Canonical basis

In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:

• In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta.
• In a polynomial ring, it refers to its standard basis given by the monomials, ${\displaystyle (X^{i})_{i}}$.
• For finite extension fields, it means the polynomial basis.
• In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix ${\displaystyle A}$, if the set is composed entirely of Jordan chains.^[1]
• In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.