Algebra representation

In

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.

Examples

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 ${\displaystyle \mathbb {C} =\mathbb {R} [x]/(x^{2}+1),}$ 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 ${\displaystyle \mathbb {C} \to \mathrm {End} (V)}$). 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.

Another important basic class of examples are representations of

K is concretely a K-vector space with k commuting operators, and is often denoted ${\displaystyle K[T_{1},\dots ,T_{k}],}$ meaning the representation of the abstract algebra ${\displaystyle K[x_{1},\dots ,x_{k}]}$ where ${\displaystyle x_{i}\mapsto T_{i}.}$

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 ${\displaystyle K[T]}$ and is used in understanding the structure of a single

.

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.

Eigenvalues and eigenvectors can be generalized to algebra representations.

The generalization of an

of an algebra representation is, rather than a single scalar, a one-dimensional representation ${\displaystyle \lambda \colon A\to R}$ (i.e., an , 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 ${\displaystyle R[T],}$ and a map of algebras ${\displaystyle R[T]\to R}$ 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 ${\displaystyle A\times M\to M}$ 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 ${\displaystyle m\in M}$ such that ${\displaystyle am=\lambda (a)m}$ for all elements ${\displaystyle a\in A,}$ for some linear functional ${\displaystyle \lambda }$ – note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar multiplication.

Because a weight is a map to a commutative ring, the map factors through the abelianization of the algebra ${\displaystyle {\mathcal {A}}}$ – equivalently, it vanishes on the derived algebra – in terms of matrices, if ${\displaystyle v}$ is a common eigenvector of operators ${\displaystyle T}$ and ${\displaystyle U}$, then ${\displaystyle TUv=UTv}$ (because in both cases it is just multiplication by scalars), so common eigenvectors of an algebra must be in the set on which the algebra acts commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the

. In this particularly simple and important case of the polynomial algebra ${\displaystyle \mathbf {F} [T_{1},\dots ,T_{k}]}$ in a set of commuting matrices, a weight vector of this algebra is a of the matrices, while a weight of this algebra is simply a ${\displaystyle k}$-tuple of scalars ${\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{k})}$ corresponding to the eigenvalue of each matrix, and hence geometrically to a point in ${\displaystyle k}$-space. These weights – in particularly their geometry – are of central importance in understanding the .

As an application of this geometry, given an algebra that is a quotient of a polynomial algebra on ${\displaystyle k}$ generators, it corresponds geometrically to an algebraic variety in ${\displaystyle k}$-dimensional space, and the weight must fall on the variety – i.e., it satisfies the defining equations for the variety. This generalizes the fact that eigenvalues satisfy the characteristic polynomial of a matrix in one variable.

• Representation theory
• Intertwiner
• Representation theory of Hopf algebras
• Lie algebra representation
• Schur’s lemma
• Jacobson density theorem
• Double commutant theorem

References

• Richard S. Pierce. Associative algebras. Graduate texts in mathematics, Vol. 88, Springer-Verlag, 1982,
  ISBN 978-0-387-90693-5