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.

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

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

The concrete definition

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

• A finite partially ordered set ${\displaystyle \Lambda }$.
• A ${\displaystyle R}$-linear anti-automorphism ${\displaystyle i:A\to A}$ with ${\displaystyle i^{2}=id_{A}}$.
• For every ${\displaystyle \lambda \in \Lambda }$ a non-empty, finite set ${\displaystyle M(\lambda )}$ 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 ${\displaystyle \lambda \in \Lambda }$ 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 ${\displaystyle C}$ is a ${\displaystyle R}$-basis of ${\displaystyle A}$.
2. ${\displaystyle i(C_{\mathfrak {st}}^{\lambda })=C_{\mathfrak {ts}}^{\lambda }}$ for all elements of the basis.
3. For every ${\displaystyle \lambda \in \Lambda }$, ${\displaystyle {\mathfrak {s}},{\mathfrak {t}}\in M(\lambda )}$ and every ${\displaystyle a\in A}$ 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 ${\displaystyle a}$,${\displaystyle {\mathfrak {u}}}$ and ${\displaystyle {\mathfrak {s}}}$ but not on ${\displaystyle {\mathfrak {t}}}$. Here ${\displaystyle A(<\lambda )}$ denotes the ${\displaystyle R}$-span of all basis elements with upper index strictly smaller than ${\displaystyle \lambda }$.

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 ${\displaystyle R}$-algebras with ${\displaystyle i^{2}=id}$ (just called "involution" from now on).

A cell ideal of ${\displaystyle A}$ w.r.t. ${\displaystyle i}$ is a two-sided ideal ${\displaystyle J\subseteq A}$ 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 ${\displaystyle R}$-module and an isomorphism

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

of ${\displaystyle A}$-${\displaystyle A}$-bimodules such that ${\displaystyle \alpha }$ and ${\displaystyle i}$ 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 ${\displaystyle A}$ w.r.t. ${\displaystyle i}$ is defined as a direct decomposition

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

into free ${\displaystyle R}$-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 ${\displaystyle A}$
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 ${\displaystyle \Lambda }$) 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 ${\displaystyle A}$.

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

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

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 ${\displaystyle \Lambda }$.

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

Representations

Cell modules and the invariant bilinear form

Assume ${\displaystyle A}$ is cellular and ${\displaystyle (\Lambda ,i,M,C)}$ is a cell datum for ${\displaystyle A}$. Then one defines the cell module ${\displaystyle W(\lambda )}$ as the free ${\displaystyle R}$-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 ${\displaystyle A}$-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 ${\displaystyle A}$-invariant in the sense that

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

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

Simple modules

Assume for the rest of this section that the ring ${\displaystyle R}$ is a field. With the information contained in the invariant bilinear forms one can easily list all simple ${\displaystyle A}$-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 ${\displaystyle L(\lambda )}$ are absolute simple ${\displaystyle A}$-modules and every simple ${\displaystyle A}$-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 ${\displaystyle R}$-algebras are cellular.
• A ${\displaystyle R}$-algebra ${\displaystyle A}$ is cellular if and only if its opposite algebra ${\displaystyle A^{op}}$ is.
• If ${\displaystyle A}$ 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 ${\displaystyle \Lambda }$ then ${\displaystyle A(\Phi ):=\sum RC_{\mathfrak {st}}^{\lambda }}$ (where the sum runs over ${\displaystyle \lambda \in \Lambda }$ and ${\displaystyle s,t\in M(\lambda )}$) is an twosided, ${\displaystyle i}$-invariant ideal of ${\displaystyle A}$ 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 ${\displaystyle A}$ is a cellular ${\displaystyle R}$-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 ${\displaystyle S}$-algebra.
• Direct products of finitely many cellular ${\displaystyle R}$-algebras are cellular.

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

• If ${\displaystyle (A,i)}$ is a finite dimensional ${\displaystyle R}$-algebra with an involution and ${\displaystyle A=A_{1}\oplus A_{2}}$ a decomposition in twosided, ${\displaystyle i}$-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 ${\displaystyle A}$ are ${\displaystyle i}$-invariant if ${\displaystyle (A,i)}$ is cellular, an immediate corollary is that a finite dimensional ${\displaystyle R}$-algebra is cellular w.r.t. ${\displaystyle i}$ if and only if all blocks are ${\displaystyle i}$-invariant and cellular w.r.t. ${\displaystyle i}$.
• Tits' deformation theorem for cellular algebras: Let ${\displaystyle A}$ be a cellular ${\displaystyle R}$-algebra. Also let ${\displaystyle R\to k}$ be a unitary homomorphism into a field ${\displaystyle k}$ and ${\displaystyle K:=Quot(R)}$ the quotient field of ${\displaystyle R}$. Then the following holds: If ${\displaystyle kA}$ is semisimple, then ${\displaystyle KA}$ is also semisimple.

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

• If ${\displaystyle A}$ is cellular w.r.t. ${\displaystyle i}$ 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 ${\displaystyle R}$ 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 ${\displaystyle A}$ is cellular w.r.t. to the involution ${\displaystyle i}$. Then the following hold

• ${\displaystyle A}$ is split, i.e. all simple modules are absolutely irreducible.
• The following are equivalent:[1]

1. ${\displaystyle A}$ is semisimple.
2. ${\displaystyle A}$ 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 ${\displaystyle C_{A}}$ of ${\displaystyle A}$ is symmetric and positive definite.
• The following are equivalent:[6]

1. ${\displaystyle A}$ 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 ${\displaystyle A}$ is cellular.
5. ${\displaystyle \det(C_{A})=1}$.

• If ${\displaystyle A}$ is Morita equivalent to ${\displaystyle B}$ and the characteristic of ${\displaystyle R}$ is not two, then ${\displaystyle B}$ is also cellular w.r.t. an suitable involution. In particular is ${\displaystyle A}$ 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 ${\displaystyle i}$-invariant idempotent.[4]

References

1. Graham, J.J; Lehrer, G.I. (1996), "Cellular algebras", Inventiones Mathematicae, 123: 1–34, Bibcode:1996InMat.123....1G, 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 978-0-521-65378-7.
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 (3): 501–517, arXiv:math/0611941, Bibcode:2007InMat.169..501G, 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 (10): 71–75, doi:10.1090/S1079-6762-99-00063-3
7. König, S.; Xi, C.C. (1999), "Cellular algebras: inflations and Morita equivalences", Journal of the London Mathematical Society, 60 (3): 700–722, CiteSeerX 10.1.1.598.3299, doi:10.1112/s0024610799008212