In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra. Linear complex structure One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as which corresponds to i2 = −1. Then a representation of C is a real vector space V, together with an action of C on V (a map ). Concretely, this is just an action of i , as this generates the algebra, and the operator representing i (the of i in End(V)) is denoted J to avoid confusion with the identity matrix I. Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebra and its geometric counterpart, algebraic geometry. A representation of a polynomial algebra in k variables over the field K is concretely a K-vector space with k commuting operators, and is often denoted meaning the representation of the abstract algebra where A basic result about such representations is that, over an algebraically closed field, the representing matrices are simultaneously triangularisable. Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by and is used in understanding the structure of a single linear operator on a finite-dimensional vector space. Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (32)
PHYS-431: Quantum field theory I
The goal of the course is to introduce relativistic quantum field theory as the conceptual and mathematical framework describing fundamental interactions.
MATH-334: Representation theory
Study the basics of representation theory of groups and associative algebras.
PHYS-512: Statistical physics of computation
The students understand tools from the statistical physics of disordered systems, and apply them to study computational and statistical problems in graph theory, discrete optimisation, inference and m
Show more
Related lectures (120)
Statistical Physics of Clusters
Explores the statistical physics of clusters, focusing on complexity and equilibrium behavior.
Schur's Lemma and Representations
Explores Schur's lemma and its applications in representations of an associative algebra over an algebraically closed field.
Deformation and Strain Tensors
Explores deformation and strain tensors, Lagrange representation, elasticity theory, and the divergence theorem.
Show more
Related concepts (9)
Representation theory
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 their algebraic operations (for example, matrix addition, matrix multiplication).
Eigenvalues and eigenvectors
In linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Change of rings
In algebra, a change of rings is an operation of changing a coefficient ring to another. Given a ring homomorphism , there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N, one can form the induced module, formed by extension of scalars, the coinduced module, formed by co-extension of scalars, and formed by restriction of scalars. They are related as adjoint functors: and This is related to Shapiro's lemma.
Show more

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.