Teorija reprezentacije

Teorija reprezentacije proučava kako algebarske strukture „deluju” na objekte. Najjednostavniji primeri su kako simetrije pravilnih poligona, koje se sastoje od refleksija i rotacija, transformišu poligon.

Teorija reprezentacije je grana matematike koja proučava apstraktne algebarske strukture predstavljajući njihove elemente kao linearne transformacije vektorskih prostora, i proučava module za ove apstraktne algebarske strukture. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.

Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.

Representation theory is pervasive across fields of mathematics for two reasons. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory:

• illuminates and generalizes Fourier analysis via harmonic analysis,
• is connected to geometry via invariant theory and the Erlangen program,
• has an impact in number theory via automorphic forms and the Langlands program.

Second, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.

The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.

Definicije i koncepti

Let V be a vector space over a field F. For instance, suppose V is Rⁿ or Cⁿ, the standard n-dimensional space of column vectors over the real or complex numbers, respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers.

There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.

• The set of all invertible n × n matrices is a group under matrix multiplication, and the representation theory of groups analyzes a group by describing ("representing") its elements in terms of invertible matrices.
• Matrix addition and multiplication make the set of all n × n matrices into an associative algebra, and hence there is a corresponding representation theory of associative algebras.
• If we replace matrix multiplication MN by the matrix commutator MN − NM, then the n × n matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.

This generalizes to any field F and any vector space V over F, with linear maps replacing matrices and composition replacing matrix multiplication: there is a group GL(V,F) of automorphisms of V, an associative algebra End_F(V) of all endomorphisms of V, and a corresponding Lie algebra gl(V,F).

Literatura

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Spoljašnje veze

• Hazewinkel Michiel, ур. (2001). „Representation theory”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Alexander Kirillov Jr., An introduction to Lie groups and Lie algebras (2008). Textbook, preliminary version pdf downloadable from author's home page.