Representations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 April 2010

Representations

A representation of a group $G$, or $G$-module, is an action of $G$ ona vector space $V$ by automorphisms (invertile linear transformations). A representation of an algebra $A$, or $A$-module, is an action of $A$ on a vector space $V$ by endomorphisms (linear transformations). A morphism $T:V_1\to V_2$ of $A$-modules is a linear transformation such that $T\left(av\right)=aT\left(v\right),$ for all $a\in A$ and $v\in V.$ An $A$-module $M$ is simple, or irreducible, if it has no submodules except $0$ and itself.

A representation of a topological group $G,$ or $G$-module, is an action of $G$ on a topological vector space $V$ by automorphisms (continuous invertible linear transformations) such that the map $$\begin{array}{rcl}G\times V& \to & V\\ \left(g,v\right)& \mapsto & gv\end{array}$$ is continuous. When dealing with representations of topological groups all submodules are assumed to be closed subspaces.

A $*$-representation of a $*$-algebra $A$ is an action of $A$ on a Hilbert space $H$ by bounded operators such that $$⟨a{v}_1{v}_2⟩=⟨v_1,a*v_2⟩,\text{for all}{v}_1,v_2\in V,a\in A.$$ A $*$-representation of $A$ on $H$ is nondegenerate is $AV=\left\{av|a\in A,v\in V\right\}$ is dense in $V.$

A unitary representation of a topological group $G$, or $G$-module, is an action of $G$ on a Hilbert space $V$ by automorphisms (unitary continuous invertible linear transformations) such that the action $G\times V\to V$ is a continuous map.

An admissable representation of an idempotent algebra $\left(A,ℰ\right)$ is an action of $A$ on a vector space $V$ by linear transformations such that

1. $V=\underset{e\in ℰ}{\cup }eV,$
2. each $eV$ is finite dimensional.

A representation of an idempotented algebra is smooth if it satisfies (a).

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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