Representation of an algebra
Encyclopedia
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.
, which is a representation of the complex number
s C, thought of as associative algebra over the real number
s R. This algebra is realized concretely as
which corresponds to 
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 image of i in End(V)) is denoted J (to avoid confusion with the identity matrix I).
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.
In some approaches to noncommutative geometry
, the free noncommutative algebra (polynomials in non-commuting variables) plays a similar role, but the analysis is much more difficult.
The generalization of an eigenvalue of an algebra representation is, rather than a single scalar, a one-dimensional representation
(i.e., an algebra homomorphism from the algebra to its underlying ring: a linear functional
that is also multiplicative).Note that for a field, the endomorphism algebra of a one-dimensional vector space (a line) is canonically equal to the underlying field: End(L) = K, since all endomorphisms are scalar multiplication; there is thus no loss in restricting to concrete maps to the base field, rather than to abstract 1-dimensional representations. For rings there are also maps to quotients rings, which need not factor through maps to the ring itself, but again abstract 1-dimensional modules are not needed. This is known as a weight
, and the analog of an eigenvector and eigenspace are called weight vector and weight space.
The case of the eigenvalue of a single operator corresponds to the algebra
and a map of algebras 
is determined by which scalar it maps the generator T to. A weight vector for an algebra representation is a vector such that any element of the algebra maps this vector to a multiple of itself – a one-dimensional submodule (subrepresentation). As the pairing 
is bilinear, "which multiple" is an A-linear functional of A (an algebra map A → R), namely the weight. In symbols, a weight vector is a vector 
such that 
for all elements 
for some linear functional 
– note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar multiplication.
Since the base ring R is commutative, weights factor through the abelianization of the algebra A (equivalently, they vanish on the derived algebra), and thus the free commutative algebras (the polynomial algebras) are of central interest.
In the case of k commuting operators on an R-module (the polynomial algebra
), a weight is given by a k-tuple of scalars 
, and the functional is given by 
In this way weights correspond to points of k-dimensional space – and allow the use of algebraic geometry
to study them – just as eigenvalues for a single operator correspond to scalars, i.e., points on the line.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a representation of an associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
is a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
for that algebra. Here an associative algebra is a (not necessarily unital) ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
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 structureLinear complex structure
In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.Complex structures have...
, which is a representation of the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C, thought of as associative algebra over the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s R. This algebra is realized concretely as
which corresponds to 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 image of i in End(V)) is denoted J (to avoid confusion with the identity matrix I).Polynomial algebras
Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebraCommutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and its geometric counterpart, algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. 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 domainStructure theorem for finitely generated modules over a principal ideal domain
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in...
to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form.
In some approaches to noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
, the free noncommutative algebra (polynomials in non-commuting variables) plays a similar role, but the analysis is much more difficult.
Weights
Eigenvalues and eigenvectors can be generalized to algebra representations.The generalization of an eigenvalue of an algebra representation is, rather than a single scalar, a one-dimensional representation
(i.e., an algebra homomorphism from the algebra to its underlying ring: a linear functionalLinear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
that is also multiplicative).Note that for a field, the endomorphism algebra of a one-dimensional vector space (a line) is canonically equal to the underlying field: End(L) = K, since all endomorphisms are scalar multiplication; there is thus no loss in restricting to concrete maps to the base field, rather than to abstract 1-dimensional representations. For rings there are also maps to quotients rings, which need not factor through maps to the ring itself, but again abstract 1-dimensional modules are not needed. This is known as a weight
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...
, and the analog of an eigenvector and eigenspace are called weight vector and weight space.
The case of the eigenvalue of a single operator corresponds to the algebra
and a map of algebras is determined by which scalar it maps the generator T to. A weight vector for an algebra representation is a vector such that any element of the algebra maps this vector to a multiple of itself – a one-dimensional submodule (subrepresentation). As the pairing is bilinear, "which multiple" is an A-linear functional of A (an algebra map A → R), namely the weight. In symbols, a weight vector is a vector such that for all elements for some linear functional – note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar multiplication.Since the base ring R is commutative, weights factor through the abelianization of the algebra A (equivalently, they vanish on the derived algebra), and thus the free commutative algebras (the polynomial algebras) are of central interest.
In the case of k commuting operators on an R-module (the polynomial algebra
), a weight is given by a k-tuple of scalars , and the functional is given by In this way weights correspond to points of k-dimensional space – and allow the use of algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
to study them – just as eigenvalues for a single operator correspond to scalars, i.e., points on the line.
See also
- Module (mathematics)Module (mathematics)In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
- Representation theory of Hopf algebras
- Lie algebra representation