An introduction to nonassociative algebras
Code  Completion  Credits  Range  Language 

XP01UNA  ZK  4  2+1  Czech 
 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: CayleyDickson 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 PoincaréBirkhoffWitt 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: CayleyDickson 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 PoincaréBirkhoffWitt 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 timetable 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)