This article is about the theory of representations a concise introduction to pure mathematics martin liebeck pdf algebraic structures by linear transformations and matrices. Not to be confused with group presentation.
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. 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. Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood.
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. Secondly, 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.
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.
There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras. Lie algebra, leading to a representation theory of Lie algebras. There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication.
G and g1g2 is the product in G. This doesn’t hold for the matrix commutator and also there is no identity element for the commutator.
This approach is both more concise and more abstract. When V is of finite dimension n, one can choose a basis for V to identify V with Fn and hence recover a matrix representation with entries in the field F. G and v in V. Equivariant maps for representations of an associative or Lie algebra are defined similarly.