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.
Contents
History
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
The concrete definition
A cell datum for
and satisfying the following conditions:
- The image of
C is aR -basis ofA . -
i ( C s t λ ) = C t s λ - For every
λ ∈ Λ ,s , t ∈ M ( λ ) and everya ∈ A the equation
This definition was originally given by Graham and Lehrer who invented cellular algebras.
The more abstract definition
Let
A cell ideal of
-
i ( J ) = J . - There is a left ideal
Δ ⊆ J that is free as aR -module and an isomorphism
A cell chain for
into free
-
i ( U k ) = U k -
J k := ⨁ j = 1 k U j A -
J k / J k − 1 A / J k − 1
Now
Polynomial examples
A cell-chain in the sense of the second, abstract definition is given by
Matrix examples
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
Further examples
Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t. to the involution that maps the standard basis as
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).
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
Cell modules and the invariant bilinear form
Assume
where the coefficients
These modules generalize the Specht modules for the symmetric group and the Hecke-algebras of type A.
There is a canonical bilinear form
for all indices
One can check that
for all
for all
Simple modules
Assume for the rest of this section that the ring
Let
These theorems appear already in the original paper by Graham and Lehrer.
Persistence properties
If
-
( A , i ) is cellular. -
( A 1 , i ) and( A 2 , i ) are cellular.
If one further assumes
Other properties
Assuming that
-
A is semisimple. -
A is split semisimple. -
∀ λ ∈ Λ : W ( λ ) is simple. -
∀ λ ∈ Λ : ϕ λ
-
A is quasi-hereditary (i.e. its module category is a highest-weight category). -
Λ = Λ 0 - All cell chains of
( A , i ) have the same length. - All cell chains of
( A , j ) have the same length wherej : A → A is an arbitrary involution w.r.t. whichA is cellular. -
det ( C A ) = 1 .