Algebra representation

The representation theory is a branch of mathematics that deals with the representation of algebras on vector spaces. In this way, any associative algebras are associated by means of homomorphisms with algebras of operators. The research focuses on the structure of such homomorphisms and their classification. The representation theory of an algebra is the theory of its modules are equivalent. More specific representation theories treat groups, Lie algebras or C * - algebras.

We consider in the following, for simplicity algebras with unit element 1 If you have an algebra without identity, then you adjungiere one.

Definitions

It had a body and a - algebra. A representation of an algebra homomorphism is, one - vector space and the algebra of all linear operators on, more precisely it is called a representation of on.

The vector space dimension is called the dimension of. Finite-dimensional representations are also called matrix representations, because by choosing a vector space basis can write each element of a matrix. Injective representations are called faithful.

Two representations and are called equivalent if there is a vector space isomorphism with for all. But one also writes for short.

The so defined equivalence is an equivalence relation on the class of all representations. The formation of concepts in representation theory are designed so that they are preserved to an equivalent representation of the transition, dimension and loyalty are the first examples.

Examples

• The Nullhomomorphismus which maps each Algebrenelement to the zero operator, is called a null representation or trivial representation.
• The identity map is a faithful representation of on.
• It is the algebra of rellwertigen ( continuous ) functions. Then

• If an algebra, so, where is defined by a representation of. This special presentation is also called the left regular representation, since it maps to the set of all left multiplications by elements of. The formula shows the fidelity of the left regular representation, in particular, each algebra a faithful representation.

The multiplicativity of the left regular representation means for all and that is for all and that is nothing more than for all. This consideration makes the role of the associative law clearly.

Direct sums

Are and two representations, as defined

Obviously again a representation of where componentwise operates on the direct sum, ie for all. This representation is called the direct sum of and and is denoted by.

This construction can obviously be generalized for direct sums of any number of summands. Is a family of representations including

Partial representations

Be a presentation. A subspace is called invariant (more precisely - invariant ) if for all.

Is evidently

Again a representation of which is called the constraint of up and labeled.

If a subspace complementary to, which is also invariant, then clearly, the equivalence is mediated by the isomorphism.

The invariant subspaces of the left regular representation of an algebra are precisely the left ideals of the algebra.

Further illustrations

An important object of study of the representation theory is decomposition of representations as a sum of partial representations. Of course, we are interested in representations which can not be further decomposed. This leads casually on the following word:

Irreducible representations

A display means irreducible if there are, and no further invariant subspaces of. For an equivalent characterization, see Lemma of Schur. A representation is called completely reducible if it is a direct sum of irreducible representations are equivalent.

The above example of a two-dimensional representation of is obviously equivalent to the direct sum of two one-dimensional and hence irreducible representations. The identical representation of the matrix algebra on a -dimensional irreducible representation, from which one can show that it is the only up to equivalence. A common goal of the representation theory is the classification of all equivalence classes of irreducible representations of a given algebra.

Non- degenerate representations

A representation of an algebra on the vector space is called non- degenerate if for always follows for all.

If any representation, then

Obviously invariant subspaces, is also called the null space of the representation. It is based on the projection and the corresponding complement. Since the zero display and non- degenerate is, we have the result that each display is the sum of a non- zero and degenierten representation. Often one considers therefore only non- degenerate representations and assumes no restriction on.

Cyclic representations

Is an illustration of cyclic if there is with a, the vector is a cyclic vector. If any image and so apparently is an invariant subspace and is a cyclic representation with a cyclic vector. One often calls for even that is not in the null space in order to avoid trivial.

Connection with moduli

Is a non- degenerate representation, as is the definition of a module. The non- degeneracy is needed for all the other module axioms are easily leads to the back of Homomorphieeigenschaften.

Vice versa, a module, it is a vector space by the scalar multiplication explained. If we define a linear operator by the formula, the result is obviously a representation.

In this design, two representations are equivalent if and only if the associated moduli are isomorphic. The representation theory of the algebra is therefore equivalent to the theory of moduli. The partial representations correspond to submodules, an irreducible representation corresponds to a simple module, a completely reducible representation of a semi- simple module. Cyclic representations correspond to generated by an element modules. The left regular representation to the corresponding module is nothing more than itself

If you only have a ring without the operation of a body, so you can only talk about -modules. The theory of modules over a ring is in this sense a generalization of the representation theory of algebras on rings.

Group representations

If a group, the group algebra is an algebra that contains the group of invertible elements of a subgroup isomorphic to the one identified with. Therefore, each non- degenerate representation of the group algebra by restricting supplies to a representation of the group. Is, conversely, a group display, as is provided by a display of the group ring. In this sense, the representation theory of groups of treated here representation theory of algebras filed under.

Representations of Lie algebras

Although Lie algebras are not associative, but one is to homomorphisms on subalgebras of interest, where the Lie bracket is mapped to the commutator, ie, where for all. An associated universal enveloping algebra construction leads to the universal, so that the representations of Lie algebras are placed in relation to the issues here representations of associative algebras.

Hilbert space representations

For the investigation of Banach *-algebras, in particular of C * - algebras and group algebras of locally compact groups, one looks for representations that reflect the topological relationships and the involution. This leads to the informal investigation of representations on Hilbert spaces, which again in turn leads to classes of such algebras, such as the important concept of type IC *-algebra, which can be defined by the representation theory of C * - algebra. The fact that C *-algebras have faithful Hilbert space representation is known as the Gelfand - Neumark.