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 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 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.
Read more about this topic: Algebra Representation
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