Nonassociative algebra

2023/2024
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Course director:
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Prerequisites

There are no prerequisites.

Content (Syllabus outline)

Important types of nonnassociative algebras (alternating algebras, Jordan algebras).
The definition of Lie algebra. Ideals and homomorphisms. Solvable and nilpotent Lie algebras.
Lie's and Cartan's Theorems. The Killing form. Completely irreducible representations. Representations of sl(2,F). Root subspace decomposition.
Root systems. Simple roots and the Weyl group. Classification of (finite-dimensional) simple Lie algebras.
Universal envelloping algebra. Theorem Poicaré-Birkhoff-Witt.
Representation theory of simlpe Lie algebras.

Readings

K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Rings that are nearly associative, Academic Press, 1982.
J. E. Humphreys: Introduction to Lie Algebras and Representation Theory, Springer, New York-Berlin, 1997.
J. P. Serre: Complex Semisimple Lie Algebras, Springer, Berlin, 2001.
W. A. de Graaf: Lie Algebras : Theory and Algorithms, North Holland, Amsterdam, 2000.

Objectives and competences

Student meets the fundamental notions and theorems of the nonassociative algebra.

Intended learning outcomes

Knowledge and understanding: Understanding of basic concepts and theorems of noncommutative algebra, and their role in some other areas.
Application: In other mathematical areas.
Reflection: Understanding the theory on the basis of examples and applications.
Transferable skills: Formulation and solution of problems using abstract methods.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Tomaž Košir:
BERNIK, Janez, DRNOVŠEK, Roman, KOKOL-BUKOVŠEK, Damjana, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. On semitransitive jordan algebras of matrices. Journal of algebra and its applications, ISSN 0219-4988, 2011, vol. 10, iss. 2, str. 319-333. [COBISS-SI-ID 15908697]
GRUNENFELDER, Luzius, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. Maximal Jordan algebras of matrices with bounded number of eigenvalues. Israel journal of mathematics, ISSN 0021-2172, 2002, vol. 128, str. 53-75. [COBISS-SI-ID 11625305]
GRUNENFELDER, Luzius, GURALNICK, Robert M., KOŠIR, Tomaž, RADJAVI, Heydar. Permutability of characters on algebras. Pacific journal of mathematics, ISSN 0030-8730, 1997, let. 178, št. 1, str. 63-70. [COBISS-SI-ID 7437145]