cellular algebra

{{about|the cellular algebras of Graham and Lehrer|the cellular algebras of Weisfeiler and Lehman|coherent 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.{{citation|title= Cellular algebras|first1 = J.J|last1= Graham|first2= G.I.|last2 = Lehrer|journal= Inventiones Mathematicae|volume= 123|year=1996|pages=1–34|doi=10.1007/bf01232365|bibcode=1996InMat.123....1G|s2cid = 189831103}} 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.

{{cite journal| last1 = Weisfeiler| first1 = B. Yu.| authorlink1 = Boris Weisfeiler| last2 = A. A. | first2 = Lehman| year = 1968| title = Reduction of a graph to a canonical form and an algebra which appears in this process| journal = Scientific-Technological Investigations| volume = 9| series = 2| pages = 12–16| language = Russian}}{{cite journal |last1=Higman |first1=Donald G. |title=Coherent algebras |journal=Linear Algebra and Its Applications |date=August 1987 |volume=93 |page=209-239 |doi=10.1016/S0024-3795(87)90326-0 |doi-access=free |hdl=2027.42/26620 |hdl-access=free }}{{cite book|first=Peter J.|last=Cameron|authorlink=Peter Cameron (mathematician)|title = Permutation Groups|url=https://archive.org/details/permutationgroup0000came|url-access=registration|publisher = Cambridge University Press | series = London Mathematical Society Student Texts (45) | year = 1999 | isbn = 978-0-521-65378-7}}

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 an $R$ -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 $i^2 = \operatorname{id}_A$ .

:* 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.

= The more abstract definition =

Let $i:A\to A$ be an anti-automorphism of $R$ -algebras with $i^2 = \operatorname{id}$ (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

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

::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

$i(U_k)=U_k$
$J_k:=\bigoplus_{j=1}^k U_j$ 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.{{citation|title= On the structure of cellular algebras|first1= S.|last1= König|first2= C.C.|last2= Xi|journal= Algebras and Modules II. CMS Conference Proceedings|year=1996|pages=365–386}} 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 $i = \operatorname{id}$ 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

: $0 \subseteq (x^{n-1}) \subseteq (x^{n-2}) \subseteq \ldots \subseteq (x^1) \subseteq (x^0)=R[x]/(x^n)$

= 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 $T_w\mapsto T_{w^{-1}}$ .{{citation|title= Hecke algebras of finite type are cellular|first1= Meinolf|last1= Geck|journal= Inventiones Mathematicae|volume= 169|issue= 3|year= 2007|pages=501–517|doi=10.1007/s00222-007-0053-2|arxiv= math/0611941|bibcode= 2007InMat.169..501G|s2cid= 8111018}} 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 $\mathcal{O}$ of a semisimple Lie algebra.

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 $\lbrace C_\mathfrak{s} \mid \mathfrak{s} \in M(\lambda)\rbrace$ 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 $\Lambda_0:=\lbrace \lambda\in\Lambda \mid \phi_\lambda\neq 0\rbrace$ 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.

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 $A^{\text{op}}$ 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 a two-sided, $i$ -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 induced 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 two-sided, $i$ -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 $i$ -invariant 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 $K:=\operatorname{Quot}(R)$ 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:

$A$ 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:{{citation|title= Cellular algebras and quasi-hereditary algebras: A comparison|first1= S.|last1= König|first2= C.C.|last2= Xi|journal= Electronic Research Announcements of the American Mathematical Society|volume= 5|issue= 10|date= 1999-06-24|pages=71–75|doi= 10.1090/S1079-6762-99-00063-3|doi-access= free}}

$A$ 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. a suitable involution. In particular $A$ is cellular (to some involution) if and only if its basic algebra is.{{citation|title= Cellular algebras: inflations and Morita equivalences|first1= S.|last1= König|first2= C.C.|last2= Xi|journal= Journal of the London Mathematical Society|volume= 60|issue= 3|year= 1999|pages=700–722|doi=10.1112/s0024610799008212|citeseerx= 10.1.1.598.3299|s2cid= 1664006}}
Every idempotent $e\in A$ is equivalent to $i(e)$ , i.e. $Ae\cong Ai(e)$ . If $\operatorname{char}(R) \neq 2$ then in fact every equivalence class contains an $i$ -invariant idempotent.

References

Category:Algebras

Category:Representation theory