Editing Canonical basis (section)

==Representation theory==
The canonical basis for the irreducible representations of a quantized enveloping algebra of
type <math>ADE</math> and also for the plus part of that algebra was introduced by Lusztig <ref>{{harvtxt|Lusztig|1990}}</ref> by
two methods: an algebraic one (using a braid group action and PBW bases) and a topological one
(using intersection cohomology). Specializing the parameter <math>q</math> to <math>q=1</math> yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was
not known earlier. Specializing the parameter <math>q</math> to <math>q=0</math> 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;<ref>{{harvtxt|Kashiwara|1990}}</ref> it is sometimes called the [[crystal basis]].
The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara <ref>{{harvtxt|Kashiwara|1991}}</ref> (by an algebraic method) and by Lusztig <ref>{{harvtxt|Lusztig|1991}}</ref> (by a topological method).

There is a general concept underlying these bases:

Consider the ring of integral [[Laurent polynomial]]s <math>\mathcal{Z}:=\mathbb{Z}\left[v,v^{-1}\right]</math> with its two subrings <math>\mathcal{Z}^{\pm}:=\mathbb{Z}\left[v^{\pm 1}\right]</math> and the automorphism <math>\overline{\cdot}</math> defined by <math>\overline{v}:=v^{-1}</math>.

A ''precanonical structure'' on a free <math>\mathcal{Z}</math>-module <math>F</math> consists of
* A ''standard'' basis <math>(t_i)_{i\in I}</math> of <math>F</math>,
* An interval finite [[partial order]] on <math>I</math>, that is, <math>(-\infty,i] := \{j\in I \mid j\leq i\}</math> is finite for all <math>i\in I</math>,
* A dualization operation, that is, a bijection <math>F\to F</math> of order two that is <math>\overline{\cdot}</math>-[[semilinear map|semilinear]] and will be denoted by <math>\overline{\cdot}</math> as well.

If a precanonical structure is given, then one can define the <math>\mathcal{Z}^{\pm}</math> submodule <math display="inline">F^{\pm} := \sum \mathcal{Z}^{\pm} t_j</math> of <math>F</math>.

A ''canonical basis of the precanonical structure is then a <math>\mathcal{Z}</math>-basis <math>(c_i)_{i\in I}</math> of <math>F</math> that satisfies:''
* <math>\overline{c_i}=c_i</math> and
* <math>c_i \in \sum_{j\leq i} \mathcal{Z}^+ t_j \text{ and } c_i \equiv t_i \mod vF^+</math>
for all <math>i\in I</math>.

One can show that there exists at most one canonical basis for each precanonical structure.<ref>{{harvtxt|Lusztig|1993|p=194}}</ref> A sufficient condition for existence is that the polynomials <math>r_{ij}\in\mathcal{Z}</math> defined by <math display="inline">\overline{t_j}=\sum_i r_{ij} t_i</math> satisfy <math>r_{ii}=1</math> and <math>r_{ij}\neq 0 \implies i\leq j</math>.

A canonical basis induces an isomorphism from <math>\textstyle F^+\cap \overline{F^+} = \sum_i \mathbb{Z}c_i</math> to <math>F^+/vF^+</math>.

=== Hecke algebras ===
Let <math>(W,S)</math> be a [[Coxeter group]]. The corresponding [[Iwahori-Hecke algebra]] <math>H</math> has the standard basis <math>(T_w)_{w\in W}</math>, the group is partially ordered by the [[Bruhat order]] which is interval finite and has a dualization operation defined by <math>\overline{T_w}:=T_{w^{-1}}^{-1}</math>. This is a precanonical structure on <math>H</math> that satisfies the sufficient condition above and the corresponding canonical basis of <math>H</math> is the [[Kazhdan–Lusztig basis]]

: <math>C_w' = \sum_{y\leq w} P_{y,w}(v^2) T_w</math>

with <math>P_{y,w}</math> being the [[Kazhdan–Lusztig polynomial]]s.