The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer. However, the terminology had previously been used by Weisfeiler and Lehman in the Soviet Union in the 1960s, to describe what are also known as association schemes.
Definitions
Let be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also be a -algebra.
This definition was originally given by Graham and Lehrer who invented cellular algebras.
The more abstract definition
Let be an anti automorphism of -algebras with . A cell ideal of w.r.t. is a two-sided ideal such that the following conditions hold:
.
There is a left ideal that is free as a -module and an isomorphism
A cell chain for w.r.t. is defined as a direct decomposition into free -submodules such that
is a two-sided ideal of
is a cell ideal of w.r.t. to the induced involution.
Now is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent. Every basis gives rise to cell chains and choosing a basis of every left ideal one can construct a corresponding cell basis for.
A cell-chain in the sense of the second, abstract definition is given by
Matrix examples
is cellular. A cell datum is given by and
For the basis one chooses the standard matrix units, i.e. is the matrix with all entries equal to zero except the -th entry which is equal to 1.
A cell-chain is given by In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset.
Assume is cellular and is a cell datum for. Then one defines the cell module as the free -module with basis and multiplication where the coefficients are the same as above. Then becomes an -left module. These modules generalize the Specht modules for the symmetric group and the Hecke-algebras of type A. There is a canonical bilinear form which satisfies for all indices. One can check that is symmetric in the sense that for all and also -invariant in the sense that for all,.
Simple modules
Assume for the rest of this section that the ring is a field. With the information contained in the invariant bilinear forms one can easily list all simple -modules: Let and define for all. Then all are absolute simple -modules and every simple -module is one of these. These theorems appear already in the original paper by Graham and Lehrer.
Properties of cellular algebras
Persistence properties
Tensor products of finitely many cellular -algebras are cellular.
Direct products of finitely many cellular -algebras are cellular.
If is an integral domain then there is a converse to this last point:
If is a finite dimensional -algebra with an involution and a decomposition in twosided, -invariant ideals, then the following are equivalent:
is cellular.
and are cellular.
Since in particular all blocks of are -invariant if is cellular, an immediate corollary is that a finite dimensional -algebra is cellular w.r.t. if and only if all blocks are -invariant and cellular w.r.t..
Tits' deformation theorem for cellular algebras: Let be a cellular -algebra. Also let be a unitary homomorphism into a field and the quotient field of. Then the following holds: If is semisimple, then is also semisimple.
If one further assumes to be a local domain, then additionally the following holds:
If is cellular w.r.t. and is an idempotent such that, then the Algebra is cellular.
Other properties
Assuming that is a field and is cellular w.r.t. to the involution. Then the following hold
All cell chains of have the same length where is an arbitrary involution w.r.t. which is cellular.
.
If is Morita equivalent to and the characteristic of is not two, then is also cellular w.r.t. an suitable involution. In particular is cellular if and only if its basic algebra is.
Every idempotent is equivalent to, i.e.. If then in fact every equivalence class contains an -invariant idempotent.
