In abstract algebra, a cellular algebra is a finite-dimensional associative algebra A with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of A.
The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer.
[1] 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 coherent algebras.
be a fixed commutative ring with unit.
In most applications this is a field, but this is not needed for the definitions.
consisting of This definition was originally given by Graham and Lehrer who invented cellular algebras.
such that the following conditions hold: A cell chain for
is defined as a direct decomposition into free
is called a cellular algebra if it has a cell chain.
One can show that the two definitions are equivalent.
[5] Every basis gives rise to cell chains (one for each topological ordering of
) and choosing a basis of every left ideal
one can construct a corresponding cell basis for
and A cell-chain in the sense of the second, abstract definition is given by
and A cell-chain (and in fact the only 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
Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t.
to the involution that maps the standard basis as
[6] This includes for example the integral group algebra of the symmetric groups as well as all other finite Weyl groups.
A basic Brauer tree algebra over a field is cellular if and only if the Brauer tree is a straight line (with arbitrary number of exceptional vertices).
[5] Further examples include q-Schur algebras, the Brauer algebra, the Temperley–Lieb algebra, the Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category
of a semisimple Lie algebra.
Then one defines the cell module
These modules generalize the Specht modules for the symmetric group and the Hecke-algebras of type A.
There is a canonical bilinear form
Assume for the rest of this section that the ring
With the information contained in the invariant bilinear forms one can easily list all simple
These theorems appear already in the original paper by Graham and Lehrer.
is an integral domain then there is a converse to this last point: If one further assumes
to be a local domain, then additionally the following holds: Assuming that
is a field (though a lot of this can be generalized to arbitrary rings, integral domains, local rings or at least discrete valuation rings) and