Equivariate illustration

In mathematics, an equivariate mapping is a mapping that commutes with the action of a group .

definition

Let it be a group and sets on which left operations of ${\displaystyle G}$ ${\displaystyle X,Y}$ ${\displaystyle G}$

${\displaystyle G\times X\to X,\quad (g,x)\mapsto g\cdot x}$ and ${\displaystyle G\times Y\to Y,\quad (g,y)\mapsto g\cdot y}$

are defined. A function is called - equivariant , - mapping or also equivariant for short if the following applies: ${\displaystyle f\colon X\to Y}$ ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle f(g\cdot x)=g\cdot f(x)}$for everyone . ${\displaystyle g\in G,x\in X}$

That means that for each the diagram ${\displaystyle g\in G}$

commutes .

An equivalent definition is: The group operates on the set of mappings via ${\displaystyle G}$ ${\displaystyle X\to Y}$

${\displaystyle f\mapsto (x\mapsto gf(g^{-1}x))}$.

Then a mapping is equivariant if and only if it remains fixed under this operation. ${\displaystyle f\colon X\to Y}$ ${\displaystyle G}$

ρ equivariance

The term -equivariance is often used in mathematics for a representation ${\displaystyle \rho }$

${\displaystyle \rho \colon G\to \mathrm {GL} (V)}$

or more generally for an effect

${\displaystyle \rho \colon G\to \mathrm {Aut} (Y)}$

used. In this context, mapping a G-set to -equivariant is called if and only if ${\displaystyle f\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \rho }$

${\displaystyle f(g\cdot x)=\rho (g)\cdot f(x)}$applies to all . ${\displaystyle g\in G,x\in X}$

Representation theory and Schur's lemma

Let and be vector spaces over a body and let be the effect of on and linear, i.e. H. there are representations ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle \rho \colon G\rightarrow \mathrm {GL} (X),\tau \colon G\rightarrow \mathrm {GL} (Y)}$

With

${\displaystyle g\cdot x=\rho (g)x,g\cdot y=\tau (g)y}$

for everyone . ${\displaystyle x\in X,y\in Y,g\in G}$

Equivariate maps are then maps with ${\displaystyle f}$

${\displaystyle f(\rho (g)x)=\tau (g)(f(x))}$

for all and . Equivariant mappings are in the context of representation theory also Vertauschungsoperatoren ( English intertwining operator called). ${\displaystyle g}$ ${\displaystyle x}$

Equivariate mappings between irreducible representations are described by Schur's lemma :

• If and are two irreducible representations , then each G-equivariant map is either 0 or an isomorphism. ${\displaystyle \rho \colon G\to \mathrm {GL} (V)}$ ${\displaystyle \tau \colon G\to \mathrm {GL} (W)}$ ${\displaystyle f\colon V\rightarrow W}$

• If a finite-dimensional vector space over an algebraically closed field , e.g. B. the complex numbers , is and and again are irreducible representations, then every G-equivariant mapping is the multiplication by a scalar: there is such that it holds for all : ${\displaystyle V=W}$ ${\displaystyle K}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \rho \colon G\to \mathrm {GL} (V)}$ ${\displaystyle \tau \colon G\to \mathrm {GL} (V)}$ ${\displaystyle f\colon V\rightarrow V}$ ${\displaystyle \lambda \in K}$ ${\displaystyle v\in V}$

${\displaystyle f(v)=\lambda v}$.

Similarly, for Hilbert space representations of topological groups as they are considered in harmonic analysis , i.e. continuous homomorphisms of a topological group in the unitary group on a possibly infinite-dimensional Hilbert space provided with the weak operator topology , every continuous linear (a generalization to closed dense defined is possible) commutation operator (equivariate mapping) between two irreducible representations is a multiple of an isometry . The (continuous) commutation operators between a unitary representation and themselves form a Von Neumann algebra .

Group algebras

Representations of a group on a vector space can be continued linearly to a representation of group algebra , thus becoming a module. If there are two representations that we regard as modules in this sense , then a mapping is -equivariant if and only if it is -linear. ${\displaystyle \rho \colon G\to \mathrm {GL} (V)}$ ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle \rho \colon K(G)\to \mathrm {GL} (V)}$ ${\displaystyle K(G)}$ ${\displaystyle V}$ ${\displaystyle K(G)}$ ${\displaystyle \rho \colon G\to \mathrm {GL} (V),\tau \colon G\to \mathrm {GL} (W)}$ ${\displaystyle K(G)}$ ${\displaystyle f\colon V\to W}$ ${\displaystyle G}$ ${\displaystyle K(G)}$

The same applies to representations of any algebras ( see also here ).

The equivalent mappings between two representations form a vector space. ${\displaystyle G}$

For a fixed group and a solid body , the -representations of and the -equivariant maps form the objects and morphisms of an enriched category over the category of the -vector spaces provided with the usual tensor product . It is ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle K}$

${\displaystyle \mathrm {id} _{V}\colon K\to \mathrm {Hom} (V,V)}$   given by   and ${\displaystyle (\mathrm {id} _{V}(k))(v)=kv}$

${\displaystyle \cdot _{UVW}\colon \mathrm {Hom} (V,W)\otimes \mathrm {Hom} (U,V)\to \mathrm {Hom} (U,W)}$   is given by   . ${\displaystyle \cdot _{UVW}(f\otimes g)=fg}$

topology

A G-space is a topological space X with a continuous action of group G. A G-map is an equivariate continuous map between two G-spaces. ${\displaystyle f:X\rightarrow Y}$

For example, acts upon by turns around the zero point. By ${\displaystyle G=SO(2)}$ ${\displaystyle X=Y=\mathbb {R} ^{2}}$

${\displaystyle f(x,y)=(-x,-y)}$

given reflection is -equivariant. ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}$ ${\displaystyle SO(2)}$

Two G-maps are called G-homotop if there is a G-map ${\displaystyle f_{0},f_{1}\colon X\rightarrow Y}$

${\displaystyle H\colon X\times \left[0,1\right]\rightarrow Y}$

With

${\displaystyle H(x,0)=f_{0}(x),H(x,1)=f_{1}(x)}$

for everyone there. (Here G acts through .) The set of G homotopy classes of G mappings is denoted by. ${\displaystyle x\in X}$ ${\displaystyle X\times \left[0,1\right]}$ ${\displaystyle g\cdot (x,t)=(g\cdot x,t)}$ ${\displaystyle f:X\rightarrow Y}$ ${\displaystyle \left[X,Y\right]_{G}}$

The equivariant homotopy groups of a G-space X are defined by

${\displaystyle \pi _{n}^{G}(X):=\left[S^{n},X\right]_{G}}$.

One has an isomorphism , where the set is the fixed points of the G-effect. ${\displaystyle \pi _{n}^{G}(X,x)\cong \pi _{n}(X^{G})}$ ${\displaystyle X^{G}=\left\{x\in X:g\cdot x=x\forall g\in G\right\}}$

The equivariant homology groups of a G-space X are defined by

${\displaystyle H_{n}^{G}(X;\mathbb {Z} ):=H_{*}(X\times _{G}EG;\mathbb {Z} )}$,

where EG is a weakly contractible topological space with a free G-effect . If the G-effect on X is also free, then is . ${\displaystyle H_{n}^{G}(X;\mathbb {Z} )\cong H_{n}(X/G;\mathbb {Z} )}$

The equivariant K-theory of a compact G-space X is defined as the quotient of the free Abelian group on the isomorphism classes of complex G-vector bundles over X according to the subgroup generated by elements of the form . For example, the complex presentation ring is the group . ${\displaystyle K_{G}(X)}$ ${\displaystyle [E\oplus F]-[E]-[F]}$ ${\displaystyle K_{G}(Punkt)=R(G)}$ ${\displaystyle G}$

Generalizations

More generally, one considers group operations on objects of any category , these are then homomorphisms from a group to the automorphism group of an object. Correspondingly, one also considers semigroup operations (this includes, for example, algebra representations ) as homomorphisms in the endomorphism semigroup of an object. An equivariant mapping is then required to be a morphism between the two objects on which the group acts. Since these are no longer necessarily mappings, one also speaks of ( -) equivariant morphisms in the general case . ${\displaystyle G}$ ${\displaystyle G}$

On the other hand, a group can be viewed as a special monoid and hence a special category with a single object . A functor is then the equivalent of a -link operation on, and natural transformations between such functors correspond to equivariate mappings. ${\displaystyle G}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle *}$ ${\displaystyle F\colon {\mathcal {C}}\to \mathbf {Set} }$ ${\displaystyle G}$ ${\displaystyle F(*)}$

Individual evidence

1. ↑ Graeme Segal: Equivariant K-theory ( Memento from June 22, 2010 in the Internet Archive )