Cellular algebra

This article is about the cellular algebras of Graham and Lehrer. For the cellular algebras of Weisfeiler and Lehman, see association scheme.

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.^[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 association schemes. ^[2]^[3]

Definitions

Let $R$ be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also $A$ be a $R$ -algebra.

The concrete definition

A cell datum for $A$ is a tuple $(\Lambda ,i,M,C)$ consisting of

A finite partially ordered set $\Lambda$ .
A $R$ -linear anti-automorphism $i:A\to A$ with $i^{2}=id_{A}$ .
For every $\lambda\in\Lambda$ a non-empty, finite set $M(\lambda )$ of indices.
An injective map

C:{\dot {\bigcup }}_{\lambda \in \Lambda }M(\lambda )\times M(\lambda )\to A

The images under this map are notated with an upper index

\lambda\in\Lambda

and two lower indices

{\mathfrak {s}},{\mathfrak {t}}\in M(\lambda )

so that the typical element of the image is written as

C_{\mathfrak {st}}^{\lambda }

and satisfying the following conditions:

The image of $C$ is a $R$ -basis of $A$ .
$i(C_{\mathfrak {st}}^{\lambda })=C_{\mathfrak {ts}}^{\lambda }$ for all elements of the basis.
For every $\lambda\in\Lambda$ , ${\mathfrak {s}},{\mathfrak {t}}\in M(\lambda )$ and every $a\in A$ the equation

aC_{\mathfrak {st}}^{\lambda }\equiv \sum _{{\mathfrak {u}}\in M(\lambda )}r_{a}({\mathfrak {u}},{\mathfrak {s}})C_{\mathfrak {ut}}^{\lambda }\mod A(<\lambda )

with coefficients

r_{a}({\mathfrak {u}},{\mathfrak {s}})\in R

depending only on

a

{\mathfrak {u}}

and

{\mathfrak {s}}

but not on

\mathfrak{t}

. Here

A(<\lambda )

denotes the

R

-span of all basis elements with upper index strictly smaller than

\lambda

This definition was originally given by Graham and Lehrer who invented cellular algebras.^[1]

The more abstract definition

Let $i:A\to A$ be an anti automorphism of $R$ -algebras with $i^{2}=id$ (just called "involution" from now on).

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

$i(J)=J$ .
There is a left ideal $\Delta \subseteq J$ that is free as a $R$ -module and an isomorphism

\alpha :\Delta \otimes _{R}i(\Delta )\to J

A

A

-bimodules such that

\alpha

and

i

are compatible in the sense that

\forall x,y\in \Delta :i(\alpha (x\otimes i(y)))=\alpha (y\otimes i(x))

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

A=\bigoplus _{k=1}^{m}U_{k}

into free $R$ -submodules such that

$i(U_{k})=U_{k}$
$J_{k}:=\bigoplus _{j=1}^{k}U_{j}$ is a two-sided ideal of $A$
$J_{k}/J_{k-1}$ is a cell ideal of $A/J_{k-1}$ w.r.t. to the induced involution.

Now $(A,i)$ is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent.^[4] Every basis gives rise to cell chains (one for each topological ordering of $\Lambda$ ) and choosing a basis of every left ideal $\Delta /J_{k-1}\subseteq J_{k}/J_{k-1}$ one can construct a corresponding cell basis for $A$ .

Examples

Polynomial examples

$R[x]/(x^{n})$ is cellular. A cell datum is given by $i=id$ and

$\Lambda :=\lbrace 0,\ldots ,n-1\rbrace$ with the reverse of the natural ordering.
$M(\lambda ):=\lbrace 1\rbrace$
$C_{11}^{\lambda }:=x^{\lambda }$

A cell-chain in the sense of the second, abstract definition is given by

0\subseteq (x^{n-1})\subseteq (x^{n-2})\subseteq \ldots \subseteq (x^{1})\subseteq (x^{0})=R

Matrix examples

$R^{d\times d}$ is cellular. A cell datum is given by $i(A)=A^{T}$ and

$\Lambda :=\lbrace 1\rbrace$
$M(1):=\lbrace 1,\dots ,d\rbrace$
For the basis one chooses $C_{st}^{1}:=E_{st}$ the standard matrix units, i.e. $C_{st}^{1}$ 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

0\subseteq R^{d\times d}

In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset $\Lambda$ .

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 $T_{w}\mapsto T_{w^{-1}}$ .^[5] 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).^[4]

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 ${\mathcal {O}}$ of a semisimple Lie algebra.^[4]

Representations

Cell modules and the invariant bilinear form

Assume $A$ is cellular and $(\Lambda ,i,M,C)$ is a cell datum for $A$ . Then one defines the cell module $W(\lambda )$ as the free $R$ -module with basis $\lbrace C_{\mathfrak {s}}|{\mathfrak {s}}\in M(\lambda )\rbrace$ and multiplication

aC_{\mathfrak {s}}:=\sum _{\mathfrak {u}}r_{a}({\mathfrak {u}},{\mathfrak {s}})C_{\mathfrak {u}}

where the coefficients $r_{a}({\mathfrak {u}},{\mathfrak {s}})$ are the same as above. Then $W(\lambda )$ becomes an $A$ -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 $\phi _{\lambda }:W(\lambda )\times W(\lambda )\to R$ which satisfies

C_{\mathfrak {st}}^{\lambda }C_{\mathfrak {uv}}^{\lambda }\equiv \phi _{\lambda }(C_{\mathfrak {t}},C_{\mathfrak {u}})C_{\mathfrak {sv}}^{\lambda }\mod A(<\lambda )

for all indices $s,t,u,v\in M(\lambda )$ .

One can check that $\phi _{\lambda }$ is symmetric in the sense that

\phi _{\lambda }(x,y)=\phi _{\lambda }(y,x)

for all $x,y\in W(\lambda )$ and also $A$ -invariant in the sense that

\phi _{\lambda }(i(a)x,y)=\phi _{\lambda }(x,ay)

for all $a\in A$ , $x,y\in W(\lambda )$ .

Simple modules

Assume for the rest of this section that the ring $R$ is a field. With the information contained in the invariant bilinear forms one can easily list all simple $A$ -modules:

Let $\Lambda _{0}:=\lbrace \lambda \in \Lambda |\phi _{\lambda }\neq 0\rbrace$ and define $L(\lambda ):=W(\lambda )/\operatorname {rad} (\phi _{\lambda })$ for all $\lambda \in \Lambda _{0}$ . Then all $L(\lambda)$ are absolute simple $A$ -modules and every simple $A$ -module is one of these.

These theorems appear already in the original paper by Graham and Lehrer.^[1]

Properties of cellular algebras

Persistence properties

Tensor products of finitely many cellular $R$ -algebras are cellular.
A $R$ -algebra $A$ is cellular if and only if its opposite algebra $A^{op}$ is.
If $A$ is cellular with cell-datum $(\Lambda ,i,M,C)$ and $\Phi \subseteq \Lambda$ is an ideal (a downward closed subset) of the poset $\Lambda$ then $A(\Phi ):=\sum RC_{\mathfrak {st}}^{\lambda }$ (where the sum runs over $\lambda\in\Lambda$ and $s,t\in M(\lambda )$ ) is an twosided, $i$ -invariant ideal of $A$ and the quotient $A/A(\Phi )$ is cellular with cell datum $(\Lambda \setminus \Phi ,i,M,C)$ (where i denotes the induces involution and M,C denote the restricted mappings).
If $A$ is a cellular $R$ -algebra and $R\to S$ is a unitary homomorphism of commutative rings, then the extension of scalars $S\otimes _{R}A$ is a cellular $S$ -algebra.
Direct products of finitely many cellular $R$ -algebras are cellular.

If $R$ is an integral domain then there is a converse to this last point:

If $(A,i)$ is a finite dimensional $R$ -algebra with an involution and $A=A_{1}\oplus A_{2}$ a decomposition in twosided, $i$ -invariant ideals, then the following are equivalent:

$(A,i)$ is cellular.
$(A_{1},i)$ and $(A_{2},i)$ are cellular.

Since in particular all blocks of $A$ are $i$ -invariant if $(A,i)$ is cellular, an immediate corollary is that a finite dimensional $R$ -algebra is cellular w.r.t. $i$ if and only if all blocks are $i$ -invariante and cellular w.r.t. $i$ .
Tits' deformation theorem for cellular algebras: Let $A$ be a cellular $R$ -algebra. Also let $R\to k$ be a unitary homomorphism into a field $k$ and $K:=Quot(R)$ the quotient field of $R$ . Then the following holds: If $kA$ is semisimple, then $KA$ is also semisimple.

If one further assumes $R$ to be a local domain, then additionally the following holds:

If $A$ is cellular w.r.t. $i$ and $e\in A$ is an idempotent such that $i(e)=e$ , then the Algebra $eAe$ is cellular.

Other properties

Assuming that $R$ 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 $A$ is cellular w.r.t. to the involution $i$ . Then the following hold

$A$ is split, i.e. all simple modules are absolutely irreducible.
The following are equivalent:^[1]

$A$ is semisimple.
$A$ is split semisimple.
$\forall \lambda \in \Lambda :W(\lambda )$ is simple.
$\forall \lambda \in \Lambda :\phi _{\lambda }$ is nondegenerate.

The Cartan matrix $C_A$ of $A$ is symmetric and positive definite.
The following are equivalent:^[6]

$A$ is quasi-hereditary (i.e. its module category is a highest-weight category).
$\Lambda =\Lambda _{0}$ .
All cell chains of $(A,i)$ have the same length.
All cell chains of $(A,j)$ have the same length where $j:A\to A$ is an arbitrary involution w.r.t. which $A$ is cellular.
$\det(C_{A})=1$ .

If $A$ is Morita equivalent to $B$ and the characteristic of $R$ is not two, then $B$ is also cellular w.r.t. an suitable involution. In particular is $A$ cellular (to some involution) if and only if its basic algebra is.^[7]
Every idempotent $e\in A$ is equivalent to $i(e)$ , i.e. $Ae\cong Ai(e)$ . If $char(R)\neq 2$ then in fact every equivalence class contains an $i$ -invariant idempotent.^[4]

References

1 2 3 4 Graham, J.J; Lehrer, G.I. (1996), "Cellular algebras", Inventiones Mathematicae, 123: 1–34, doi:10.1007/bf01232365
↑ Weisfeiler, B. Yu.; A. A., Lehman (1968). "Reduction of a graph to a canonical form and an algebra which appears in this process". Scientific-Technological Investigations. 2 (in Russian). 9: 12–16.
↑ Cameron, Peter J. (1999). Permutation Groups. London Mathematical Society Student Texts (45). Cambridge University Press. ISBN 0-521-65378-9.
1 2 3 4 König, S.; Xi, C.C. (1996), "On the structure of cellular algebras", Algebras and modules II. CMS Conference Proceedings: 365–386
↑ Geck, Meinolf (2007), "Hecke algebras of finite type are cellular", Inventiones Mathematicae, 169: 501–517, doi:10.1007/s00222-007-0053-2
↑ König, S.; Xi, C.C. (1999-06-24), "Cellular algebras and quasi-hereditary algebras: A comparison", Electronic Research Announcements of the American Mathematical Society, 5: 71–75
↑ König, S.; Xi, C.C. (1999), "Cellular algebras: inflations and Morita equivalences", Journal of the London Mathematical Society, 60: 700–722, doi:10.1112/s0024610799008212

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