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
generated under addition can be characterized by an appeal to the theory of convex polyhedra. Under mild conditions, we show that every
\oplus-automorphism of such a set is induced by a permutation of the coordinates. As a corollary we obtain a result of Dokuchaev et al
about \oplus-automorphisms of the cone of exponential matrices (AKA integral quasi-semi-metrics). The polyhedral viewpoint also yields
a simple proof of a result by the same authors characterizing the automorphisms of the additive monoid of exponential matrices.

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
generated under addition can be characterized by an appeal to the theory of convex polyhedra. Under mild conditions, we show that every
\oplus-automorphism of such a set is induced by a permutation of the coordinates. As a corollary we obtain a result of Dokuchaev et al
about \oplus-automorphisms of the cone of exponential matrices (AKA integral quasi-semi-metrics). The polyhedral viewpoint also yields
a simple proof of a result by the same authors characterizing the automorphisms of the additive monoid of exponential matrices.

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.
This is a joint work with M. Dokuchaev, V. Kirichenko and G. Kudryavtseva.

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.
This is a joint work with M. Dokuchaev, V. Kirichenko and G. Kudryavtseva.

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.
This is a joint work with M. Dokuchaev, V. Kirichenko and G. Kudryavtseva.

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.
Neste trabalho descrevemos \mathcal{Z} como o produto direto de dois subgrupos que denominamos unidades de primeira especie e
unidades de segunda especie de \mathcal{Z}.
Exibimos, tambem, em alguns casos, um conjunto de geradores multiplicativamente independente para os dois subgrupos acima citados (ambos sao grupos abelianos livres finitamente gerados).
Em parceria com Prof. Juan Jacobo Simon (Universidad de Murcia - Murcia, Espanha)

Lecturer

Title

Date

Time

Room

Jairo Z. Gonçalves (IME-USP)

Free groups in a normal subgroup of the field of
fractions of a skew polynomial ring (part IV)

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
fractions of a skew polynomial ring (part III)

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
fractions of a skew polynomial ring (part II)

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
fractions of a skew polynomial ring

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
of a polynomial ring in one variable and a new proof

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:
Let D be a division ring with center Z.  If D is finitely generated  (as a division ring)  over Z and D has infinite dimension over Z, then D contains a noncommutative free Z-algebra.
In many of the examples for which the conjecture is known to be valid, the division ring D contains a (noncommutative) free group algebra over Z, not only a free Z-algebra. Note that, in general, if X is the set of  free generators of a free algebra inside a division ring, X  may not be a set of free generators of a free group Z-algebra. Given a division ring D with a valuation v, we obtain sufficient conditions for the existence of noncommutative free group algebras in D. These conditions involve the graded division ring grad_v(D) associated to the filtration induced by the valuation.

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:
Let D be a division ring with center Z.  If D is finitely generated  (as a division ring)  over Z and D has infinite dimension over Z, then D contains a noncommutative free Z-algebra.
In many of the examples for which the conjecture is known to be valid, the division ring D contains a (noncommutative) free group algebra over Z, not only a free Z-algebra. Note that, in general, if X is the set of  free generators of a free algebra inside a division ring, X  may not be a set of free generators of a free group Z-algebra. Given a division ring D with a valuation v, we obtain sufficient conditions for the existence of noncommutative free group algebras in D. These conditions involve the graded division ring grad_v(D) associated to the filtration induced by the valuation.

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:
Let D be a division ring with center Z.  If D is finitely generated  (as a division ring)  over Z and D has infinite dimension over Z, then D contains a noncommutative free Z-algebra.
In many of the examples for which the conjecture is known to be valid, the division ring D contains a (noncommutative) free group algebra over Z, not only a free Z-algebra. Note that, in general, if X is the set of  free generators of a free algebra inside a division ring, X  may not be a set of free generators of a free group Z-algebra. Given a division ring D with a valuation v, we obtain sufficient conditions for the existence of noncommutative free group algebras in D. These conditions involve the graded division ring grad_v(D) associated to the filtration induced by the valuation.

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.
This is a joint work with Ruy Exel.

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.
This is a joint work with Ruy Exel.

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.
This is a joint work with Ruy Exel.

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.
This is a joint work with Ruy Exel.

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
Comp. 2014), we considered this problem for group algebras of certain solvable groups. In our talks we intend to investigate this question even further. For example, in the case in which the division ring D is generated over its center k by a torsion free nilpotent group G \leq D^{\bullet} = D \ {0}.

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
Comp. 2014), we considered this problem for group algebras of certain solvable groups. In our talks we intend to investigate this question even further. For example, in the case in which the division ring D is generated over its center k by a torsion free nilpotent group G \leq D^{\bullet} = D \ {0}.