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, .
- For finite extension fields, it means the polynomial basis.
- In representation theory, Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations
Famous quotes containing the words canonical and/or basis:
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)
“Self-alienation is the source of all degradation as well as, on the contrary, the basis of all true elevation. The first step will be a look inward, an isolating contemplation of our self. Whoever remains standing here proceeds only halfway. The second step must be an active look outward, an autonomous, determined observation of the outer world.”
—Novalis [Friedrich Von Hardenberg] (1772–1801)