Cellular algebra

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 ${\displaystyle i^{2}=\operatorname {id} _{A}}$.
• 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]

Let ${\displaystyle i:A\to A}$ be an anti-automorphism of ${\displaystyle R}$-algebras with ${\displaystyle i^{2}=\operatorname {id} }$ (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.[5] 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 ${\displaystyle A}$.

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

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

${\displaystyle R^{\,d\times d}}$ 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 ${\displaystyle \Lambda }$.

Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t. to the involution that maps the standard basis as ${\displaystyle T_{w}\mapsto T_{w^{-1}}}$.[6] 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).[5]

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.[5]

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 ${\displaystyle \lbrace C_{\mathfrak {s}}\mid {\mathfrak {s}}\in M(\lambda )\rbrace }$ 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 ${\displaystyle a\in A}$,${\displaystyle x,y\in W(\lambda )}$.

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 ${\displaystyle \Lambda _{0}:=\lbrace \lambda \in \Lambda \mid \phi _{\lambda }\neq 0\rbrace }$ 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]

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

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:[7]

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. a suitable involution. In particular ${\displaystyle A}$ is cellular (to some involution) if and only if its basic algebra is.[8]
• Every idempotent ${\displaystyle e\in A}$ is equivalent to ${\displaystyle i(e)}$, i.e. ${\displaystyle Ae\cong Ai(e)}$. If ${\displaystyle \operatorname {char} (R)\neq 2}$ then in fact every equivalence class contains an ${\displaystyle i}$-invariant idempotent.[5]