An introduction to superalgebras.
Code  Completion  Credits  Range 

XP01USA  ZK  4  2+1 
 Lecturer:
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

The basic course in the theory of superalgebras. We introduce notions of a graded algebra, superalgebra, Grassmann envelope of a superalgebra. Consider varieties of superalgebras and identities in superalgebras. We pay a big attention on the variety of alternative and Jordan superalgebras.
 Requirements:
 Syllabus of lectures:

1. Basic concepts: graded algebra, tensor product of superalgebras, Clifford and Grassmann superalgebras, Grassmann envelope, varieties of superalgebras, simple superalgebras.
2. Free superalgebras: the free nonassociative algebra, relatively free algebras, homogeneous varieties, identities in superalgebras, dimensions of algebras, free superalgebras.
3. Nilpotency and solvability: the Baer radical, semidirect product, PI algebras, criterium of nilpotency, bimodules and birepresentations.
4. Elements of structure theory for alternative and Jordan superalgebras.
 Syllabus of tutorials:
 Study Objective:
 Study materials:

1. Kemer, A.R.: Varieties and Z_2graded algebras, (in Russian) Izv.Akad.Nauk SSSR, Ser.Mat. 48(5) 1984, 10421059, (in English) Math USSR Izvestiya, 25(2) 1985, 359374
2. Kemer, A.R.: Solution of the finite basis problem for identities of associative algebras, Soviet Math Dokl, 37(1) 1988, 6064
3. Osborn, J.: Varieties of algebras, Adv in Math 8(1972) 163369.
4. I.P.Shestakov: Alternative and Jordan Superalgebras, Siberian Advances in Mathematics 9 (1999), no.2, 8399.
5. M.VaughangLie: Superalgebras and dimensions of algebras, Internat. J.Algebra and Computation 8(1998), no.1, 97125.
 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)