Cellular algebra

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.

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 coherent algebras.

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 an -algebra.

The concrete definition

A cell datum for is a tuple consisting of

* A finite partially ordered set .

* A -linear anti-automorphism with .

* For every a non-empty finite set of indices.

* An injective map

::

:The images under this map are notated with an upper index and two lower indices so that the typical element of the image is written as .

and satisfying the following conditions:

1. The image of is a -basis of .
2. for all elements of the basis.
3. For every , and every the equation

::

:with coefficients depending only on , and but not on . Here denotes the -span of all basis elements with upper index strictly smaller than .

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 (just called "involution" from now on).

A cell ideal of w.r.t. is a two-sided ideal such that the following conditions hold:

1. .
2. There is a left ideal that is free as a -module and an isomorphism

::

:of --bimodules such that and are compatible in the sense that

::

A cell chain for w.r.t. is defined as a direct decomposition

into free -submodules such that

2. is a two-sided ideal of
3. is a cell ideal of w.r.t. 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 (one for each topological ordering of ) and choosing a basis of every left ideal one can construct a corresponding cell basis for .

Examples

Polynomial examples

is cellular. A cell datum is given by and

• with the reverse of the natural ordering.

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 (s,t)-th entry which is equal to 1.

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 with respect to the involution that maps the standard basis as . 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).

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.

Representations

Cell modules and the invariant bilinear form

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.
• A -algebra is cellular if and only if its opposite algebra is.
• If is cellular with cell-datum and is an ideal (a downward closed subset) of the poset then (where the sum runs over and ) is a two-sided, -invariant ideal of and the quotient is cellular with cell datum (where i denotes the induced involution and M, C denote the restricted mappings).
• If is a cellular -algebra and is a unitary homomorphism of commutative rings, then the extension of scalars is a cellular -algebra.
• 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 two-sided, -invariant ideals, then the following are equivalent:

1. is cellular.
2. 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 (though a lot of this can be generalized to arbitrary rings, integral domains, local rings or at least discrete valuation rings) and is cellular w.r.t. the involution . Then the following hold

• is split, i.e. all simple modules are absolutely irreducible.
• The following are equivalent:

1. is semisimple.
2. is split semisimple.
3. is simple.
4. is nondegenerate.

• The Cartan matrix of is symmetric and positive definite.
• The following are equivalent:

1. is quasi-hereditary (i.e. its module category is a highest-weight category).
2. .
3. All cell chains of have the same length.
4. All cell chains of have the same length where is an arbitrary involution w.r.t. which is cellular.
5. .

• If is Morita equivalent to and the characteristic of is not two, then is also cellular w.r.t. a suitable involution. In particular is cellular (to some involution) 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.

References