|
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.〔Classic texts on representation theory include and . Other excellent sources are and .〕 In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. 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.〔For the history of the representation theory of finite groups, see . For algebraic and Lie groups, see .〕 Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood.〔There are many textbooks on vector spaces and linear algebra. For an advanced treatment, see .〕 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.〔.〕 A feature of representation theory is its pervasiveness in mathematics. There are two sides to this. 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.〔, .〕 The second aspect is the diversity of 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.〔See the previous footnotes and also .〕 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. A ''representation'' should not be confused with a ''presentation''. ==Definitions and concepts== Let ''V'' be a vector space over a field F.〔 For instance, suppose ''V'' is R''n'' or C''n'', 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 analyses 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 EndF(''V'') of all endomorphisms of ''V'', and a corresponding Lie algebra gl(''V'',F). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Representation theory」の詳細全文を読む スポンサード リンク
|