Representation theory

In the representation theory of groups or elements of general algebras by means of homomorphisms on linear transformations of vector spaces (matrices ) are mapped. The representation theory has applications in almost all areas of mathematics and theoretical physics. A presentation of theoretical set of Langlands was a major step for Wiles ' proof of the Great set of Fermat and representation theory provided the theoretical background for the prediction that quarks existieren.Auch for the purely algebraic study of groups or algebras the representation by matrices is often useful.

Types of representations

Classically representation theory employed with homomorphisms for groups and vector spaces (the general linear group), see

• Presentation ( group).

More generally, the representation theory of rings and algebras is considered, which contains the representation theory of groups as a special case ( since each representation of a group induces a representation of its group ring ), see this

• Representation ( algebra).

In physics, in addition to discrete groups of solid state physics especially representations of Lie groups of importance, such as the rotation group and the groups of the Standard Model. Here one additionally requires that representations should be smooth homomorphisms, see

• Representation ( Lie group ).

The Lie's sets convey a correspondence between representations of Lie groups and the induced representations of their Lie algebras. For the representation theory of Lie algebras see

• Representation ( Lie algebra ).

Lie algebras are not associative, which is why their representation theory is not a special case of the representation theory of associative algebras. But you can assign any Lie algebra to its universal enveloping algebra, which is an associative algebra.

Basic concepts

Below is a group Lie group or algebra and a representation of, ie a group, Lie groups and algebras homomorphism into the algebra of linear transformations of a vector space (the image in the case of groups or Lie groups - natural isomorphisms even is ).

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.

Partial representations

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

Is evidently

Again a representation of which are called for by the restriction and is designated by.

Is complementary to the subspace, which is also invariant, as is true, wherein the equivalence is mediated by the isomorphism.

Direct sums

Are and two representations, as defined

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 be arbitrarily for direct sums of many summands generalize. Is a family of representations including

Irreducibility, complete reducibility, Ausreduzierung

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 " product" (better: tensor product ) of two ( irreducible ) representations is ia reducible and can " ausreduziert " be according to components of the irreducible representations, with specific coefficients such as the Clebsch -Gordan coefficients of the angular momentum physics arise. This is a particularly important aspect for the applications in physics.

History

In the 17th and 18th Representation Theory and Harmonic Analysis of abelian groups as, for example, or in connection with Euler products and Fourier transformations occurred ( in multiplicative characters in the form of separation of functions). This one does not work with the illustrations, but with their multiplicative characters. Frobenius defined in 1896 first ( without explicitly take on representations regarding ) a term multiplicative characters for non-Abelian groups, Burnside and Schur developed its definitions then again on the basis of matrix representations and Emmy Noether finally gave essentially the modern definition by means of linear maps a vector space, which later needed in quantum mechanics study of infinite-dimensional representations enabled.

In 1900, the representation theory of the symmetric and alternating groups of Frobenius and Young was drafted. 1913 proved the law of Cartan highest weight, which classifies the irreducible representations of complex semisimple Lie algebras. Schur observed in 1924 that one can extend the representation theory of finite groups to compact groups by invariant integration, the representation theory of compact Lie groups related was then developed by Weyl. The Proven hair and von Neumann existence and uniqueness of the hair - measure then allowed the early 30s, the extension of this theory to compact topological groups. Further developments will concern the application of the representation theory of locally compact groups such as the Heisenberg group in quantum mechanics, the theory of locally compact abelian groups with applications in algebraic number theory ( Harmonic Analysis on Adelen ) and later the Langlands program.