An introduction to nonassociative algebras

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The basic course in the theory of nonassociative algebra. We introduce the otions of free nonassociative algebra, tensor algebra, bimodules and irepresentations for algebras in a variety. We pay a big attention on the ariety of alternative algebras and composition algebras. We define Lie, alcev and Jordan algebras, their universal enveloping algebras.

Requirements:

Syllabus of lectures:

1. Some basic concepts: the free nonassociative algebra, tensor algebra, symmetric algebra. Grassmann algebra.

2. Varieties of algebras. Bimodules and birepresentations for algebras in a variety.

3.Alternative algebras: nilpotent algebras, the radical, semisimple algebras. The Artin theorem. The Kleinfeld theorem.

4. Composition algebras: Cayley-Dickson process. Generalized theorem of Hurwitz. Quaternions and octonions.

5. Split composition algebras.

6. Speciality problem for Lie, Malcev, and Jordan algebras. Universal enveloping algebras. The Poincaré-Birkhoff-Witt Theorem.

Syllabus of tutorials:

1. Some basic concepts: the free nonassociative algebra, tensor algebra, symmetric algebra. Grassmann algebra.

2. Varieties of algebras. Bimodules and birepresentations for algebras in a variety.

3. Alternative algebras: nilpotent algebras, the radical, semisimple algebras. The Artin theorem. The Kleinfeld theorem.

4. Composition algebras: Cayley-Dickson process. Generalized theorem of Hurwitz. Quaternions and octonions.

5. Split composition algebras.

6. Speciality problem for Lie, Malcev, and Jordan algebras. Universal enveloping algebras. The Poincaré-Birkhoff-Witt Theorem.

Study Objective:

Study materials:

1. N.Jacobson: Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ., Vol. XXXIX, Am. Math. Soc., Providence, 1968.

2. R.D.Schafer: An introduction to nonassociative algebras, Corrected reprint of the 1966 original. Dover Publications, Inc., New York, 1995.

3. K.A.Zhevlakov, A.M.Slinko, I.P.Shestakov, A.I.Shirshov: Rings that are nearly associative, Moscow, Nauka, 1978; English transl.: Academic Press,

N.Y. 1982.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Doctoral studies, daily studies (compulsory elective course)
• Doctoral studies, combined studies (compulsory elective course)
• Doctoral studies, structured daily studies (compulsory elective course)
• Doctoral studies, structured combined studies (compulsory elective course)