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In mathematics, a '''canonical basis''' is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
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* 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 [[monomial]]s, <math>(X^i)_i</math>.
* For finite extension fields, it means the [[polynomial basis]].
 
==Representation theory==
 
In representation theory there are several basis that are called "canonical", e.g. Lusztig's canonical basis and closely related Kashiwara's [[crystal basis]] in quantum groups and their representations. There is a general concept underlying these basis:
 
Consider the ring of integral [[Laurent polynomial]]s <math>\mathcal{Z}:=\mathbb{Z}[v,v^{-1}]</math> with its two subrings <math>\mathcal{Z}^{\pm}:=\mathbb{Z}[v^{\pm 1}]</math> and the automorphism <math>\overline{.}</math> that is defined by <math>v\mapsto 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>,
* A [[partial order]] on <math>I</math> that is interval finite, i.e. <math>(-\infty,i] := \{j\in I | j\leq i\}</math> is finite for all <math>i\in I</math>,
* A dualization operation, i.e. a bijection <math>F\to F</math> of order two that is <math>\overline{.}</math>-[[semilinear transformation|semilinear]] and will be denoted by <math>\overline{.}</math> as well.
 
If a precanonical structure is given, then one can define the <math>\mathcal{Z}^{\pm}</math> submodule <math>F^{\pm} := \sum \mathcal{Z}^{\pm} t_j</math> of <math>F</math>.
 
A ''canonical basis at <math>v=0</math>'' 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</math>and <math>c_i \equiv t_i \mod vF^+</math>
for all <math>i\in I</math>. A ''canonical basis at <math>v=\infty</math>'' is analogously defined to be a basis <math>(\widetilde{c}_i)_{i\in I}</math> that satisfies
* <math>\overline{\widetilde{c}_i}=\widetilde{c}_i</math> and
* <math>\widetilde{c}_i \in \sum_{j\leq i} \mathcal{Z}^- t_j</math> and <math>\widetilde{c}_i \equiv t_i \mod v^{-1}F^-</math>
for all <math>i\in I</math>. The naming "at <math>v=\infty</math>" alludes to the fact <math>\lim_{v\to\infty} v^{-1} =0</math> and hence the "specialization" <math>v\mapsto\infty</math> corresponds to quotienting out the relation <math>v^{-1}=0</math>.
 
One can show that there exists at most one canonical basis at v=0 (and at most one at <math>v=\infty</math>) for each precanonical structure. A sufficient condition for existence is that the polynomials <math>r_{ij}\in\mathcal{Z}</math> defined by <math>\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 at v=0 (<math>v=\infty</math>) induces an isomorphism from <math>F^+\cap \overline{F^+} = \sum_i \mathbb{Z}c_i</math> to <math>F^+/vF^+</math>  (<math>F^{-} \cap \overline{F^{-}}=\sum_i \mathbb{Z}\widetilde{c}_i \to F^{-}/v^{-1} F^{-}</math> respectively).
 
=== Examples ===
 
==== Quantum groups ====
 
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at <math>v=0</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> at <math>v=0</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.
 
==See also==
*[[Canonical form]]
*[[Normal form (mathematics)|Normal form (disambiguation)]]
*[[Polynomial basis]]
*[[Normal basis]]
*[[Change of bases]]
 
==References==
 
*{{Citation | last1=Deng | first1=Bangming | last2=Ju |first2=Jie| last3=Parshall|first3=Brian | last4=Wang|first4=Jianpan | title=Finite Dimensional Algebras and Quantum Groups | url=http://books.google.de/books?id=k9aUvxN2w2MC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical surveys and monographs | isbn=9780821875315 | year=2008 | volume=150}}
 
{{mathdab}}
 
[[Category:Linear algebra]]
[[Category:Abstract algebra]]
 
[[fr:Base canonique]]
[[pt:Base canônica]]

Latest revision as of 10:58, 23 December 2014

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