Category Theory and Representation Theory
| Description: Category Theory and Representation Theory Quiz | |
| Number of Questions: 15 | |
| Created by: Aliensbrain Bot | |
| Tags: category theory representation theory mathematics |
What is a category?
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A collection of objects and morphisms between them
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A set of elements and operations on those elements
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A group of transformations on a set
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A ring with a unit
A category consists of a collection of objects and a collection of morphisms between those objects. The morphisms are also called arrows.
What is a functor?
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A map between two categories that preserves the structure of the categories
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A function between two sets that preserves the operations on the sets
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A group homomorphism between two groups
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A ring homomorphism between two rings
A functor is a map between two categories that preserves the structure of the categories. This means that it maps objects to objects and morphisms to morphisms in a way that respects the composition of morphisms.
What is a natural transformation?
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A map between two functors that preserves the structure of the functors
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A function between two functions that preserves the operations on the functions
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A group homomorphism between two group homomorphisms
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A ring homomorphism between two ring homomorphisms
A natural transformation is a map between two functors that preserves the structure of the functors. This means that it maps objects to objects and morphisms to morphisms in a way that respects the composition of morphisms.
What is a representation of a group?
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A homomorphism from the group to the group of invertible linear transformations of a vector space
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A function from the group to the set of all linear transformations of a vector space
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A group homomorphism from the group to the group of units of a ring
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A ring homomorphism from the group to the group of invertible elements of a field
A representation of a group is a homomorphism from the group to the group of invertible linear transformations of a vector space. This means that it maps group elements to linear transformations in a way that respects the group operation.
What is a character of a group?
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A function from the group to the complex numbers that is constant on conjugacy classes
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A function from the group to the real numbers that is constant on conjugacy classes
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A group homomorphism from the group to the group of units of a ring
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A ring homomorphism from the group to the group of invertible elements of a field
A character of a group is a function from the group to the complex numbers that is constant on conjugacy classes. This means that it takes the same value on all elements of a conjugacy class.
What is the Schur orthogonality relations?
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A set of equations that relate the characters of a group
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A set of equations that relate the representations of a group
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A set of equations that relate the elements of a group
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A set of equations that relate the morphisms of a category
The Schur orthogonality relations are a set of equations that relate the characters of a group. These equations can be used to prove a number of important results about group representations.
What is the Maschke's theorem?
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A theorem that states that every finite-dimensional representation of a group is completely reducible
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A theorem that states that every finite-dimensional representation of a group is semisimple
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A theorem that states that every finite-dimensional representation of a group is indecomposable
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A theorem that states that every finite-dimensional representation of a group is simple
Maschke's theorem states that every finite-dimensional representation of a group is completely reducible. This means that it can be decomposed into a direct sum of irreducible representations.
What is the Wedderburn's theorem?
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A theorem that states that every finite division ring is a field
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A theorem that states that every finite-dimensional division algebra is a field
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A theorem that states that every finite-dimensional algebra is semisimple
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A theorem that states that every finite-dimensional algebra is simple
Wedderburn's theorem states that every finite division ring is a field. This means that it is a ring with no zero divisors and every nonzero element has a multiplicative inverse.
What is the Jacobson radical of a ring?
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The largest nilpotent ideal of a ring
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The largest radical ideal of a ring
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The largest prime ideal of a ring
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The largest maximal ideal of a ring
The Jacobson radical of a ring is the largest nilpotent ideal of the ring. This means that it is an ideal that consists of all elements that are nilpotent.
What is the Artin-Wedderburn theorem?
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A theorem that states that every semisimple ring is a direct product of simple rings
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A theorem that states that every semisimple algebra is a direct product of simple algebras
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A theorem that states that every finite-dimensional semisimple algebra is a direct product of simple algebras
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A theorem that states that every finite-dimensional semisimple ring is a direct product of simple rings
The Artin-Wedderburn theorem states that every semisimple ring is a direct product of simple rings. This means that it can be decomposed into a direct product of rings that have no proper ideals.
What is the Brauer-Thrall theorem?
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A theorem that states that every finite-dimensional division algebra over a field is a central simple algebra
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A theorem that states that every finite-dimensional central simple algebra over a field is a division algebra
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A theorem that states that every finite-dimensional algebra over a field is a central simple algebra
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A theorem that states that every finite-dimensional central simple algebra over a field is an algebra
The Brauer-Thrall theorem states that every finite-dimensional division algebra over a field is a central simple algebra. This means that it is a simple algebra that is also a centralizer of itself.
What is the Noether-Skolem theorem?
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A theorem that states that every finitely generated module over a principal ideal domain is a direct sum of cyclic modules
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A theorem that states that every finitely generated module over a Euclidean domain is a direct sum of cyclic modules
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A theorem that states that every finitely generated module over a Dedekind domain is a direct sum of cyclic modules
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A theorem that states that every finitely generated module over a field is a direct sum of cyclic modules
The Noether-Skolem theorem states that every finitely generated module over a principal ideal domain is a direct sum of cyclic modules. This means that it can be decomposed into a direct sum of modules that are generated by a single element.
What is the Hilbert basis theorem?
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A theorem that states that every ideal in a Noetherian ring is finitely generated
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A theorem that states that every ideal in a Dedekind domain is finitely generated
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A theorem that states that every ideal in a principal ideal domain is finitely generated
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A theorem that states that every ideal in a Euclidean domain is finitely generated
The Hilbert basis theorem states that every ideal in a Noetherian ring is finitely generated. This means that it can be generated by a finite number of elements.
What is the Krull-Schmidt theorem?
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A theorem that states that every finitely generated module over a Noetherian ring is a direct sum of indecomposable modules
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A theorem that states that every finitely generated module over a Dedekind domain is a direct sum of indecomposable modules
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A theorem that states that every finitely generated module over a principal ideal domain is a direct sum of indecomposable modules
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A theorem that states that every finitely generated module over a Euclidean domain is a direct sum of indecomposable modules
The Krull-Schmidt theorem states that every finitely generated module over a Noetherian ring is a direct sum of indecomposable modules. This means that it can be decomposed into a direct sum of modules that cannot be further decomposed into smaller modules.
What is the Wedderburn-Artin theorem?
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A theorem that states that every semisimple ring is a direct product of simple rings
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A theorem that states that every semisimple algebra is a direct product of simple algebras
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A theorem that states that every finite-dimensional semisimple algebra is a direct product of simple algebras
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A theorem that states that every finite-dimensional semisimple ring is a direct product of simple rings
The Wedderburn-Artin theorem states that every semisimple ring is a direct product of simple rings. This means that it can be decomposed into a direct product of rings that have no proper ideals.