Sumários
15. Semisimple Lie algebras
13 novembro 2020, 11:00 • Pedro Resende
Simple Lie algebras. Derived series and lower central series of a Lie algebra, solvable Lie algebras and nilpotent Lie algebras. Reductive Lie algebras and semisimple Lie algebras (defined to be reductive Lie algebras with trivial center, following Brian Hall's book). Decomposition of a reductive Lie algebra as the direct sum of its center and a semisimple Lie algebra. Decomposition of a semisimple Lie algebra as the direct sum of uniquely determined simple Lie subalgebras. Semisimplicity of the complexification of the Lie algebra of a simply connected matrix Lie group. This lecture followed sections 3.2 and 7.1 of Brian Hall's book.
14. Representations of Lie groups and Lie algebras
12 novembro 2020, 11:00 • Pedro Resende
Matrix representation of weights and roots with respect to abelian Lie subalgebras equipped with the trace inner product. Weyl group of an abelian Lie subalgebra. Example — sl(3;C): symmetries of the set of weights and roots; weight diagrams for several irreducible representations; proof that the weights of an irreducible representation are the integral elements in the convex hull of the orbit of the highest weight under the action of the Weyl group which can be obtained from the highest weight by subtracting an integer combination of roots. This lecture followed sections 6.6–6.8 of Brian Hall's book.
13. Representations of Lie groups and Lie algebras
6 novembro 2020, 11:00 • Pedro Resende
Proof of the classification of the irreducible representations of sl(3;C) (based on sections 6.4–6.5 of Brian Hall's book): highest weight cyclic representations and their identification with the irreducible representations; construction of irreducible representations as invariant subspaces of tensor powers of the standard representation and its dual. Example: construction of the irreducible representation with highest weight (1,1).
12. Representations of Lie groups and Lie algebras
5 novembro 2020, 11:00 • Pedro Resende
The reordering lemma for representations of Lie algebras. Cyclic generators versus irreducible representations. Representations of SU(3) (introduction). Basis of sl(3;C) and commutation relations of the basis elements. Irreducibility of the standard representation. Weights, weight vectors, weight spaces, weight multiplicity. Roots and root vectors. Positive simple roots. Partial order on the set of weights; highest weights. Statement of the classification theorem for the irreducible representations of sl(3;C) and SU(3). Dominant integral elements. Beginning of the proof of the classification theorem: direct sum decomposition of irreducible representations in terms of weight spaces. The material for this lecture is chapter 6 of Brian Hall's book up to and including Proposition 6.9, plus Proposition 6.12.
11. Representations of Lie groups and Lie algebras
30 outubro 2020, 11:00 • Pedro Resende
Corollaries of Schur's lemma. Brief overview of Haar measures for locally compact Hausdorff groups, and of the construction of Haar measures via left/right invariant volume forms in the case of Lie groups. Proof that any representation of a compact Hausdorff group G is unitary with respect to an inner product obtained by integration on G. Corollary: every finite dimensional representation of a compact Hausdorff group (in particular a compact Lie group) is completely reducible. Representations of SU(2), su(2), and sl(2;C); classification of the irreducible representations. Representations of SO(3). This lecture is based on chapter 4 of Brian Hall (sections 4.5–4.7 plus Theorem 4.28 and Example 4.10).