Restricted Lie algebras and Groups 2025/26
Math PhD School UniMIB-UniPV-UniCatt-INdAM
Restricted Lie algebras and Groups 2025/26
Math PhD School UniMIB-UniPV-UniCatt-INdAM
The course is devoted to the study of p-restricted Lie algebras, where p denotes a prime number., and some applications in group theory.
After defining a p-restricted Lie algebra over a field of characteristic p, we will focus on the restricted universal envelope of a p-restricted Lie algebra, and prove its existence and uniqueness — roughly speaking, every p-restricted Lie algebra is a subalgebra of a canonical associative algebra. In order to do that, we need first to study and prove the analogous results for “usual” Lie algebras and their universal envelopes.
The study of p-restricted Lie algebras finds an application in group theory. As we shall see, one may associate a p-restricted Lie algebra over the finite field Z/pZ to a group: such a Lie algebra can be seen as a “linearization” of the group, and used to study the structure of the group.
The prerequisite is basic algebra (in particular, groups and associative algebras). The knowledge of Lie algebras — in particular, the attendance of M. Avitabile’s course “Introduction to Lie algebras” — helps but it is not a prerequisite.
When: February–March 2026 (tot.: 24 hours).
Where: the course will be delivered entirely on-line.
Content:
A refresh of free associative algebras, graded algebras and Lie algebras
The universal envelope of a Lie algebra and Poincare-Birkhoff-Witt Theorem ́
p-restricted Lie algebras: definition and examples
The restricted universal envelope of a p-restricted Lie algebra
The p-Zassenhaus filtration of a group and the p-restricted Lie algebra associated to a group
Jennings’ Theorem — the bridge between groups and restricted Lie algebras
Examples with groups
References:
L. Bartholdi, H. Haerer, Th. Schick, "Right angled Artin groups and partial commutation, old and new", Enseign. Math. 66 (2020), no. 1-2.
R. Carter, Lie algebras of finite and affine type, Cambridge stud. adv. math. vol. 96, CUP, 2005.
J. Dixon, M. du Sautoy, A. Mann, D. Segal, Analytic pro-p groups 2nd ed., Cambridge stud. adv. math. vol. 61, CUP, 1999.
N. Jaconson, Lie algebras, Dover, 1979.
A. Lichtman, "On Lie algebras of free products of groups", J. Pure Appl. Math. 18 (1980), no. 1.