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

The canonical basis for the irreducible representations of a quantized enveloping algebra of type ${\displaystyle ADE}$ and also for the plus part of that algebra was introduced by Lusztig by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology). Specializing the parameter ${\displaystyle q}$ to ${\displaystyle q=1}$ yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter ${\displaystyle q}$ to ${\displaystyle q=0}$ yields something like a shadow of a basis. This shadow (but not the basis itself) for the case of irreducible representations was considered independently by Kashiwara; it is sometimes called the crystal basis. The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara (by an algebraic method) and by Lusztig (by a topological method).

There is a general concept underlying these bases:

Consider the ring of integral Laurent polynomials ${\displaystyle {\mathcal {Z}}:=\mathbb {Z} \left[v,v^{-1}\right]}$ with its two subrings ${\displaystyle {\mathcal {Z}}^{\pm }:=\mathbb {Z} \left[v^{\pm 1}\right]}$ and the automorphism ${\displaystyle {\overline {\cdot }}}$ defined by ${\displaystyle {\overline {v}}:=v^{-1}}$.

A precanonical structure on a free ${\displaystyle {\mathcal {Z}}}$-module ${\displaystyle F}$ consists of

If a precanonical structure is given, then one can define the ${\displaystyle {\mathcal {Z}}^{\pm }}$ submodule ${\textstyle F^{\pm }:=\sum {\mathcal {Z}}^{\pm }t_{j}}$ of ${\displaystyle F}$.

A canonical basis of the precanonical structure is then a ${\displaystyle {\mathcal {Z}}}$-basis ${\displaystyle (c_{i})_{i\in I}}$ of ${\displaystyle F}$ that satisfies:

for all ${\displaystyle i\in I}$.