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.^[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:

1. The image of $C$ is a $R$-basis of $A$.
2. $i(C_\mathfrak{st}^\lambda)=C_\mathfrak{ts}^\lambda$ for all elements of the basis.
3. 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:

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

$\alpha: \Delta\otimes_R i(\Delta) \to J$

of $A$-$A$-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

1. $i(U_k)=U_k$
2. $J_k:=\bigoplus_{j=1}^k U_j$ is a two-sided ideal of $A$
3. $J_k/J_{k-1}$ 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:

1. $(A,i)$ is cellular.
2. $(A_1,i)$ 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]

1. $A$ is semisimple.
2. $A$ is split semisimple.
3. $\forall\lambda\in\Lambda: W(\lambda)$ is simple.
4. $\forall\lambda\in\Lambda: \phi_\lambda$ is nondegenerate.

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

1. $A$ is quasi-hereditary (i.e. its module category is a highest-weight category).
2. $\Lambda=\Lambda_0$.
3. All cell chains of $(A,i)$ have the same length.
4. 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.
5. $\det(C_A)=1$.

• 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. Graham, J.J; Lehrer, G.I. (1996), "Cellular algebras", Inventiones Mathematicae 123: 1–34, doi:10.1007/bf01232365
2. 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.
3. Cameron, Peter J. (1999). Permutation Groups. London Mathematical Society Student Texts (45). Cambridge University Press. ISBN 0-521-65378-9.
4. König, S.; Xi, C.C. (1996), "On the structure of cellular algebras", Algebras and modules II. CMS Conference Proceedings: 365–386
5. Geck, Meinolf (2007), "Hecke algebras of finite type are cellular", Inventiones Mathematicae 169: 501–517, doi:10.1007/s00222-007-0053-2
6. 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
7. 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