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Category Theory and Representation Theory

Category Theory and Representation Theory Quiz

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a category?

  1. A collection of objects and morphisms between them
  2. A set of elements and operations on those elements
  3. A group of transformations on a set
  4. A ring with a unit
Question 2 Multiple Choice (Single Answer)

What is a functor?

  1. A map between two categories that preserves the structure of the categories
  2. A function between two sets that preserves the operations on the sets
  3. A group homomorphism between two groups
  4. A ring homomorphism between two rings
Question 3 Multiple Choice (Single Answer)

What is a natural transformation?

  1. A map between two functors that preserves the structure of the functors
  2. A function between two functions that preserves the operations on the functions
  3. A group homomorphism between two group homomorphisms
  4. A ring homomorphism between two ring homomorphisms
Question 4 Multiple Choice (Single Answer)

What is a representation of a group?

  1. A homomorphism from the group to the group of invertible linear transformations of a vector space
  2. A function from the group to the set of all linear transformations of a vector space
  3. A group homomorphism from the group to the group of units of a ring
  4. A ring homomorphism from the group to the group of invertible elements of a field
Question 5 Multiple Choice (Single Answer)

What is a character of a group?

  1. A function from the group to the complex numbers that is constant on conjugacy classes
  2. A function from the group to the real numbers that is constant on conjugacy classes
  3. A group homomorphism from the group to the group of units of a ring
  4. A ring homomorphism from the group to the group of invertible elements of a field
Question 6 Multiple Choice (Single Answer)

What is the Schur orthogonality relations?

  1. A set of equations that relate the characters of a group
  2. A set of equations that relate the representations of a group
  3. A set of equations that relate the elements of a group
  4. A set of equations that relate the morphisms of a category
Question 7 Multiple Choice (Single Answer)

What is the Maschke's theorem?

  1. A theorem that states that every finite-dimensional representation of a group is completely reducible
  2. A theorem that states that every finite-dimensional representation of a group is semisimple
  3. A theorem that states that every finite-dimensional representation of a group is indecomposable
  4. A theorem that states that every finite-dimensional representation of a group is simple
Question 8 Multiple Choice (Single Answer)

What is the Wedderburn's theorem?

  1. A theorem that states that every finite division ring is a field
  2. A theorem that states that every finite-dimensional division algebra is a field
  3. A theorem that states that every finite-dimensional algebra is semisimple
  4. A theorem that states that every finite-dimensional algebra is simple
Question 9 Multiple Choice (Single Answer)

What is the Jacobson radical of a ring?

  1. The largest nilpotent ideal of a ring
  2. The largest radical ideal of a ring
  3. The largest prime ideal of a ring
  4. The largest maximal ideal of a ring
Question 10 Multiple Choice (Single Answer)

What is the Artin-Wedderburn theorem?

  1. A theorem that states that every semisimple ring is a direct product of simple rings
  2. A theorem that states that every semisimple algebra is a direct product of simple algebras
  3. A theorem that states that every finite-dimensional semisimple algebra is a direct product of simple algebras
  4. A theorem that states that every finite-dimensional semisimple ring is a direct product of simple rings
Question 11 Multiple Choice (Single Answer)

What is the Brauer-Thrall theorem?

  1. A theorem that states that every finite-dimensional division algebra over a field is a central simple algebra
  2. A theorem that states that every finite-dimensional central simple algebra over a field is a division algebra
  3. A theorem that states that every finite-dimensional algebra over a field is a central simple algebra
  4. A theorem that states that every finite-dimensional central simple algebra over a field is an algebra
Question 12 Multiple Choice (Single Answer)

What is the Noether-Skolem theorem?

  1. A theorem that states that every finitely generated module over a principal ideal domain is a direct sum of cyclic modules
  2. A theorem that states that every finitely generated module over a Euclidean domain is a direct sum of cyclic modules
  3. A theorem that states that every finitely generated module over a Dedekind domain is a direct sum of cyclic modules
  4. A theorem that states that every finitely generated module over a field is a direct sum of cyclic modules
Question 13 Multiple Choice (Single Answer)

What is the Hilbert basis theorem?

  1. A theorem that states that every ideal in a Noetherian ring is finitely generated
  2. A theorem that states that every ideal in a Dedekind domain is finitely generated
  3. A theorem that states that every ideal in a principal ideal domain is finitely generated
  4. A theorem that states that every ideal in a Euclidean domain is finitely generated
Question 14 Multiple Choice (Single Answer)

What is the Krull-Schmidt theorem?

  1. A theorem that states that every finitely generated module over a Noetherian ring is a direct sum of indecomposable modules
  2. A theorem that states that every finitely generated module over a Dedekind domain is a direct sum of indecomposable modules
  3. A theorem that states that every finitely generated module over a principal ideal domain is a direct sum of indecomposable modules
  4. A theorem that states that every finitely generated module over a Euclidean domain is a direct sum of indecomposable modules
Question 15 Multiple Choice (Single Answer)

What is the Wedderburn-Artin theorem?

  1. A theorem that states that every semisimple ring is a direct product of simple rings
  2. A theorem that states that every semisimple algebra is a direct product of simple algebras
  3. A theorem that states that every finite-dimensional semisimple algebra is a direct product of simple algebras
  4. A theorem that states that every finite-dimensional semisimple ring is a direct product of simple rings
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