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representation theory

Context

Representation theory

Mathematics

Contents

Idea

Representation theory is concerned with the study of algebraic structures via their representations, This concerns notably groups, directly or in their incarnation as group algebras, Hopf algebras or Lie algebras, and usually concerns linear representations, hence modules of these structures. But more generally representation theory also studies representations/modules/actions of generalizations of such structures, such as coalgebras via their comodules etc.

See also at geometric representation theory.

In homotopy type theory

The fundamental of representation theory have a particular natural formulation in homotopy theory and in fact in homotopy type theory, which also refines it from the study of representations of groups to that of
∞-representations of ∞-groups. This includes both discrete ∞-groups as well as geometric homotopy types such as smooth ∞-groups, the higher analog of Lie groups.

The key observation to this translation is that

  1. an ∞-group GG is equivalently given by its delooping BG\mathbf{B}G regarded with its canonical point (see at looping and delooping), hence the universal GG-principal ∞-bundle

    G * BG \array{ G &\longrightarrow& \ast \\ && \downarrow \\ && \mathbf{B}G }
  2. an ∞-action ρ\rho of GG on any geometric homotopy type VV is equivalently given by a homotopy fiber sequence of the form

    V V// ρG BG, \array{ V &\stackrel{}{\longrightarrow}& V//_\rho G \\ && \downarrow \\ && \mathbf{B}G } \,,

    hence by a VV-fiber ∞-bundle over BG\mathbf{B}G which is the ρ\rho-associated ∞-bundle to the universal GG-principal ∞-bundle (see at ∞-action for more on this).

Under this identification, the representation theory of GG is equivalently

More in detail, this yields the following identifications:

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}H
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

References

A list of texts on representation theory is maintained at

The relation to number theory and the Langlands program is discussed in

Revised on February 16, 2014 05:31:00 by Urs Schreiber (89.204.153.238)