Sumários
11 Dezembro 2025, 10:30
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Marko Stosic
Homogeneous spaces.
Groups action. Orbits and stabilizers.
Root decomposition of semisimple Lie algebras.
Positive roots, simple roots.
Classification of simple Lie algebras.
4 Dezembro 2025, 10:30
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Marko Stosic
Representations of sl(2,C) (continuation).
Highest weight representations.
Characterization of irreducible representations of sl(2,C).
Complete reducibility of representations of sl(2,C) (and of any semisimple Lie algebra).
Weights and weight decomposition of a representation of a semisimple Lie algebra.
Roots and root decomposition of a semisimple Lie algebra, as weights and weight decomposition of the adjoint representation.
Root decomposition of sl(3,C).
27 Novembro 2025, 10:30
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Marko Stosic
Killing form and semisimple Lie algebras. sl(n,C).
Real form of the complex Lie algebra.
Semisimplicity of sl(n,R), su(n), and classical matricial Lie algebras.
Representations of semisimple Lie algebras.
Irreducible representations of sl(2,C).
20 Novembro 2025, 10:30
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Marko Stosic
Solvable and nilpotent ideals of Lie algebra.
Maximal solvable ideal -- radical of a Lie algebra.
Simple and semisimple Lie algebras.
4th structural theorem for Lie algebra: quotient of a Lie algebra by its radical is a semisimple Lie algebra.
Killing form.
Examples of Killing form, in particular for Lie algebra sl(2,C).
Cartan's criterion for solvability and semisimplicity of a Lie algebra (via Killing form).
sl(2,C) is semisimple.
13 Novembro 2025, 10:30
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Marko Stosic
Classification of abstract Lie algebras.
Ideals of Lie algebras.
Solvable and nilpotent Lie algebras.