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Canonical basis

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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:

Representation theory

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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). Specializing the parameter to yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was not known earlier. Specializing the parameter to 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;[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 with its two subrings and the automorphism defined by .

A precanonical structure on a free -module consists of

  • A standard basis of ,
  • An interval finite partial order on , that is, is finite for all ,
  • A dualization operation, that is, a bijection of order two that is -semilinear and will be denoted by as well.

If a precanonical structure is given, then one can define the submodule of .

A canonical basis of the precanonical structure is then a -basis of that satisfies:

  • and

for all .

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 defined by satisfy and .

A canonical basis induces an isomorphism from to .

Hecke algebras

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Let be a Coxeter group. The corresponding Iwahori-Hecke algebra has the standard basis , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by . This is a precanonical structure on that satisfies the sufficient condition above and the corresponding canonical basis of is the Kazhdan–Lusztig basis

with being the Kazhdan–Lusztig polynomials.

Linear algebra

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If we are given an n × n matrix and wish to find a matrix 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. A diagonal matrix is a special case of a matrix in Jordan normal form. An ordinary eigenvector is a special case of a generalized eigenvector.

Every n × n matrix possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If is an eigenvalue of of algebraic multiplicity , then will have linearly independent generalized eigenvectors corresponding to .

For any given n × n matrix , 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 that are in the Jordan chain generated by are also in the canonical basis.[7]

Computation

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Let be an eigenvalue of of algebraic multiplicity . First, find the ranks (matrix ranks) of the matrices . The integer is determined to be the first integer for which has rank (n being the number of rows or columns of , that is, is n × n).

Now define

The variable designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue that will appear in a canonical basis for . 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]

Example

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This example illustrates a canonical basis with two Jordan chains. Unfortunately, it is a little difficult to construct an interesting example of low order.[9] The matrix

has eigenvalues and with algebraic multiplicities and , but geometric multiplicities and .

For we have

has rank 5,
has rank 4,
has rank 3,
has rank 2.

Therefore

Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 4, 3, 2 and 1.

For we have

has rank 5,
has rank 4.

Therefore

Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 2 and 1.

A canonical basis for is

is the ordinary eigenvector associated with . and are generalized eigenvectors associated with . is the ordinary eigenvector associated with . is a generalized eigenvector associated with .

A matrix in Jordan normal form, similar to is obtained as follows:

where the matrix is a generalized modal matrix for and .[10]

See also

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Notes

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  1. ^ Bronson (1970, p. 196)
  2. ^ Lusztig (1990)
  3. ^ Kashiwara (1990)
  4. ^ Kashiwara (1991)
  5. ^ Lusztig (1991)
  6. ^ Lusztig (1993, p. 194)
  7. ^ Bronson (1970, pp. 196, 197)
  8. ^ Bronson (1970, pp. 197, 198)
  9. ^ Nering (1970, pp. 122, 123)
  10. ^ Bronson (1970, p. 203)

References

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  • Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
  • Deng, Bangming; Ju, Jie; Parshall, Brian; Wang, Jianpan (2008), Finite Dimensional Algebras and Quantum Groups, Mathematical surveys and monographs, vol. 150, Providence, R.I.: American Mathematical Society, ISBN 9780821875315
  • Kashiwara, Masaki (1990), "Crystalizing the q-analogue of universal enveloping algebras", Communications in Mathematical Physics, 133 (2): 249–260, Bibcode:1990CMaPh.133..249K, doi:10.1007/bf02097367, ISSN 0010-3616, MR 1090425, S2CID 121695684
  • Kashiwara, Masaki (1991), "On crystal bases of the q-analogue of universal enveloping algebras", Duke Mathematical Journal, 63 (2): 465–516, doi:10.1215/S0012-7094-91-06321-0, ISSN 0012-7094, MR 1115118
  • Lusztig, George (1990), "Canonical bases arising from quantized enveloping algebras", Journal of the American Mathematical Society, 3 (2): 447–498, doi:10.2307/1990961, ISSN 0894-0347, JSTOR 1990961, MR 1035415
  • Lusztig, George (1991), "Quivers, perverse sheaves and quantized enveloping algebras", Journal of the American Mathematical Society, 4 (2): 365–421, doi:10.2307/2939279, ISSN 0894-0347, JSTOR 2939279, MR 1088333
  • Lusztig, George (1993), Introduction to quantum groups, Boston, MA: Birkhauser Boston, ISBN 0-8176-3712-5, MR 1227098
  • Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646