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![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![X, Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd)
-
and
are defined. A function is called - equivariant , - mapping or also equivariant for short if the following applies:
![f \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
for everyone .![g \ in G, x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/0207e50819089c645e02d1567e27ebf6c8efc1a5)
That means that for each the diagram
![g \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be73903416a0dd94b8cbc2268ce480810c0e62)
![Equivariant commutative diagram.svg](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Equivariant_commutative_diagram.svg/175px-Equivariant_commutative_diagram.svg.png)
commutes .
An equivalent definition is: The group operates on the set of mappings via
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/290b16963d52e4a7995aae01ee854b97a6ea10c0)
-
.
Then a mapping is equivariant if and only if it remains fixed under this operation.
![f \ colon X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9ff205beb51e7899846aeae788ae5e5546a3e)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
ρ equivariance
The term -equivariance is often used in mathematics for a representation
![{\ displaystyle \ rho \ colon G \ to \ mathrm {GL} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437271686d8f5740e2beb0171a83263b39308ff9)
or more generally for an effect
![{\ displaystyle \ rho \ colon G \ to \ mathrm {Aut} (Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b41f9ee56beb0062d3fbcc0c92c1274c37c42b)
used. In this context, mapping a G-set to -equivariant is called if and only if
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
-
applies to all .![g \ in G, x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/0207e50819089c645e02d1567e27ebf6c8efc1a5)
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![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle \ rho \ colon G \ rightarrow \ mathrm {GL} (X), \ tau \ colon G \ rightarrow \ mathrm {GL} (Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d94f9865aedf2f88e48a528daee1c9102a9410d)
With
![g \ cdot x = \ rho (g) x, g \ cdot y = \ tau (g) y](https://wikimedia.org/api/rest_v1/media/math/render/svg/9829e40007a39828d05be9559758fa35188d8a0f)
for everyone .
![x \ in X, y \ in Y, g \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/e457d2eec5bfd8e575beaf780d6f799af344e6a3)
Equivariate maps are then maps with
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![f (\ rho (g) x) = \ tau (g) (f (x))](https://wikimedia.org/api/rest_v1/media/math/render/svg/053b3e4a05dd912af0588a6d95de3b0bbf1a62a1)
for all and . Equivariant mappings are in the context of representation theory also Vertauschungsoperatoren ( English intertwining operator called).
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437271686d8f5740e2beb0171a83263b39308ff9)
![{\ displaystyle \ tau \ colon G \ to \ mathrm {GL} (W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e899ad6204a59666037826fbf9ff01c4fb82cd9)
![{\ displaystyle f \ colon V \ rightarrow W}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f3dcd027d3632531a13b19c25863347a65f995)
- 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 :
![V = W](https://wikimedia.org/api/rest_v1/media/math/render/svg/740038d36bd79466d6938d73b83fe737161fa1c6)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![{\ displaystyle \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
![{\ displaystyle \ rho \ colon G \ to \ mathrm {GL} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437271686d8f5740e2beb0171a83263b39308ff9)
![{\ displaystyle \ tau \ colon G \ to \ mathrm {GL} (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b29ab8ed1701e665826738269e0e4ff5884cf2)
![f \ colon V \ rightarrow V](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1034b01207b465b441224cec032e21154ad19c)
![\ lambda \ in K](https://wikimedia.org/api/rest_v1/media/math/render/svg/22d95fcd9b07168a8162820f7fab4d8ee43366e8)
![v \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb)
-
.
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437271686d8f5740e2beb0171a83263b39308ff9)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![K (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/479ebbb1a6f341af575193d9d58cfb52dff981e0)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![K (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/479ebbb1a6f341af575193d9d58cfb52dff981e0)
![{\ displaystyle \ rho \ colon G \ to \ mathrm {GL} (V), \ tau \ colon G \ to \ mathrm {GL} (W)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8904cdcd5fcd669b6ef3d02f2225ff3235bcb34)
![K (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/479ebbb1a6f341af575193d9d58cfb52dff981e0)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![K (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/479ebbb1a6f341af575193d9d58cfb52dff981e0)
The same applies to representations of any algebras ( see also here ).
The equivalent mappings between two representations form a vector space.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
-
given by and![{\ displaystyle (\ mathrm {id} _ {V} (k)) (v) = kv}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b630f8356d7830f8f6dbdb99ba8e9618cd576f0e)
-
is given by .![\ cdot _ {{UVW}} (f \ otimes g) = fg](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3750ba4937fca4f445fd6a8fd8280c484dad835)
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.
![f: X \ rightarrow Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b215af1e965d0595a97ad2b21f7d0cbcf6281303)
For example, acts upon by turns around the zero point. By
![G = SO (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8204aedb25126ba4281303f27540413e20590869)
![X = Y = {\ mathbb R} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73aed6ffd1415c94f7399feb08f041897780d43f)
![f (x, y) = (- x, -y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0881f76b6074592883e9b3c019b3ef769f4ea4)
given reflection is -equivariant.
![f: {\ mathbb R} ^ {2} \ to {\ mathbb R} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25c44be73c93794e647a2a66e3fc18f527395064)
![SO (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ed04ecd7bcb79dff884a9fe6594b8b431c08e5)
Two G-maps are called G-homotop if there is a G-map
![{\ displaystyle f_ {0}, f_ {1} \ colon X \ rightarrow Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08fcc20a311392cb3575fe789cbc618c13e81af4)
![{\ displaystyle H \ colon X \ times \ left [0,1 \ right] \ rightarrow Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26567553f5020f46977c598c36b6035432a31a0d)
With
![H (x, 0) = f_ {0} (x), H (x, 1) = f_ {1} (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc895e952a04769ea27a828f88b3d17b8a9a72e)
for everyone there. (Here G acts through .) The set of G homotopy classes of G mappings is denoted by.
![x \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d)
![X \ times \ left [0,1 \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/57292d99ea0f9458ef890fb409a5af9f5da33115)
![g \ cdot (x, t) = (g \ cdot x, t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/60384b4141bd9784c055939d9747aab9cb9c33ae)
![f: X \ rightarrow Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b215af1e965d0595a97ad2b21f7d0cbcf6281303)
![\ left [X, Y \ right] _ {G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abdb4877c0c987cb5d219123d41be943263291f4)
The equivariant homotopy groups of a G-space X are defined by
-
.
One has an isomorphism , where the set is the fixed points of the G-effect.
![\ pi _ {n} ^ {G} (X, x) \ cong \ pi _ {n} (X ^ {G})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c27a30142d9fcb4941d8ca2f7da9dba4582f456)
![X ^ {G} = \ left \ {x \ in X: g \ cdot x = x \ forall g \ in G \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d26cfb6d7a4d7252c312a3e2d8276c2dda26c17)
The equivariant homology groups of a G-space X are defined by
-
,
where EG is a weakly contractible topological space with a free G-effect . If the G-effect on X is also free, then is .
![H_ {n} ^ {G} (X; \ mathbb {Z}) \ cong H_ {n} (X / G; \ mathbb {Z})](https://wikimedia.org/api/rest_v1/media/math/render/svg/31af455c9784279a703f5894973fc9d236682543)
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 .
![K_ {G} (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/91d5cb733a37c8354402f034b0221fe217a80a6d)
![[E \ oplus F] - [E] - [F]](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2aa7dfc44eebd02c7fa83eb99c491b9a36f0de)
![K_ {G} (point) = R (G)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ffdcd594ef30b65a278ab082020012bbdce21c)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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 .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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.
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ mathcal {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3edab7022ca9e2976651bc59c489513ee9019)
![F \ colon {\ mathcal C} \ to {\ mathbf {Set}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9640021867ac3dd42497b533654adeb142fa5a61)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![F (*)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e068b77dace0ba49d67813f1fc72584e919c4edb)
Individual evidence
-
↑ Graeme Segal: Equivariant K-theory ( Memento from June 22, 2010 in the Internet Archive )