An introduction to superalgebras.

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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 non-associative 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_2-graded algebras, (in Russian) Izv.Akad.Nauk SSSR, Ser.Mat. 48(5) 1984, 1042-1059, (in English) Math USSR Izvestiya, 25(2) 1985, 359-374

2. Kemer, A.R.: Solution of the finite basis problem for identities of associative algebras, Soviet Math Dokl, 37(1) 1988, 60-64

3. Osborn, J.: Varieties of algebras, Adv in Math 8(1972) 163-369.

4. I.P.Shestakov: Alternative and Jordan Superalgebras, Siberian Advances in Mathematics 9 (1999), no.2, 83-99.

5. M.Vaughang-Lie: Superalgebras and dimensions of algebras, Internat. J.Algebra and Computation 8(1998), no.1, 97-125.

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)