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Representation theory

Studies algebraic structures via linear transformations.

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By representing elements of abstract algebraic objects as linear transformations on vector spaces, representation theory makes those objects more concrete.[1] This method transforms abstract algebraic questions into ones about linear algebra, a field that is thoroughly comprehended.[2] Groups, associative algebras, and Lie algebras are among the algebraic structures that can be described in this manner.[3]

Quick Facts

Field
Mathematics

Facts from the source article.

Key facts and history

Alternatively, a representation is a homomorphism from the algebraic object to the general linear group of a vector space.[4] Irreducible representations serve as building blocks for many groups, as non-irreducible representations decompose into simpler subrepresentations and quotients.[5] Unitary representations on Hilbert spaces have been applied in quantum mechanics since the 1920s, influenced by Hermann Weyl and developed by Eugene Wigner, George Mackey, and Harish-Chandra.[6] The representation theory of Lie groups is crucial in physics and chemistry, and many important groups in those fields are Lie groups.[7]

The earliest and most notable type is group representation theory, where group members are depicted using invertible matrices, and the group operation corresponds to matrix multiplication.[3] One method to define a representation employs the concept of an action, which extends the standard way matrices multiply with column vectors.[8] A representation is called irreducible when its only subrepresentations are the zero subspace and the entire space itself.[9] Élie Cartan's research led to a complete classification of the finite-dimensional representations for semisimple Lie algebras.[10]

What the research sources say

The nLab account notes that representation theory concerns linear representations of groups, group algebras, Hopf algebras, or Lie algebras, and also studies representations of generalizations such as coalgebras.[11] The fundamental concepts of representation theory have a natural formulation in homotopy theory and homotopy type theory, refining the study from representations of groups to ∞-representations of ∞-groups.[11] Under this identification, the representation theory of a group G is equivalently the homotopy theory in the slice (∞,1)-topos over the classifying space BG.[11]

References

  1. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: Classic texts on representation theory include and . Other excellent sources are and .; "representation theory in nLab", ncatlab.org (we also read that source) en.wikipedia.org
  2. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: Ronan, "linear algebra", Encyclopedia Britannica; There are many textbooks on vector spaces and linear algebra. For an advanced treatment, see . (original not read by us) en.wikipedia.org
  3. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: For the history of the representation theory of finite groups, see . For algebraic and Lie groups, see .; Etingof, "Introduction to representation theory", www-math.mit.edu, January 10, 2011 (original not read by us) en.wikipedia.org
  4. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: Teleman, "Representation Theory", math.berkeley.edu, 2005; Teleman, "Representation Theory", math.berkeley.edu, 2005 (original not read by us) en.wikipedia.org
  5. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: Humphreys, "Linear Algebraic Groups", Springer New York, 1975 (we also read that source) en.wikipedia.org
  6. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: See .; .; . (original not read by us) en.wikipedia.org
  7. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: .; . (original not read by us) en.wikipedia.org
  8. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: This material can be found in standard textbooks, such as , , , , , , and . (original not read by us) en.wikipedia.org
  9. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: The representation {0} of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor (original not read by us) en.wikipedia.org
  10. Wikipedia, "Representation theory" (CC BY-SA 4.0), which cites: . (original not read by us) en.wikipedia.org
  11. "representation theory in nLab", ncatlab.org. facts only, no text reused. ncatlab.org

Each numbered sentence above was checked against the source it cites before publication. Sentences citing Wikipedia adapt text from the English Wikipedia article Representation theory (contributors), used under CC BY-SA 4.0; this entry is available under the same licence. Other sources were used for facts only; no text or images are copied from them.

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Compiled from Wikipedia and the sources listed below. Text from Wikipedia is available under CC BY-SA 4.0; this entry is adapted from it.

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