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Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination. 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 R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination. 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
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 17E
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Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination.
Transcribed Image Text:Let R be a commutative Noetherian ring and M, N be finitely generated R-modules. a) Define the tensor product M R N and prove that it is also a finitely generated R-module. Use the properties of finitely generated modules in your proof. b) Investigate the Tor functor by computing Tor (M, N) for specific modules M and N. Provide an explicit example and perform the computation. c) Define the projective dimension of a module and prove that if M has finite projective dimension, then Torr³ (M, N) = 0 for all n greater than the projective dimension of M. Close d) Explore the flatness of modules by proving that a module M is flat if and only if Tor (M, N) = 0 for all finitely generated R-modules N. Provide a detailed proof of this equivalence. e) Using the concept of homological dimension, determine the global dimension of a polynomial ring R[×1, x2, . . ., x] over a field R. Provide a comprehensive proof supporting your determination.
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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