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
and also for the plus part of that algebra was introduced by Lusztig [2] by two methods: an algebraic one (using a braid group action and PBW bases) and a topological one (using intersection cohomology).
yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier.
This shadow (but not the basis itself) for the case of irreducible representations was considered independently by Kashiwara;[3] it is sometimes called the crystal basis.
The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara [4] (by an algebraic method) and by Lusztig [5] (by a topological method).
There is a general concept underlying these bases: Consider the ring of integral Laurent polynomials
consists of If a precanonical structure is given, then one can define the
A canonical basis of the precanonical structure is then a
One can show that there exists at most one canonical basis for each precanonical structure.
[6] A sufficient condition for existence is that the polynomials
A canonical basis induces an isomorphism from
, the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by
that satisfies the sufficient condition above and the corresponding canonical basis of
in Jordan normal form, similar to
, we are interested only in sets of linearly independent generalized eigenvectors.
A matrix in Jordan normal form is an "almost diagonal matrix," that is, as close to diagonal as possible.
is a special case of a matrix in Jordan normal form.
possesses n linearly independent generalized eigenvectors.
Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent.
linearly independent generalized eigenvectors corresponding to
, there are infinitely many ways to pick the n linearly independent generalized eigenvectors.
If they are chosen in a particularly judicious manner, we can use these vectors to show that
is similar to a matrix in Jordan normal form.
In particular, Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors
designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue
Note that Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see generalized eigenvector).
[8] This example illustrates a canonical basis with two Jordan chains.
Unfortunately, it is a little difficult to construct an interesting example of low order.
in Jordan normal form, similar to