Course Description:
The course is an introduction to the theory of polynomial identities for associative algebras and aims to show how tools from representation theory can be applied to study algebras with polynomial identities. In the first part, we will review fundamental concepts from the representation theory of finite groups, focusing in greater detail on the case of the symmetric group. Next, we will delve into PI theory. We will focus on the concepts of polynomial identities, T-ideals, and PI algebras, and explore how these can be studied using numerical invariants related to their identities. Finally, we will emphasize how the combinatorial tools discussed earlier can be used to investigate the T-ideal of identities satisfied by a given algebra.
Schedule:
Lectures:
February 9: 14:30-17:30,
February 10: 11:00-13:00,
February 11: 14:30-17:30,
February 12: 11:00-13:00,
February 13: 11:00-13:00,
February 16: 14:30-17:30.
Location:
aula A.0.6 in the Renato Ricamo building (Coppito 1).
Course materials:
Core Monographs:
E. Aljadeff, A. Giambruno, C. Procesi, A. Regev, Rings with Polynomial Identities and Finite Dimensional Representations of Algebras, Colloquium Publication, vol 66, AMS, Providence, RI, 2020.
A. Giambruno, M. Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, Vol. 122, Amer. Math. Soc., Providence, RI, 2005.
Selected Papers:
R. G. Swan, An application of graph theory to algebra, Proc. Amer. Math. Soc. 14 (1963), 367-373. DOI
R. G. Swan, Correction to “An application of graph theory to algebra’’ Proc. Amer. Math. Soc. 21 (1969), 379-380. DOI
Further Reading:
R. Diestel, Graph Theory, Graduate Texts in Mathematics, vol. 173, Springer-Verlag, Heidelberg, 1997.
B. E. Sagan, The symmetric group: Representations, combinatorial algorithms, and symmetric functions, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001.
Possible Seminar Topics (Final Exam):
Students may choose one of the following topics for their seminar presentation and are welcome to contact me to discuss the details.
Regev's Theorem (an exponential upper bound for the codimensions).
T-ideal and codimensions of upper triangular 2x2 matrices.
Varieties of polynomial growth.
Characterization of fundamental algebras through S_n-representations.
...
Students are more than welcome to propose their own seminar topics. Feel free to contact me.