Basic notions of linear algebra Linear space, subspace, factor-space Linear mappings between spaces, complexes, exact sequences Direct sum and tensor product of linear spaces, linear functionals, dual space, tensors Complexification and realization of linear spaces, real form Grading and filtration on linear space Basic algebraic systems Associative algebras, Lie algebras Mappings between algebras, homomorphisms, isomorphisms Subalgebra, ideal and factor-algebra, simple and semi-simple algebras; Abelian, nilpotent, solvable and reductive Lie algebras Short exact sequences of algebras Algebras Matn (C), Matn (R), gl (n, C), gl (n, R) Simplicity of algebras Matn (C) and Matn (R) Notions of algebra module and algebra representation Reducible, irreducible and completely reducible representations Representation homomorphisms, intertwining operator, Schur's lemma Theory of weight representations of sl (2, C) Notions of highest weight representation, Verma module and contragradient module Realization of the Verma module and its contragradient in polinomial space of one variable Notion of singular vectors and its explicit finding Finite-dimensional irreducible modules Weight representations without highest weight Extensions of representations Examples of representations not being weight Classical Lie groups and its Lie algebras Definition of groups, group homomorphisms and isomorphisms, subgroups, normal subgroups Permutation group, finite groups Bi-linear forms which are invariant under group action Classical Lie groups GL (n) , SL (n) , O (n) , SO (n) , Sp (n) ; Euclidean, Lorentz and Poincare groups; groups U (n) and SU (n) Connection between Lie group and its Lie algebra for matrix groups Lie algebras of classical Lie groups Bi-linear forms which are invariant under Lie algebra action Group action on a multitude, group representation, group orbits, adjoint and coadjoint group representation Bi-linear forms on representations of Lie algebras and groups, Killing form Cartan subgroup and subalgebras, Borel subgroup and subalgebras, top and bottom triangular subgroups and subalgebras, Gauss decomposition of classical groups and algebras Enveloping algebras of Lie algebras Free associative algebras and Lie algebras, algebra assignment by means of generatrices and relations Enveloping algebra of representation, universal enveloping algebra Poincare-Birkhoff-Witt basis Connection between representations of Lie algebra and its universal eneveloping algebra Universal eneveloping algebra of classical Lie algebras Center of universal eneveloping algebra, Casimir operators and center of U (sl (2)) , universal eneveloping algebras of Lorentz and Poincare algebras Notion of induced representation, Verma modules of classical Lie algebras Tensor product of associative algebras and representations, notion of coproduct, coassociativity and cocommutativity, commutative diagrams, bi-algebras and Hopf algebra Realization of universal enveloping algebra of Lie algebra as a space of generalized functions on group, structure of Hopf algebra on universal enveloping algebra of Lie algebra Tensor products of algebra representations and contragradient modules Tensor products of finite-dimensional irreducible representations of sl (2, C) , Klebsh-Gordan coefficients Representations of classical Lie algebras in tensor spaces Schur's duality, representation of symmetric group and Young diagram, types of symmetry Anticommutative variables, Grassmann algebras, Clifford algebras, connection between Clifford algebras and matrix algebras, complex and real cases, notion of super Lie algebra, simplest examples, trace and supertrace on associative algebra Spinors as Clifford algebra representations, charge-conjugate matrix, Majorana and Weyl spinors Realization of representations of classical Lie algebras in boson and fermion Fock spaces Elements of Cartan theory Cartansubalgebras, root vectors, root decomposition, Chevalley's basis, lattices of roots and weights Weight representations, representations of the highest and lowest weight, lattice of representation weights, character of representation, representation from " categories O " Simple roots, Dynkin diagrams, classification of simple complex Lie algebras, reconstruction of Lie algebra using Dynkin diagram Weyl group, Weyl chambers, prepotent weights, integral weights, elements of the structural theory of Verma modules , Jordan-Gelder series for Verma module, repetition factor of simple subfactors, singular vectors, Katz-Kazhdan theorem, structure of " categories O ", shift functor Elements of homological algebra Chevalley cohomologies of Lie algebras, interpretation of younger cohomological classes, central extensions, affine algebras, Virassaro algebra Hochschield cohomologies, elements of deformation theory, deformed brackets Some notions of category theory Basic computational means: diagrammatic search, long exact sequences of cohomology, algebraical homotopy, resolvents and the simplest spectral sequences Computing of cohomologies with coefficients lying in finite-dimesional irreducible representations for simple algebras and its subalgebras, Bernstein-Gelfand-Gelfand resolvent Algebraical aspects of quantization Poisson algebras, Poisson algebras deformations, notion of deformed quantization Poisson-Lie groups, Sklyanin bracket, quantum groups Quantum universal enveloping algebras of semi-simple Lie algebras, quasi-tensorial categories, universal R-matrix,Yang-Baxter equations Quantum sl(2) and its representations in the root of the identity, infinite-dimensional center