Elements of contemporary mathematics

  1. Basic definitions in group theory: grupoid, semigroup, groups; general group classifications, subgroups, cosets, quotient groups, group products, conjugation, conjugacy classes, basic theorems (Cayley, Lagrange, rearrangement theorem etc.).
  2. Group homomorphism and isomorphism, isomorphism theorems.
  3. Vector spaces, simple sums and tensor products. Rings, algebras, Lie algebras, division algebras. Abelian, associative and alternative algebras. Ideals.
  4. Group action on a set, oribt, stabilizer.
  5. Representation of groups; reducibility, unitarity, unitary representations of finite and compact groups. Tensor product of representations, representations of algebras.
  6. Schur’s lemma.
  7. Tensor operators, Eckart–Wigner theorem.
  8. Manifolds: basic definions. Orientability, coordinates, tangent and cotangent space. Vectors and tensors on manifolds.
  9. Elements of topology: homeomorphisms, simplices, orientation, simplicial complexes, boundaries and cycles, homotopy groups, homologies.
  10. Differential forms on manifolds. External algebra, external products, closed and exact forms, cohomologies: symplicial and Cˇech cohomology.
  11. Integration of differential forms over chains, Stokes theorem, relation of homology and cohomology, de Rham complex.
  12. Action of groups on manifolds. Quotient spaces and orbifolds. Covering spaces.
  13. Lie groups: basic definitions and topological properties.
  14. Fibre boundles, principal and associated fibre bundle.
  15. Connection on fibre bundles, connection forms, curvature. Connection on vector bundles, local connection forms. Metric. Linear connection, torsion, Levi-Civita connection.
  16. The Lie algebra: generators, structure constants, Killing forms, adjoint representation, simple and semi-simple Lie algebras.
  17. Covering groups, the third Lie theorem, exponentiation of the Lie algebra.
  18. Representations of Lie algebras: the Cartan-Weyl matrix, weights and roots.
  19. Master formula for roots. Geometry of roots. Representation construction: the highest weight method. Dynkin coefficients.
  20. Cartan matrix, Dynkin diagrams, P-systems, reduction rules for Dynkin diagrams.
  21. Classification of simple Lie algebras.

Construction of simple Lie subalgebras with Dynkin diagrams.

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