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Let g be a finite-dimensional semisimple Lie algebra over an algebraically closed field C. a) Prove that g is isomorphic to a direct sum of simple Lie algebras. Provide a detailed explanation using the Cartan decomposition. b) Determine the root system associated with g and prove that it satisfies the axioms of a root system. Include verification of reflection symmetries and the integrality of root combinations. c) Classify all finite-dimensional irreducible representations of g. Provide a proof using highest weight theory and the Weyl character formula. d) Investigate the universal enveloping algebra U(g) of g and prove the Poincaré-Birkhoff-Witt (PBW) theorem. Explain its significance in the representation theory of g. e) Explore the Casimir element in U (g) and prove that it acts as a scalar on every irreducible representation of g. Provide a detailed proof and discuss its implications for the structure of representations.

Let g be a finite-dimensional semisimple Lie algebra over an algebraically closed field C. a) Prove that g is isomorphic to a direct sum of simple Lie algebras. Provide a detailed explanation using the Cartan decomposition. b) Determine the root system associated with g and prove that it satisfies the axioms of a root system. Include verification of reflection symmetries and the integrality of root combinations. c) Classify all finite-dimensional irreducible representations of g. Provide a proof using highest weight theory and the Weyl character formula. d) Investigate the universal enveloping algebra U(g) of g and prove the Poincaré-Birkhoff-Witt (PBW) theorem. Explain its significance in the representation theory of g. e) Explore the Casimir element in U (g) and prove that it acts as a scalar on every irreducible representation of g. Provide a detailed proof and discuss its implications for the structure of representations.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
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Let g be a finite-dimensional semisimple Lie algebra over an algebraically closed field C. a) Prove that g is isomorphic to a direct sum of simple Lie algebras. Provide a detailed explanation using the Cartan decomposition. b) Determine the root system associated with g and prove that it satisfies the axioms of a root system. Include verification of reflection symmetries and the integrality of root combinations. c) Classify all finite-dimensional irreducible representations of g. Provide a proof using highest weight theory and the Weyl character formula. d) Investigate the universal enveloping algebra U(g) of g and prove the Poincaré-Birkhoff-Witt (PBW) theorem. Explain its significance in the representation theory of g. e) Explore the Casimir element in U (g) and prove that it acts as a scalar on every irreducible representation of g. Provide a detailed proof and discuss its implications for the structure of representations.
Transcribed Image Text:Let g be a finite-dimensional semisimple Lie algebra over an algebraically closed field C. a) Prove that g is isomorphic to a direct sum of simple Lie algebras. Provide a detailed explanation using the Cartan decomposition. b) Determine the root system associated with g and prove that it satisfies the axioms of a root system. Include verification of reflection symmetries and the integrality of root combinations. c) Classify all finite-dimensional irreducible representations of g. Provide a proof using highest weight theory and the Weyl character formula. d) Investigate the universal enveloping algebra U(g) of g and prove the Poincaré-Birkhoff-Witt (PBW) theorem. Explain its significance in the representation theory of g. e) Explore the Casimir element in U (g) and prove that it acts as a scalar on every irreducible representation of g. Provide a detailed proof and discuss its implications for the structure of representations.
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