INTRODUCTION TO HIGHER ALGEBRA AND REPRESENTATION THEORY (I.Yu.Tipunin)

 Basic notions of linear algebra
 Linear space, subspace, factorspace
 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 factoralgebra, simple and semisimple algebras; Abelian, nilpotent, solvable and reductive Lie algebras
 Short exact sequences of algebras
 Algebras Mat_{n} (C), Mat_{n} (R), gl (n, C), gl (n, R)
 Simplicity of algebras Mat_{n} (C) and Mat_{n} (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
 Finitedimensional 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
 Bilinear 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
 Bilinear forms which are invariant under Lie algebra action
 Group action on a multitude, group representation, group orbits, adjoint and coadjoint group representation
 Bilinear 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
 PoincareBirkhoffWitt 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, bialgebras 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 finitedimensional irreducible representations of sl (2, C) , KlebshGordan 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, chargeconjugate 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 , JordanGelder series for Verma module, repetition factor of simple subfactors, singular vectors, KatzKazhdan 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 finitedimesional irreducible representations for simple algebras and its subalgebras, BernsteinGelfandGelfand resolvent
 Algebraical aspects of quantization
 Poisson algebras, Poisson algebras deformations, notion of deformed quantization
 PoissonLie groups, Sklyanin bracket, quantum groups
 Quantum universal enveloping algebras of semisimple Lie algebras, quasitensorial categories, universal Rmatrix,YangBaxter equations
 Quantum sl(2) and its representations in the root of the identity, infinitedimensional center

