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 ${\displaystyle (\Lambda ,i,M,C)}$ consisting of

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

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

The images under this map are notated with an upper index $\lambda\in\Lambda$ and two lower indices ${\displaystyle {\mathfrak {s}},{\mathfrak {t}}\in M(\lambda )}$ so that the typical element of the image is written as ${\displaystyle C_{\mathfrak {st}}^{\lambda }}$.

and satisfying the following conditions:

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

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

with coefficients ${\displaystyle r_{a}({\mathfrak {u}},{\mathfrak {s}})\in R}$ depending only on $a$,${\mathfrak {u}}$ and ${\mathfrak {s}}$ but not on $\mathfrak{t}$. Here ${\displaystyle A(<\lambda )}$ 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 ${\displaystyle i:A\to A}$ be an anti automorphism of $R$-algebras with ${\displaystyle i^{2}=id}$ (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. ${\displaystyle i(J)=J}$.
2. There is a left ideal ${\displaystyle \Delta \subseteq J}$ that is free as a $R$-module and an isomorphism

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

of $A$-$A$-bimodules such that $\alpha$ and $i$ are compatible in the sense that

${\displaystyle \forall x,y\in \Delta :i(\alpha (x\otimes i(y)))=\alpha (y\otimes i(x))}$

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

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

into free $R$-submodules such that

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

Now ${\displaystyle (A,i)}$ 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 ${\displaystyle \Delta /J_{k-1}\subseteq J_{k}/J_{k-1}}$ one can construct a corresponding cell basis for $A$.

Examples

Polynomial examples

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

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

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

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

Matrix examples

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

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

${\displaystyle 0\subseteq R^{d\times d}}$

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 ${\displaystyle T_{w}\mapsto T_{w^{-1}}}$.^[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 ${\displaystyle (\Lambda ,i,M,C)}$ is a cell datum for $A$. Then one defines the cell module ${\displaystyle W(\lambda )}$ as the free $R$-module with basis ${\displaystyle \lbrace C_{\mathfrak {s}}|{\mathfrak {s}}\in M(\lambda )\rbrace }$ and multiplication

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

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

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

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

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

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

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

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

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

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 ${\displaystyle \Lambda _{0}:=\lbrace \lambda \in \Lambda |\phi _{\lambda }\neq 0\rbrace }$ and define ${\displaystyle L(\lambda ):=W(\lambda )/\operatorname {rad} (\phi _{\lambda })}$ for all ${\displaystyle \lambda \in \Lambda _{0}}$. 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 ${\displaystyle A^{op}}$ is.
• If $A$ is cellular with cell-datum ${\displaystyle (\Lambda ,i,M,C)}$ and ${\displaystyle \Phi \subseteq \Lambda }$ is an ideal (a downward closed subset) of the poset $\Lambda$ then ${\displaystyle A(\Phi ):=\sum RC_{\mathfrak {st}}^{\lambda }}$ (where the sum runs over $\lambda\in\Lambda$ and ${\displaystyle s,t\in M(\lambda )}$) is an twosided, $i$-invariant ideal of $A$ and the quotient ${\displaystyle A/A(\Phi )}$ is cellular with cell datum ${\displaystyle (\Lambda \setminus \Phi ,i,M,C)}$ (where i denotes the induces involution and M,C denote the restricted mappings).
• If $A$ is a cellular $R$-algebra and ${\displaystyle R\to S}$ is a unitary homomorphism of commutative rings, then the extension of scalars ${\displaystyle S\otimes _{R}A}$ 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 ${\displaystyle (A,i)}$ is a finite dimensional $R$-algebra with an involution and ${\displaystyle A=A_{1}\oplus A_{2}}$ a decomposition in twosided, $i$-invariant ideals, then the following are equivalent:

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

• Since in particular all blocks of $A$ are $i$-invariant if ${\displaystyle (A,i)}$ 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 ${\displaystyle R\to k}$ be a unitary homomorphism into a field $k$ and ${\displaystyle K:=Quot(R)}$ the quotient field of $R$. Then the following holds: If ${\displaystyle kA}$ is semisimple, then ${\displaystyle KA}$ 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 ${\displaystyle e\in A}$ is an idempotent such that ${\displaystyle i(e)=e}$, then the Algebra ${\displaystyle eAe}$ 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. ${\displaystyle \forall \lambda \in \Lambda :W(\lambda )}$ is simple.
4. ${\displaystyle \forall \lambda \in \Lambda :\phi _{\lambda }}$ 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. ${\displaystyle \Lambda =\Lambda _{0}}$.
3. All cell chains of ${\displaystyle (A,i)}$ have the same length.
4. All cell chains of ${\displaystyle (A,j)}$ have the same length where ${\displaystyle j:A\to A}$ is an arbitrary involution w.r.t. which $A$ is cellular.
5. ${\displaystyle \det(C_{A})=1}$.

• 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 ${\displaystyle e\in A}$ is equivalent to ${\displaystyle i(e)}$, i.e. ${\displaystyle Ae\cong Ai(e)}$. If ${\displaystyle char(R)\neq 2}$ 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