral partial
action such as the topological freeness and minimality. Then we
apply our results to discuss the simplicity of the shift algebra
O. The C* version of the construction is also elaborated, and it
is shown that the C*-algebra O is isomorphic to the so-called
Carlsen-Matsumoto C*-algebra. |
Lecturer |
Title |
Date |
Time |
Room |
Antonio Giambruno (Università di Palermo, Italia) |
Computing codimensions of finite dimensional algebras |
March 15, 2016 |
14:30-15:30 |
268-A |
Abstract: Let A be a PI-algebra over a field of characteristic zero and V the variety it generates.The sequence of codimensions of A is a numerical sequence measuring, for every n, the dimension of the space of multilinear elements of degree n of a relatively free algebra of V. The computation of such sequence is a quite hard problem in general, but the computation of its asymptotics has been achieved in a quite general setting. An essential tool has been the representation theory of the symmetric group. Here we present a new purely combinatorial method that allows to compute the asymptotics of such sequence in case of finite dimensional fundamental algebras. |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
More on Lewin's conjecture (II) |
March 1, 2016 |
14:30-15:30 |
268-A |
Abstract: J.
Lewin conjectured that: if D is the field of fractions of the
group algebra kG of the torsion free nilpotent group G over the
field k, and if x and y are elements of G that do not commute,
then 1+x and 1+y generate a free subgroup. In a joint paper with
Mandel and Shirvani (J. Algebra, 1999), we proved that this is
true when char k \neq 2. More recently, in collaboration with A.
I. Lichtman (Internat. J. Algebra |
Lecturer |
Title |
Date |
Time |
Room |
Jairo Z. Gonçalves (IME-USP) |
More on Lewin's conjecture |
February 23, 2016 |
14:30-15:30 |
268-A |
Abstract: J.
Lewin conjectured that: if D is the field of fractions of the
group algebra kG of the torsion free nilpotent group G over the
field k, and if x and y are elements of G that do not commute,
then 1+x and 1+y generate a free subgroup. In a joint paper with
Mandel and Shirvani (J. Algebra, 1999), we proved that this is
true when char k \neq 2. More recently, in collaboration with A.
I. Lichtman (Internat. J. Algebra |