SUBJECT
Rings and algebras
lecture + practical
master
2+3
Semesters 1-4
Autumn/Spring semester
Asociative rings and algebras. Constructions: polynomials, formal power series, linear operators, group algebras, free algebras, tensor algebras, exterior algebras. Structure theory: the radical, direct and semidirect decompositions. Chain conditions. The Hilbert Basis Theorem, the Hopkins theorem.
Categories and functors. Algebraic and topological examples. Natural transformations. The concept of categorical equivalence. Covariant and contravariant functors. Properties of the Hom and tensor functors (for non-commutative rings). Adjoint functors. Additive categories, exact functors. The exactness of certain functors: projective, injective and flat modules.
Homolgical algebra. Chain complexes, homology groups, chain homotopy. Examples from algebra and topology. The long exact sequence of homologies.
Commutative rings. Ideal decompositions. Prime and primary ideals. The prime spectrum of a ring. The Nullstellensatz of Hilbert.
Lie algebras. Basic notions, examples, linear Lie algebras. Solvable and nilpotent Lie algebras. Engel’s theorem. Killing form. The Cartan subalgebra. Root systems and quadratic forms. Dynkin diagrams, the classification of semisimple complex Lie algebras. Universal enveloping algebra, the Poincaré–Birkhoff–Witt theorem.
- Cohn, P.M.: Algebra I-III. Hermann, 1970, Wiley 1989, 1990.
- Jacobson, N.: Basic Algebra I-II. Freeman, 1985, 1989.
- Humphreys, J.E.: Introduction to Lie algebras and representation theory. Springer, 1980.