Page 611 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 611
ALGEBRA
On the length of matrix algebras
Olga Markova, ov_markova@mail.ru
Moscow State University, Russian Federation
By the length of a finite system of generators for a finite-dimensional algebra over an arbitrary
field we mean the least positive integer k such that the products of length not exceeding k
span this algebra (as a vector space). The maximum length for the systems of generators of an
algebra is referred to as the length of the algebra. Apart from intrinsic algebraic importance,
the length function has applications, for example, in computing methods of the mechanics of
isotropic continua and matrix rational procedures.
The length evaluation can be a difficult problem, since, for example, the length of the full
matrix algebra is still unknown (Paz’s Problem, 1984). Paz conjectured that the length of any
generating set for the algebra of n by n matrices is at most 2n−2. The question about the length
determination was later extended on different matrix sets and subalgebras by Laffey (1986).
First we present a survey of our results on the main properties of length function. In par-
ticular, we provide a construction of series of matrix algebras demonstrating that a length of a
subalgebra can be larger than the length of the algebra and the difference of their lengths can be
arbitrary large. This result partially explains the difficulty of length evaluation.
In this talk we will also show that Paz’s conjecture holds under the assumption that the
generating set contains a nonderogatory matrix or a matrix with minimal polynomial of degree
n − 1. We will also present linear bounds for the length of generating sets that include a matrix
with some restrictions on its Jordan normal form. Having an upper bound, we also provide
examples of matrix sets of different type which length achieve the bound 2n − 2.
This talk is based on joint research with Alexander Guterman (Moscow State University),
Thomas Laffey and Helena Šmigoc (University College Dublin).
On middle Bol loops and the total multiplication groups
Parascovia Sirbu, syrbuviv@yahoo.com
Moldova State University, Republic of Moldova
Let (Q, ·) be a loop and a ∈ Q. The mappings x → ax, x → xa, x → a/x are denoted
by La, Ra, Da, respectively. The multiplication and total multiplication groups M lt(Q) =
{La, Ra; a ∈ Q} and T M lt(Q) = {La, Ra, Da; a ∈ Q} of (Q, ·) along with their subgroups
Inn(Q) and T Inn(Q) - the stabilizers of the unit in M lt(Q) and T M lt(Q), respectively, are
important tools in studying such properties of loops as normality of subloops, solvability, nilpo-
tency etc. It is known that M lt(Q) is invariant under the isotopy of loops (A. Albert) while
T M lt(Q) is invariant under the isostrophy of loops (announced by the author and A. Drapal).
Characterizations of the mentioned groups for some classes of loops with inverse properties
are obtained, including their representation, general properties and systems of generators for
T Inn(Q). Necessary and sufficient conditions when T M lt(Q) is nilpotent are given.
An open problem regarding middle Bol loops is if this class includes the class of loops with
universal (i.e. invariant under the isotopy of loops) flexibility. If this conjecture is true, then
the loops with invariant flexibility under the isostrophy are Moufang loops. It is shown that
commutative loops with invariant flexibility under the isostrophy are Moufang loops.
609
On the length of matrix algebras
Olga Markova, ov_markova@mail.ru
Moscow State University, Russian Federation
By the length of a finite system of generators for a finite-dimensional algebra over an arbitrary
field we mean the least positive integer k such that the products of length not exceeding k
span this algebra (as a vector space). The maximum length for the systems of generators of an
algebra is referred to as the length of the algebra. Apart from intrinsic algebraic importance,
the length function has applications, for example, in computing methods of the mechanics of
isotropic continua and matrix rational procedures.
The length evaluation can be a difficult problem, since, for example, the length of the full
matrix algebra is still unknown (Paz’s Problem, 1984). Paz conjectured that the length of any
generating set for the algebra of n by n matrices is at most 2n−2. The question about the length
determination was later extended on different matrix sets and subalgebras by Laffey (1986).
First we present a survey of our results on the main properties of length function. In par-
ticular, we provide a construction of series of matrix algebras demonstrating that a length of a
subalgebra can be larger than the length of the algebra and the difference of their lengths can be
arbitrary large. This result partially explains the difficulty of length evaluation.
In this talk we will also show that Paz’s conjecture holds under the assumption that the
generating set contains a nonderogatory matrix or a matrix with minimal polynomial of degree
n − 1. We will also present linear bounds for the length of generating sets that include a matrix
with some restrictions on its Jordan normal form. Having an upper bound, we also provide
examples of matrix sets of different type which length achieve the bound 2n − 2.
This talk is based on joint research with Alexander Guterman (Moscow State University),
Thomas Laffey and Helena Šmigoc (University College Dublin).
On middle Bol loops and the total multiplication groups
Parascovia Sirbu, syrbuviv@yahoo.com
Moldova State University, Republic of Moldova
Let (Q, ·) be a loop and a ∈ Q. The mappings x → ax, x → xa, x → a/x are denoted
by La, Ra, Da, respectively. The multiplication and total multiplication groups M lt(Q) =
{La, Ra; a ∈ Q} and T M lt(Q) = {La, Ra, Da; a ∈ Q} of (Q, ·) along with their subgroups
Inn(Q) and T Inn(Q) - the stabilizers of the unit in M lt(Q) and T M lt(Q), respectively, are
important tools in studying such properties of loops as normality of subloops, solvability, nilpo-
tency etc. It is known that M lt(Q) is invariant under the isotopy of loops (A. Albert) while
T M lt(Q) is invariant under the isostrophy of loops (announced by the author and A. Drapal).
Characterizations of the mentioned groups for some classes of loops with inverse properties
are obtained, including their representation, general properties and systems of generators for
T Inn(Q). Necessary and sufficient conditions when T M lt(Q) is nilpotent are given.
An open problem regarding middle Bol loops is if this class includes the class of loops with
universal (i.e. invariant under the isotopy of loops) flexibility. If this conjecture is true, then
the loops with invariant flexibility under the isostrophy are Moufang loops. It is shown that
commutative loops with invariant flexibility under the isostrophy are Moufang loops.
609
