Regular representation

In mathematics, one defines for various mathematical structures, the left- regular and right- regular representation. These are of particular importance in representation theory, including harmonic analysis and representation theory of Banach algebras in functional analysis. They can be independent of specific properties of a structure explicitly construct a simple manner from the operations of the structure, thus ensuring 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 regular representation ) is then a representation on the vector space, which acts by multiplication from the left, ie

The right regular representation acts by analogy, from the right:

It is a Antihomomorphismus, generally it is not therefore display of, but an operation right, and thus illustration of the opposing algebra. The right regular representation is nothing more than the left regular representation of commutative and vote for the left regular of the same.

Does the algebra a neutral element with respect to multiplication, these two representations are injective. In general this need not be the case, for a ring with the multiplication as these representations are equal to the zero mapping.

Summing up representations of the moduli, so that the left regular representation is simply construed as a left module.

Representation of a Banach algebra

Let now a (associative ) Banach algebra. The left and the right regular representation are then defined as in the algebraic case, the respect an element of the algebra associated linear endomorphism now even steadily the standard algebra. Because by

Is the operator limited and its operator norm is maximum. Therefore, the depiction itself constantly and even a contraction when equipping the definition of itself with the norm of the algebra and the target amount with the operator norm. The same is true for the right regular representation, since it can be regarded as the opposite regular left Banach algebra.

Does the Banach algebra an approximation of the one, which is the case for every C * - algebra, then the (left) regular representation is injective, because otherwise the core of the presentation contained an element different from that multiplies thus with any other element of the Algebra, including the elements of the approximate one, would.

Representation of a group and the group algebra

Now let a group and an arbitrary body. The left and the right regular representation are now on the vector space, which is freely generated by the set of group elements, that is, each group element is identified with a vector, so that all together form a basis of. The left regular representation is then defined as

Being defined by a mapping only to the basic elements and should be continued to a linear map on. This linear map is invertible, since it has the sequel to as inverse. The right regular representation is defined as analog

Which is also a representation. Summing up the elements of a finite carrier in functions ( this provides an explicit construction of the free object ), then the action of the two representations is carried

Given.

The room can be equipped with a multiplication, whereby it is the so-called group algebra. Each representation of a group can be uniquely continued to a representation of the group algebra (with identification of the group elements with basis vectors ). In the case of the left regular representation, this is by

Given, that is, is the above-defined left regular representation of the group algebra. The right regular representation of a group can also be regarded as a homomorphism on the opposite group, as

Where the multiplication on the right side is to read. To do this you simply concatenated the group isomorphism with the right regular representation in the above sense. This representation can then continue on and the right regular representation in the above sense is obtained as a homomorphism.

A unitary representation of a topological group

In the harmonic analysis we consider unitary representations of locally compact topological groups in Hilbert spaces. Such a group can be equipped with a left Hair measure, the space of square integrable functions on the group with respect to this measure is then a Hilbert space, on which the left regular representation can be defined as above as

Is a well defined operator, since due to the invariance of the measure hair same functions are mapped to turn to levels equal to zero functions to zero amounts. The display is unitary, that is, is always a unitary operator, since it is due to the invariance isometric, and has as the inverse. Moreover, it is continuous when equipping with the weak operator topology. For this, it suffices to show that the mapping

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.

The right regular representation is defined on the space with the corresponding right to Hair measure ( for measurable quantities applies ):

Equivalently, can the representation using the modular function as

Define. For unimodular groups, the modular function is the same and and the right regular representation is thus simple in this case

In any case, the left- regular and right- regular representation unitarily equivalent via the unitary Vertauschungsoperators. The left and the right regular representation are injective.

Physical example

The regular representation of (usually or ) allows for simple cases of quantum mechanics (room for a particle without spin) and also classical field theories describing the symmetry of the space under translations: Quantum mechanical states or even classical fields can be interpreted as a square-integrable functions, shifts of the space act on these as unitary operators.

Group algebra

For any locally compact topological group with a left Hair measure you define the group algebra with the convolution as a product. This forms a Banach algebra with approximate one and with a suitable involution even a Banach * - algebra. After Faltungsungleichung Young are both square integrable, from her also follows that the mapping with respect to the standard is limited. It can thus be the Hilbert space representation

Define, called the left regular representation. This is a contraction after the Faltungsungleichung by Young and like any * - homomorphism of a Banach *-algebra into a C * - algebra. For compact is a ( two-sided ) ideal of the group algebra, and it is simply on the restriction of the left regular representation of a Banach algebra in the above sense under change of the standard.

On the other hand, can be the left regular ( unitary ) representation of the group to a representation of the group algebra as "continue", for that in the weak sense

Applies, that is, for being

This " sequel " is the above representation of what the identical name justifies. If the group is finite, and its topology is discrete, so this just above continuation of the left regular representation corresponds to a group on the algebraic group algebra. If the group is unimodular, can also be the general " continuation" of the right regular representation as to identify the algebraic case. In any case, this "sequel " is unitarily equivalent to the left regular representation of the group algebra.

Two-sided regular representation

For a locally compact topological group is also defined the two-sided regular representation, which can be interpreted as two-sided group operation on the space by concatenating the left and the right regular representation:

Partial representations and decompositions

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

The irreducible representations of the part of the left regular representation form the so-called discrete series. An irreducible representation if and only belongs to the discrete series if it has a nontrivial square integrable matrix coefficients, that is, there exist vectors from the representation space of, so that the function

Square integrable with respect to a measure is left hair. In the unimodular case, all matrix coefficients are square integrable then already, generally at least from a dense subspace and random. Tighten the square integrable matrix coefficients of the elements of the discrete series a dense subspace of, so can the two-sided representation as a direct sum

Disassemble, with respect to unitary equivalence will each elected only one representative of such irreducible sub-representation in the sum and denote the outer tensor product and the contragredient representation. In particular, for compact groups is any irreducible representation element of the discrete series and their multiplicity, with which it is contained in the left regular representation, is equal to the finite dimension of its representation space, see for the particular case of Peter -Weyl theorem.

In contrast, the left regular representation of non-compact abelian groups has no irreducible sub-representation. Here and in other cases it is possible a decomposition of the two-sided representation into irreducibles as a direct integral. For each of the second axiom of countability fulfilling, unimodular type - 1 group, the two-sided regular representation is unitarily equivalent to the direct integral

Where a deputy system of the irreducible representations of and the Plancherel measure, respectively. if and only lies in the discrete series if, hence the name. The unitary Vertauschungsoperatoren between those direct sums and integrals are just the generalized Fourier transforms and their inversions, which can be understood as diagonalization of the two-sided representation. Under those conditions then can also be left -and right- regular representation as a direct integrals decompose:

It denotes the trivial representation.

For example, there is the regular representation of as a direct integral over all irreducible representations, which just correspond to the characters, with respect to the Lebesgue measure with a scale factor.