Regular representation

In mathematics , the left-regular and right- regular representation are defined for different types of mathematical structures . These are of particular importance in representation theory , including harmonic analysis and the representation theory of Banach algebras in functional analysis . They can be explicitly and easily constructed from the operations of the structure independently of concrete properties of a structure and thus ensure the existence of rich, non-trivial representations in the general case.

Representation of an algebra

Be an associative algebra over a field . The left-regular representation (or even just the regular representation ) is then a representation on the vector space that acts through multiplication from the left, that is ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mathbf {L} }$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \mathbf {L} \colon {\mathcal {A}}\to L({\mathcal {A}}),(\mathbf {L} (A))B=AB}$.

The right-regular representation works analogously from the right: ${\displaystyle \mathbf {R} }$

${\displaystyle \mathbf {R} \colon {\mathcal {A}}\to R({\mathcal {A}}),(\mathbf {R} (A))B=BA}$

This is an antihomomorphism , so in general it is not a representation of , but a right operation and thus representation of the opposite algebra . The right-regular representation is nothing else than the left-regular representation of and for commutatives agrees with the left-regular representation of . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}^{\operatorname {op} }}$ ${\displaystyle {\mathcal {A}}^{\operatorname {op} }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

If algebra has a neutral element with regard to multiplication, then these two representations are injective . In general, this does not have to be the case; for an algebra with multiplication, for example, these representations are equal to the zero mapping . ${\displaystyle AB=0}$

Summarizing representations of as - modules , so the left regular representation is nothing more than a left module over interpreted. ${\displaystyle A}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

Representation of a Banach algebra

Now be an (associative) Banach algebra . The left- and right-regular representation are then defined just as in the algebraic case, the linear endomorphism assigned to an element of the algebra now even being continuous with respect to the norm of the algebra. Because by ${\displaystyle {\mathcal {A}}}$

${\displaystyle \|\mathbf {L} (A)B\|=\|AB\|\leq \|A\|\|B\|}$

the operator is bounded and its operator norm is maximal . Therefore, the representation itself is continuous and even a contraction if the definition set is equipped with the norm of algebra and the target set with the operator norm. This also applies to the right-regular representation, since it can be understood as the left-regular representation of the opposite Banach algebra. ${\displaystyle \mathbf {L} (A)}$ ${\displaystyle \|A\|}$

If the Banach algebra has an approximation of one , which is the case for every C * -algebra , then the (left) regular representation is injective, because otherwise the core of the representation would contain an element different from , which is thus multiplied by every other element of the Algebra, including the elements of approximate unity, would result. ${\displaystyle 0}$ ${\displaystyle 0}$

Representation of a group and group algebra

Now be a group and any body . The left and right regular representations are now based on the vector space , which is generated free from the set of group elements, i.e. each group element is identified with a vector so that all together form a basis of . The left-regular representation is then defined as ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K(G)}$ ${\displaystyle K(G)}$

${\displaystyle \mathbf {L} \colon G\to GL(K(G)),g\mapsto (h\mapsto gh)}$,

whereby a mapping is only defined for the basic elements and is to be continued to a linear mapping on . This linear mapping is invertible because it has the continuation of on as the inverse. Similarly, the legal representation is defined as ${\displaystyle h\mapsto gh}$ ${\displaystyle K(G)}$ ${\displaystyle h\mapsto g^{-1}h}$ ${\displaystyle K(G)}$

${\displaystyle \mathbf {R} \colon G\to GL(K(G)),g\mapsto (h\mapsto gh^{-1})}$,

which is also a representation. If one understands the elements of as functions with a finite support (this provides an explicit construction of the free object), then the effect of the two representations is through ${\displaystyle K(G)}$ ${\displaystyle G\to K}$

${\displaystyle \forall g,h\in G,f\in K(G)\quad (\mathbf {L} (g)(f))(h)=f(g^{-1}h)\wedge (\mathbf {R} (g)(f))(h)=f(hg)}$

given.

The room can be equipped with a multiplication, which makes it a so-called group algebra . Each representation of a group can be uniquely extended to a representation of the group algebra (with identification of the group elements with basis vectors). In the case of the regular left display, this is through ${\displaystyle K(G)}$

${\displaystyle \mathbf {L} \colon K(G)\to L(K(G)),a\mapsto (b\mapsto ab)}$

given, that is, it is the left-regular representation of group algebra defined above. The right-regular representation of a group can also be understood as a homomorphism on the opposite group , as ${\displaystyle G^{\operatorname {op} }}$

${\displaystyle \mathbf {R} \colon G^{\operatorname {op} }\to GL(K(G)),g\mapsto (h\mapsto hg)}$,

where the multiplication can be read on the right-hand side in . To do this, simply link the group isomorphism with the right-regular representation in the above sense. This representation can then be continued and the right-regular representation in the above sense is obtained as a homomorphism . ${\displaystyle G}$ ${\displaystyle G^{\operatorname {op} }\to G,g\mapsto g^{-1}}$ ${\displaystyle K(G^{\operatorname {op} })=K(G)^{\operatorname {op} }}$ ${\displaystyle K(G)^{\operatorname {op} }\to K(G)}$

Unitary representation of a topological group

In the harmonic analysis one considers unitary representations of locally compact topological groups in Hilbert spaces. Such a group can be equipped with a left hair measure , the space of the square-integrable functions on the group with regard to this measure is then a Hilbert space on which the left-regular representation can be defined as above ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle L^{2}(G,\mu )}$

${\displaystyle \mathbf {L} \colon G\to U(L^{2}(G,\mu )),(\mathbf {L} (x)(f))(y)=f(x^{-1}y)}$.

${\displaystyle \mathbf {L} (x)}$is on a well-defined operator , since due to the invariance of the hair amount up to zero amounts are in turn mapped to the same functions on the same quantities to zero functions. The representation is unitary, that is, it is always a unitary operator , because it is isometric due to the invariance and has an inverse. In addition, it is constant if you equip it with the weak operator topology . For this it suffices to show that the figure ${\displaystyle L^{2}(G,\mu )}$ ${\displaystyle \mathbf {L} (x)}$ ${\displaystyle \mathbf {L} (x^{-1})}$ ${\displaystyle U(L^{2}(G,\mu ))}$

${\displaystyle x\mapsto \langle \mathbf {L} (x)f,f\rangle }$

for continuous functions with compact support is continuous in the neutral element of the group, which follows from the fact that continuous functions with compact support are always uniformly continuous . ${\displaystyle f}$

The rather regular representation is on the room with the to the right associated Hair measure defined (for measurable amounts applies ): ${\displaystyle L^{2}(G,{\tilde {\mu }})}$ ${\displaystyle \mu }$ ${\displaystyle {\tilde {\mu }}}$ ${\displaystyle E\subseteq G}$ ${\displaystyle {\tilde {\mu }}(E)=\mu (E^{-1})}$

${\displaystyle \mathbf {R} \colon G\to U(L^{2}(G,{\tilde {\mu }})),(\mathbf {R} (x)(f))(y)=f(yx)}$

Equivalently, the representation using the modular function can be expressed as ${\displaystyle \Delta }$

${\displaystyle \mathbf {R} \colon G\to U(L^{2}(G,\mu )),(\mathbf {R} (x)(f))(y)=\Delta (x)^{\frac {1}{2}}f(yx)}$

define. For unimodular groups the modular function is the same and and the right regular representation is simple so in this case ${\displaystyle 1}$ ${\displaystyle \mu ={\tilde {\mu }}}$

${\displaystyle \mathbf {R} \colon G\to U(L^{2}(G,\mu )),(\mathbf {R} (x)(f))(y)=f(yx)}$.

In any case, the left-regular and right-regular representations are unitarily equivalent by means of the unitary commutation operator . The left and right regular representation are injective. ${\displaystyle L^{2}(G,\mu )\to L^{2}(G,{\tilde {\mu }}),f\mapsto (x\mapsto f(x^{-1}))}$

Physical example

The regular representation of the (usually or ) allows in simple cases of quantum mechanics (space for a particle, without spin ) and also classical field theories the description of the symmetry of the space under translations : Quantum mechanical states or also classical fields can be understood as square-integrable functions, displacements of space act on it as unitary operators. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n=3}$ ${\displaystyle n=4}$

Group algebra L ¹

For each locally compact topological group with a left hair measure , the group algebra is defined with the convolution as the product. This forms a Banach algebra with approximate unity and with a suitable involution even a Banach - * algebra. According to Young's convolution inequality , both and are square-integrable, from which it also follows that the mapping is bounded with respect to the -norm. The Hilbert space representation can thus be used ${\displaystyle \mu }$ ${\displaystyle L^{1}(G,\mu )}$ ${\displaystyle *}$ ${\displaystyle f\in L^{1}(G,\mu ),g\in L^{2}(G,\mu )}$ ${\displaystyle f*g}$ ${\displaystyle g*f}$ ${\displaystyle g\mapsto f*g}$ ${\displaystyle L^{2}}$

${\displaystyle \mathbf {L} \colon L^{1}(G,\mu )\to {\mathcal {B}}(L^{2}(G,\mu )),f\mapsto (g\mapsto f*g)}$

define, called left-regular representation. According to Young's convolution inequality and like every * homomorphism of a Banach - * algebra into a C * algebra, this is a contraction. For compact is a (two-sided) ideal of group algebra and it is simply a matter of restricting the left-regular representation of a Banach algebra in the above sense to changing the norm. ${\displaystyle G}$ ${\displaystyle L^{2}(G,\mu )}$ ${\displaystyle L^{1}(G,\mu )}$ ${\displaystyle L^{2}(G,\mu )}$

On the other hand, the left-regular (unitary) representation of the group can be "continued" into a representation of group algebra in such a way that for in the weak sense ${\displaystyle G}$ ${\displaystyle \mathbf {L} }$ ${\displaystyle f\in L^{1}(G,\mu )}$

${\displaystyle \mathbf {L} (f)=\int _{G}f(x)\mathbf {L} (x)\mathrm {d} \mu (x)}$

holds, d. H. for is ${\displaystyle u,v\in L^{2}(G,\mu )}$

${\displaystyle \langle \mathbf {L} (f)u,v\rangle =\int _{G}f(x)\langle \mathbf {L} (x)u,v\rangle \mathrm {d} \mu (x)}$.

This “continuation” is precisely the above representation , which justifies the identical designation. If the group is finite and its topology discrete , then this corresponds to the continuation of the left-regular representation of a group on the algebraic group algebra . If the group is unimodular, the general “continuation” of the legal representation can also be identified as in the algebraic case. In any case, this “continuation” is unitarily equivalent to the left-regular representation of group algebra . ${\displaystyle f\mapsto (g\mapsto f*g)}$ ${\displaystyle G}$ ${\displaystyle K(G)}$ ${\displaystyle f\mapsto (g\mapsto g*f)}$ ${\displaystyle L^{1}(G,\mu )}$

Two-sided regular representation

For a locally compact topological group, one also defines the two-sided regular representation , which can be understood as a two-sided group operation , on space by concatenating the left and right regular representation: ${\displaystyle L^{2}(G,\mu )}$

${\displaystyle (\mathbf {L} \times \mathbf {R} )\colon G\times G\to U(L^{2}(G,\mu )),(g,h)\mapsto \mathbf {L} (g)\mathbf {R} (h)=\mathbf {R} (h)\mathbf {L} (g)}$

Partial representations and decompositions

The question of generalizations of the Fourier transform and Plancherel's theorem in harmonic analysis is closely related to the decomposition of the left, right and two-sided regular representation into irreducible representations .

The irreducible partial representations of the left-regular representation form the so-called discrete series . An irreducible representation belongs to the discrete series if and only if it has a nontrivial square-integrable matrix coefficient , that is, there are vectors from the representation space of , so that the function ${\displaystyle \pi }$ ${\displaystyle u,v\neq 0}$ ${\displaystyle \pi }$

${\displaystyle x\mapsto \langle \pi (x)u,v\rangle }$

is square integrable with respect to a left hair measure. In the unimodular case, all matrix coefficients are then already square-integrable, generally at least for from a dense sub-vector space and arbitrary. If the square-integrable matrix coefficients of the elements of the discrete series span a dense subspace of , the two-sided representation can be expressed as a direct sum ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle L^{2}(G,\mu )}$

${\displaystyle \bigoplus _{\pi {\text{ irr. Teildarstellung von }}\mathbf {L} }\pi \otimes {\bar {\pi }}}$

decompose, whereby with regard to unitary equivalence only one representative of such irreducible partial representations is chosen in the sum and denotes the outer tensor product and the contra-redeemed representation . Particularly for compact groups, every irreducible representation is an element of the discrete series and its multiplicity with which it is contained in the left-regular representation is equal to the finite dimension of its representation space , see Peter-Weyl's theorem for this special case . ${\displaystyle \otimes }$ ${\displaystyle {\bar {\pi }}}$

In contrast, the left-regular representation of non-compact Abelian groups has no irreducible partial representations. Here and in other cases, however, the two-sided representation can be broken down into irreducibles as a direct integral . For every unimodular type 1 group that satisfies the second countability axiom , the two-sided regular representation is unitarily equivalent to the direct integral

${\displaystyle \int _{\hat {G}}^{\oplus }\pi \otimes {\bar {\pi }}\mathrm {d} {\tilde {\mu }}(\pi )}$,

where a representative system of the irreducible representations of and denotes the Plancherel measure. is in the discrete series if and only if , hence the name. The unitary commutation operators between those direct sums or integrals are precisely the generalized Fourier transforms and their inversions, which can be understood as the diagonalization of the two-sided representation. Under the given conditions, left and right regular representations can also be decomposed as direct integrals: ${\displaystyle {\hat {G}}}$ ${\displaystyle G}$ ${\displaystyle {\tilde {\mu }}}$ ${\displaystyle \pi }$ ${\displaystyle \mu (\pi )>0}$

${\displaystyle \mathbf {L} \cong \int _{\pi \in {\hat {G}}}^{\oplus }I\otimes {\bar {\pi }}\mathrm {d} {\tilde {\mu }}(\pi )}$

${\displaystyle \mathbf {R} \cong \int _{\pi \in {\hat {G}}}^{\oplus }\pi \otimes I\mathrm {d} {\tilde {\mu }}(\pi )}$

Here denote the trivial representation . ${\displaystyle I}$

For example, the regular representation of the results as a direct integral over all irreducible representations, which precisely correspond to the characters , with respect to the Lebesgue measure with a scaling factor. ${\displaystyle \mathbb {R} ^{n}}$

See also

• Indirect group
• A representation as a regular permutation group is in the theory of finite groups a representation of the group as a group of permutations of a finite set M , with the additional requirement that apart from the one element, no group element may have a fixed element in M in the operation under consideration ( regular ). For details, see permutation group # Left and Right Regular Representation .

Web links

• AI Shtern: Regular representation . Entry in the online version of the Encyclopaedia of Mathematics .

Individual evidence

1. ↑ Steven H. Weintraub: Representation Theory of Finite Groups: Algebra and Arithmetic . AMS , 2003, ISBN 0-8218-3222-0 , pp. 5 .
2. ^ William Arveson : A Short Course on Spectral Theory . Springer , New York 2002, ISBN 0-387-95300-0 , pp. 13 , doi : 10.1007 / b97227 .
3. ↑ Gerald Budge Folland: A Course in Abstract Harmonic Analysis . CRC Press , 1995, ISBN 0-8493-8490-7 , pp. 68-69 .
4. ↑ Folland, p. 73.
5. ^ Paul Garrett: Some facts about discrete series (holomorphic, quaternionic). (PDF; 86 kB) December 18, 2004, accessed October 7, 2012 . 
6. ↑ Folland, p. 234.
7. ↑ Jacques Dixmier : C * -Algebras . North-Holland, 1977, ISBN 0-7204-0762-1 , pp. 368 .
8. ^ Mitsuo Sugiura: Unit Representations and Harmonic Analysis . 2nd Edition. North-Holland, 1990, ISBN 0-444-88593-5 , pp. 122 .