Banca de DEFESA: Hercules de Carvalho Bezerra

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : Hercules de Carvalho Bezerra
DATE: 27/02/2023
TIME: 14:00
LOCAL: videoconferência
TITLE:

Just infinite restricted Lie algebras

KEY WORDS:

Lie algebras, restricted Lie algebras,
self-similar algebras, nil-algebras, graded algebras, fine gradings, 
$p$-groups, growth, Poisson algebras

PAGES: 85
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

In this work, we build examples analogous to Grigorchuk and Gupta-Sidki groups, 
which play an important role in modern group theory as they are natural examples of 
self-similar finitely generated periodic groups, in the field of restricted Lie algebras.

In 2021, Petrogradsky and Shestakov constructed an example of just-infinite, 3-generated, 
Lie superalgebra Q over an arbitrary field, which gives rise to an associative closure, 
a Poisson superalgebra, and two Jordan superalgebras.
Due to the way these five superalgebras were constructed, it was possible to obtain a clear monomial basis, 
in addition to study the structure, growth, and other properties of each one of them.

Now, we construct a restricted Lie algebra L, over a field of any positive characteristic p, 
which gives rise to an associative closure A, and a Poisson algebra P.
We present in the work the following properties:
L and A are N^3-graded by multidegree in the generators.
We exhibit a monomial basis of L, and show that L and A have slow polynomial growth.
We also prove that the Lie algebra L is just infinite, in addition to being a nil algebra.
We show that the lattice points of Z^3 corresponding  to Z^3-graded components of L, A, 
and the restricted enveloping algebra without unit u=u(L) belong to a paraboloid type body of rotation.
Using this observation we prove that L, A, and u are direct sums of two locally nilpotent subalgebras 
and there are infinitely many such decompositions. 
We call L, A and P fractal algebras because these contain infinite copies of themselves.

BANKING MEMBERS:
Interno - 1601562 - ALEX CARRAZEDO DANTAS
Externo à Instituição - DESSISLAVA HRISTOVA KOCHLOUKOVA
Interno - 1984613 - IGOR DOS SANTOS LIMA
Externo à Instituição - IVAN CHESTAKOV - USP
Presidente - 2018215 - VICTOR PETROGRADSKIY