Algebra representation

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 ${\displaystyle \lambda :A\to R}$ (i.e., an algebra homomorphism from the algebra to its underlying ring: a linear functional that is also multiplicative). 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 ${\displaystyle R[T],}$ 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 polynomial algebras. 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 simultaneous eigenvector 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 representation theory of Lie algebras, specifically the finite-dimensional representations of semisimple Lie algebras.

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.