In algebra, a crystal base or canonical base is a base of a representation, such that generators of a quantum group or semisimple Lie algebra have a particularly simple action on it. Crystal bases were introduced by Kashiwara (1990) and Lusztig (1990) (under the name of canonical bases).
As a consequence of its defining relations, the quantum group can be regarded as a Hopf algebra over the field of all rational functions of an indeterminate q over , denoted .
For simple root and non-negative integer , define
In an integrable module , and for weight , a vector (i.e. a vector in with weight ) can be uniquely decomposed into the sums
where , , only if , and only if .
Linear mappings can be defined on by
Let be the integral domain of all rational functions in which are regular at (i.e. a rational function is an element of if and only if there exist polynomials and in the polynomial ring such that , and ). A crystal base for is an ordered pair , such that
- is a free -submodule of such that
- is a -basis of the vector space over
- and , where and
To put this into a more informal setting, the actions of and are generally singular at on an integrable module . The linear mappings and on the module are introduced so that the actions of and are regular at on the module. There exists a -basis of weight vectors for , with respect to which the actions of and are regular at for all i. The module is then restricted to the free -module generated by the basis, and the basis vectors, the -submodule and the actions of and are evaluated at . Furthermore, the basis can be chosen such that at , for all , and are represented by mutual transposes, and map basis vectors to basis vectors or 0.
A crystal base can be represented by a directed graph with labelled edges. Each vertex of the graph represents an element of the -basis of , and a directed edge, labelled by i, and directed from vertex to vertex , represents that (and, equivalently, that ), where is the basis element represented by , and is the basis element represented by . The graph completely determines the actions of and at . If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets and such that there are no edges joining any vertex in to any vertex in ).
For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate Kac–Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac–Moody algebra.
It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base.
Tensor products of crystal bases
Let be an integrable module with crystal base and be an integrable module with crystal base . For crystal bases, the coproduct , given by
is adopted. The integrable module has crystal base , where . For a basis vector , define
The actions of and on are given by
The decomposition of the product two integrable highest weight modules into irreducible submodules is determined by the decomposition of the graph of the crystal base into its connected components (i.e. the highest weights of the submodules are determined, and the multiplicity of each highest weight is determined).
- Jantzen, Jens Carsten (1996), Lectures on quantum groups, Graduate Studies in Mathematics, 6, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0478-0, MR 1359532
- Kashiwara, Masaki (1990), "Crystalizing the q-analogue of universal enveloping algebras", Communications in Mathematical Physics, 133 (2): 249–260, doi:10.1007/bf02097367, ISSN 0010-3616, MR 1090425
- Lusztig, G. (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, MR 1035415