On the structure

of cellular algebras

by

Steen Konig and Changchang Xi

Intro duction

Recently Graham and Lehrer have intro duced a new class of alge

bras which they called cellular algebras These are dened by the existence

of a certain basis with very sp ecial prop erties motivated by the Kazhdan

Lusztig basis of Hecke algebras Graham and Lehrer have shown that various

imp ortant classes of algebras are cellular in particular ArikiKoikeHecke

algebras and Brauer centralizer algebras

The aim of this pap er is to study the structure of cellular algebras in

particular in the case of the ground ring being a eld this is assumed in

sections four to seven We start by shortly describing the two classes of

motivating examples just mentioned In section three we review some of the

results of Graham and Lehrer and giveanequivalent form of their denition

and some examples

In section four weinvestigate the cell ideals which are the building blo cks

of cellular algebras In particular we show that there are two dierent

sorts of such ideals one being familiar from and thus providing a close

connection to the theory of quasihereditary algebras In section ve we

show that the involution o ccuring in the denition of cellular algebras must

x isomorphism classes of simple mo dules This is a strong restriction as

we illustrate by the example of Brauer tree algebras

In section six we collect some homological prop erties for later use In

section seven we give an inductive construction of cellular algebras This

Mathematics Sub ject Classication Primary E G C G Sec

ondary E G F M R

Both authors have obtained supp ort from the Volkswagen Foundation Researchin

Pairs Programme of the Mathematical Research Institute Ob erwolfach

provides us with a metho d to pro duce many examples which in particular

show that there are no restrictions on the cell ideals eg with resp ect to

endomorphism rings which is a strong contrast to the situation for quasi

hereditary algebras We also give a b ound on the Lo ewy length of a cellular

algebra in terms of the number of cell ideals Moreover we determine the

global dimension of certain cellular algebras

This pap er is the rst in a series of three pap ers In the subsequentpaper

we dene integral cellular algebras This is a smaller class of algebras

still containing the examples of but with much stronger prop erties than

cellular algebras in general The pap er contains a new characterization

of cellular algebras which gives another inductive construction of cellular

algebras and which leads to a generalization of Ho chschild cohomology

Wewould like to thank Sebastian Oehms for a helpful discussion

Motivating examples

In of Lie algebras and algebraic groups several classes

of nite dimensional asso ciative algebras play imp ortantroles Most of them

are dened by generators and relations The motivation for studying cellular

algebras or related classes of algebras like quasihereditary algebras is

to understand these sp ecial classes of examples by putting them into an

axiomatic framework and thus to reveal hidden structure In this section we

give a list of those examples which at presentseemtobethemostinteresting

ones Details are given only for those examples which are not already covered

by the literature on quasihereditary algebras

Ariki Koike Hecke algebras

Hecke algebras of typ e A are wellknown deformations of the group algebra

of the for a survey see eg They have b een studied

and used intensively in many areas of mathematics For the other classical

Coxeter groups typ e B C and so on Hecke algebras have b een dened as

e been made to dene Hecke algebras well Recently several attempts hav

in a more general context for so called complex reection groups The so

called cyclotomic Hecke algebras are closely related to a system of results

and conjectures which has b een built around Broues conjecture on derived

equivalences for blo cks with ab elian defect group For more information

on this topic and more general on the generic approach to mo dular rep

resentation theory of nite groups of Lie typ e the reader is referred to

One large class of cyclotomic Hecke algebras has b een studied by Ariki and

Koike in a direct wayby deforming group algebras of wreath pro ducts

Denition Ariki and Koike Let R bethering Zq q u u

r

for some natural number r Then the Hecke algebra H cal led an Ariki

nr

Koike Hecke algebra is dened over R by the fol lowing generators and

relations

Generators t a a a

 n

Relations t u t u t u

 r



a q a q i n

i

i

ta ta a ta t

   

a a a a j i j j

i j j i

a a a a a a i n

i i i i i i

Sp ecializing r and u resp ectively r and u one gets

back the classical Hecke algebras of typ es A and B resp ectively

This Hecke algebra can be seen as deformation of the group algebra of

the wreath pro duct Zr Z o Ariki and Koike show in that H is

n nr

n

a free R mo dule of rank nr and they classify the simple representations

of the semisimple algebra fracR H A criterion deciding for which

R nr

values of q and the u s the sp ecialized Hecke algebra is semisimple is given

i

by Ariki An imp ortant application of ArikiKoike Hecke algebras is

Arikis pro of of a generalization of the LLTconjecture which gives an

explicit way of computing characters of Hecke algebras see and

Brauer algebras

These algebras not to be confused with Brauer tree algebras arise in the

representation theory of orthogonal and symplectic groups in the following

way whichishow Brauer came ab out studying them For analogy let

us rst recall a basic prop erty of the representation theory of the group

G GL k It acts naturally on the ndimensional k vector space V

n

r

hence also on the r fold tensor pro duct V The centralizer algebra of the

Gaction on the tensor pro duct that is the set of all k endomorphisms of

r r

V which satisfy gv gv for all v V is a quotient of and in

many cases equal to the group algebra of the symmetric group This

r

fact had been used by Schur for relating the representation theories in

characteristic zero of the general linear and the symmetric group Nowitis

natural to try what happ ens when one replaces GL k by one of its classical

n

semisimple subgroups say an orthogonal or a symplectic one which of course

r

still acts on the tensor space V Call the corresp onding centralizer algebra

D n where n is the number of the group in the series containing it

r

Now Brauer observed that this algebra is a quotientofandin many cases

isomorphic to another algebra say B xwherex is sp ecialised to a natural

r

number in case of the orthogonal group and to a negative integer in the

symplectic case which is dened as follows and which is now called the

Brauer algebra or the Brauer centralizer algebra

an indeterminate x and a natural number r Then B x Fix a eld k

r

has a basis consisting of all diagrams which consist of r vertices divided

into ordered sets the r top vertices and the r b ottom vertices and r

edges such that each edge b elongs to exactly vertices and each vertex

b elongs to exactly one edge Multiplication of basis elements is dened by

concatenating diagrams Assume we are given two basis elements say a and

b First draw an edge from b ottom vertex i of a to top vertex i of b for

each i r This pro duces a diagram which is almost of the desired

form except that there may b e cycles not attached to any of the new top

and b ottom vertices Denote the number of these cycles by d Then delete

all cycles the result is a basis element say c Now the pro duct ab is by

d

denition x c

Of course for a eld element say n the Brauer algebra B n is dened

r

by using n instead of x that is by forming the quotient of B x mo dulo

x n

details the reader is referred to Wenzls pap er For examples and more

Brauer algebras have b een studied extensively by Hanlon and Wales in

a series of long pap ers with many explicit results Based on

their computations they conjectured that B n is semisimple if n is not an

r

integer Wenzl proved this conjecture He also studied applications to

knot theory

Other examples

There are other algebras which also can be dened by diagrams and are

of use in knot theory We mention Temp erleyLieb algebras and Jones

annular algebras The algebras in b oth classes are smaller and thus more

op en to explicit computations than the Brauer algebras

Other imp ortant classes of examples which we want to cover are the

blo cks of the BernsteinGelfandGelfand category O of nite dimensional

semisimple complex Lie algebras and the generalized Schur algebras asso

ciated with rational or p olynomial representations of semisimple algebraic

groups in describing characteristic The reader is referred to for more

information on these two classes of algebras which are the main examples

of quasihereditary algebras

Denitions and basic prop erties

First we recall the original denition of Graham and Lehrer and explain

how they use it Then wegive an equivalent denition whichwe will use for

lo oking at the structure of cellular algebras

Denition Graham and Lehrer Let R be a

An associative R algebra A is cal led a cellular algebra with cel l datum

I M C i if the fol lowing conditions are satised

C The nite set I is partial ly ordered Associated with each I

where S T there is a nite set M The algebra A has an R basis C

ST

runs through al l elements of M M for al l I



C The map i is an R line ar antiautomorphism of A with i id

which sends C to C

ST TS

C For each I and S T M and each a A the product

P

 

aC can be written as r U S C r where r is a linear

a

ST U M  UT

combination of basis elements with upper index strictly smal ler than

coecients r U S R do not depend on T and where the

a



In the following an R linear antiautomorphism i of A with i id will

b e called an involution

Using this denition Graham and Lehrer could showthatthe following

examples mentioned in the previous section t into this context

Theorem Graham and Lehrer The fol lowing algebras arecel lu

lar

a ArikiKoike Hecke algebras

b Brauers algebras

c TemperleyLieb algebras

d Jones annular algebras

The pro ofs for the rst and the second class of algebras are quite tricky

and complicated and use nontrivial combinatorial to ols like Kazhdan

Lusztig basis and RobinsonSchensted algorithm the remaining classes of

examples are similar to sp ecializations of the Brauer algebras

Other examples of cellular algebras are discussed in

n

Let us give an easy example The algebra A k xx is cellular with

n n

cell chain x x x A and involution i id Note

however that it is not cellular with resp ect to the involution x x In

fact the cell chain would ha ve to be the same as b efore but one do es not

nd a basis which is xed under this involution see section for details

This indicates that the involution is not just an extra condition but there is

a quite subtle interplaybetween the involution and the other conditions

We remark that the niteness of the index set I is not required in the

denition in However it is assumed in most of their results and it is

satised in all of their examples Thus it seems to b e convenient to include

this condition in the denition and thus to exclude innite dimensional

algebras like k xwhichwould b e cellular otherwise We will recall b elow the

results of on classifying simple mo dules of a cellular algebra Roughly

they giv e a bijection between isomorphism classes of simple mo dules and

those cell ideals which satisfy an additional condition The example k x

shows that this cannot work without assuming I to be nite There are

however remarkable examples of quantum groups namely Lusztigs U see

part four of which are cellular with innite index set In the general

case of innite index set it seems natural also to lo ok at completions of

cellular algebras see

From the denition of cellular algebras it follows directly that they are

ltered bya chain of ideals

et A becel lular with cel l da Prop osition Graham and Lehrer L

tum I M C i Fixanindex I Then the R span of al l basis elements

C for is a twosided ideal in A

ST

The ideal in the prop osition will b e denoted by J If is minimal

this ideal is called a cell ideal Varying pro duces a chain of ideals called

a cell chain By J we denote the sum of the ideals J with

Prop osition Graham and Lehrer As a left module the quotient

J J is a direct sum of copies of a module which has basis

C where S runs and T is xed in particular the isomorphism class of

ST

this module does not depend on the choice of T

The mo dule in the prop osition will b e called standard mo dule since

we will see later on that this notion extends the familiar notion of standard

mo dules of a quasihereditary algebra In this mo dule is called a cell

mo dule Of course there is a dual version of the prop osition stating that as

J is a direct sum of copies of the mo dule i a right mo dule J

where the right action of A is via the involution i

Although J is a direct sum of copies of as a left mo dule it do es not

have to b e generated byany of these copies as a twosided ideal Let A be



the noncommutative algebra k a b c d r ad k a b c d with four

generators and radical square zero Dene i to send b to c and c to b and to

x a and d Then the radical J of A is the fourdimensional space with basis

a b c d In fact J is a cell ideal as one can checkbycho osing ka kb

hence i ka kc But because of radAJ JradA this cell

ideal cannot b e generated byany prop er subspace

n

The example k xx shows that standard mo dules iandj hav

ing dierent indices i j may be isomorphic Moreover there may be

nontrivial extensions b etween them

The main use Graham and Lehrer make of these notions is for construct

ing simple Amo dules For the ab ove classes of algebras this is usually a

hard problem see for partial solutions Since the co ecients in ex

pressing the pro duct C C as a linear combination of basis elements do

ST UV

not dep end on the indices S and V one can dene a bilinear form send

ing S V to the co ecient of C in this expression Graham and Lehrer

SV

show that in case of being nonzero the standard mo dule has a

unique simple quotient say L All simple Amo dules arise in this way

and for dierent one gets dierent simple mo dules Since the radical

of the standard mo dule equals the radical of the asso ciated bilinear form

the question of nding a basis of L is reduced to a problem of linear

doing explicit computations in each algebra so one has the p ossibilityof

individual case which of course do es not solve the problem of nding

general formulae eg for the dimensions of simple mo dules

Now we rephrase the denition of cellular algebras and show that the

new deniton is equivalent to the denition of

Denition Let A be an R algebra where R is a commutative Noethe

rian Assume there is an antiautomorphism i on A with



i id A twosided ideal J in A is cal led a cell ideal if and only if

iJ J and there exists a left ideal J such that is nitely gen

er ated and free over R and that there is an isomorphism of Abimodules

J i where i J is the iimage of making the

R

fol lowing diagram commutative

J i

R

x y iy ix i





i J

R

The algebra A with the involution i is cal led cellular if and only if

  

there is an R module decomposition A J J J for some n

n



j

  

with iJ J for each j and such that setting J J gives a chain

j

j j

l

l 

A each of them of twosided ideals of A J J J J

  n



xed by i and for each j j n the quotient J J J is a cel l

j j

j

ideal with respect to the involution inducedbyi on the quotient of AJ

j

From now on we always will assume that the ring R is a commutative

No etherian integral domain This is not necessary for the denitions How

ever all the examples of cellular algebras which we know of satisfy this

assumption Moreover the basic problems of the theory of cellular algebras

for example to compute decomp osition numb ers make this assumption nat

ural In particular the notion of rank will b e available

We note one immediate consequence of this denition A cellular al

gebra is nitely generated and free as an Rmo dule

The two denitions of cel lular algebras are equivalent Pro of Assume

that A is cellular in the sense of Graham and Lehrer Fix a minimal index

By C Dene J to be the R span of the basis elements C

ST

this is a twosided ideal By C it is xed by the involution i Fix any

index T Dene as remarked ab ove as the R span of C where S

ST

varies Dening by sending C to C iC gives the required

UV UT VT

isomorphism Thus J is a cell ideal Continuing by induction it follows

that A is cellular in the sense of the new denition

Conversely if a cell ideal say J in the sense of the new denition is given

wecho ose any R basis say fC g of and denote by C the inverse image

S ST

Since is a left mo dule C is satised C under of C iC

S T

follows from the required commutative diagram This nishes the pro of for

those basis elements o ccuring in a cell ideal Induction on the length of the

chain of ideals J provides us with a cellular basis of the quotient algebra

j

AJ Cho osing any preimages in A of these basis elements together with a

basis of J as ab ovewe pro duce a cellular basis of A

At this p oint one may ask the following questions Is there a universal

prop erty of the standard mo dules Are standard mo dules always inde

comp osable Do their endomorphism rings have sp ecial prop erties Later

on we will see that the answer to each of these questions is no in general

but yes in the examples in which one is interested as will b e shown in

Since the denition of cellular algebras uses induction one has to ask

which algebras provide the induction start

Prop osition Let A be an algebra with a cel l ideal J which is equal to

A Then A is isomorphic to a ful l matrix ring over the ground ring R

Pro of The assumption says that A has an involution i and can be

written as i for some left ideal Hence there is an R isomorphism

R

A Hom A A Hom iA Hom iHom A

A A R R A



Denote the R rank of the free R mo dule b y m Then A has R rank m

and as left mo dule A is isomorphic to m copies of Hence Hom A

A

whichisasubmoduleofthe R free mo dule Hom A hence torsionfree

R

But by the ab ove isomorphism it cannot have larger has R rank at least m

rank Thus the Aendomorphism ring E whichagain is torsionfree of

has rank one and contains R Now E is a subring of fracR which equals

End fracR which sends in to itself bymultiplication

R

fracR A

R

hence E is equal to R

The pro of shows that for any cellular algebra the standard mo dules

of maximal index are indecomp osable and have endomorphism ring R Of

course the full matrix ring over R is cellular so the converse of the prop osi

tion also is true We remark that the prop osition also follows from corollary

in Conversely our argument reproves this result

We nish this section by giving another class of examples

Prop osition Let A be a nite dimensional commutative algebraoveran

algebraical ly closed eld k Then A is cel lular with respect to the involution

i id

Pro of Since A is commutative its maximal semisimple quotient is

commutative as well Thus the simple Amo dules are onedimensional

The left so cle of A coincides of course with the right so cle thus any one

dimensional direct summand of it is a cell ideal Now we can pro ceed by

induction

Cell ideals

In this section we have a closer lo ok at cell ideals It turns out that they

are of two dierent kinds one of them b eing familiar from the theory of

quasihereditary algebras Part of this information phrased dierently is

contained in sections two and three of From now on we assume R

to be a eld

Prop osition Let A be an R algebra R k any eld with an invo

lution i and J a cel l ideal Then J satises one of the fol lowing mutual ly

exclusive conditions

A J has square zero

B There exists a primitive idempotent e in A such that J is generated



by e as a twosided ideal In particular J J Moreover eAe equals

Re R and multiplication in A provides an isomorphism of Abimodules

Ae eA J

R

Pro of By assumption J has an R basis C whose pro ducts satisfy

ST

the rule C If all the pro ducts C C are zero then we are in situation

ST UV

A Thus wemay assume that there is one such pro duct which is not zero

Since the co ecients do not dep end on S or V thepro duct C C also

UT UT

is not zero But by or a direct comparison of the twoways writing

this pro duct as a linear combination of basis elements using C and its

dual this pro duct is a scalar multiple of C Hence there is an idemp otent

UT

in J whichthus cannot b e nilp otent

So J contains a primitive idemp otent say eandAe is a left ideal which

is contained in J The cell ideal J asaleftAmo dule is a direct sum of copies

of a standard mo dule But Ae is a submo dule hence a direct summand

of the left ideal J Ae J e It follows that Ae is a direct summand of

whic hwe can decomp ose into Ae M for some Amo dule M Because of

m m

J i we can decomp ose J as left mo dule into Ae M where m

R

is the R dimension of whic h equals the R dimension of i Of course

m

Ae is contained in the trace X of Ae inside J that is the sum of all images

of homomorphisms Ae J This trace X is contained in the trace AeA

of Ae in A But the dimension of AeA is less than or equal to the pro duct

Ae with the dimension say n of eA The number n of the dimension of

equals the dimension of Aie whichby the same argument also is a direct

summand of Since m is less than or equal to n there must b e equality

i ieA hence also Ae and there must be equality in all of the

ab ove inequalities In particular J equals AeA and also Ae eA Since

R

multiplication Ae eA always is surjective it must be an isomorphism

R

As this multiplication factors over Ae eA it follows that eAe must be

eAe

equal to Re R

We note that here we need the assumption that R k is a eld

Case B just says that J is a heredity ideal generated by a primitive

idemp otent Conversely a heredityidealJ whichisxedbyaninvolution

i and generated by a primitive idemp otent e clearly is a cell ideal The

following corollary is already known see eg

Corollary Let A be a quasihereditary algebra with an involution i x

ing a complete set of primitive orthogonal idempotents Then A is cel lular

being idempotent ie of Conversely a cel lular algebra with al l cel l ideals

type B is quasihereditary

An ideal J xed by an involution i and generated by a primitive idempo

tent e xed by i is a cel l ideal if and only if it is a heredity ideal

There are two subtle p oints in this context whichwewould liketomen

tion A cell ideal which is a heredity ideal must b e generated by a primitive

idemp otent But it can happ en that no such idemp otent is xed by i More

over a heredity ideal which is xed by an involution i need not be a cell

ideal Examples which in fact use simple algebras are given in

Here a warning is necessary The indexing of cell ideals of a cellular

algebra unfortunately is precisely the converse of the usual indexing for

ideals in a hereditychain of a quasihereditary algebra so passing from one

theory to the other one should replace by

An alternative pro of and a generalization of this result has b een given

already in the recent preprint by Du and Rui A similar result for

integral quasihereditary algebras is given in The latter can b e used to

see that integral Schur algebras are cellular

In general a quasihereditary algebra do es not have to admit anyinvolu

tion But there is a large class of examples which are b oth quasihereditary

and cellular This includes in particular blo cks of category O and Schur

algebras More examples can b e found in

With a quasihereditary algebra A there come two series of smaller

quasihereditary algebras having the form AJ or eAe resp ectively Here

i

the idemp otents e have to be chosen in a sp ecial way which is prescrib ed

the partial order For cellular algebras there is a much larger class by

of idemp otents with eAe cellular

Prop osition Let A becel lular with respect to an involution i Let e be

an idempotent in A whichisxed by i Then eAe is cel lular with respect to

the restriction of i

Pro of Clearly i is an involution of eAe For a cell ideal J of A we



have to show that eJ e is a cell ideal of eAe Putting e hence

  

i ie one gets the isomorphism eJ e i

k

We note that this result do es not need the assumption of k b eing a eld

An application of this result is an alternative pro of that Hecke algebras

of typ e A in particular the group algebras of the symmetric groups are

cellular

Corollary Hecke algebras of type A are cel lular

n

Pro of Integral Schur algebras are quasihereditary and there is an

involution i xing the ideals in the usual heredity chain Now the Hecke

algebras of typ e A can b e written as eAe for A some integral Schur algebra

n

i and e some idemp otent xed by

The involution

The involution o ccuring in the denition of cellular algebras plays a crucial

role It is notjustan additional datum but it makes the other data in the

denition much more subtle In this section weshow in particular that the

involution necessarily xes isomorphism classes of simple mo dules As an

application we classify the cellular Brauer tree algebras This implies that

for the known classes of cellular algebras which are connected to nite group

theorylike the Hecke algebras only a sp ecial kind of Brauer tree can o ccur

In the case of Hecke algebras this reproves a result of Geck

A generalization of cellular algebras called standardly based obtained

by omitting the involution in the denition has been studied by Du and

Rui see the class of algebras obtained in this wayismuch larger than

the class of cellular algebras In fact any nite dimensional algebra A over

a p erfect eld is standardly based A minimal with resp ect to inclusion

op

twosided ideal I of A is a minimal A A submo dule of A Hence I is

k

op

simple as A A mo dule and thus I has the form I S T for some

k

simple left Amo dule S and some simple right Amo dule T

We start with an easy example showing that it dep ends on the choice

of the involution whether an algebra is cellular or not In fact for A



k xx k any eld of characteristic dierent from two we ma y consider

two involutions i id and i x x In the rst case A is clearly



cellular the cell chain b eing x A In the second case however A

is not cellular Assume to the contrarythatJ is a nonzero cell ideal with

resp ect to i Then the dimension of J has to be a square thus it equals



one So J must b e the ideal generated by x and equals J and also i



But the square

J i

R

y z iz iy i





i J

R

cannot be made commutative by any since the left hand vertical map

sends x to x whereas the righthandvertical map sends a generator y z

to itself

Prop osition Let A bea cel lular algebra with involution i and e a prim

itive idempotent of A Then ie is equivalent to e

Pro of See for a similar argument Fix a chain of cell ideals

J J A and assume that e lies in J but not in J Since

n j j

the ideals in a cell chain are xed by i ie also lies in J but not in J

j j

hence we can pass to a cellular quotient algebra of A and assume that e and

ie b oth are contained in a cell ideal say J which has the form i

R

Since J contains an idemp otent it is a heredity ideal and equals Ae by

prop osition But Aie is a direct summand of J hence also of which

implies that Ae and Aiemust b e isomorphic

Corollary Let A bedirected that is the isomorphism classes of simple

Amodules can be ordered in such a way that Ext LiLj implies

A

i j Then A is cel lular for some involution if and only if it is split

semisimple

Pro of We know already that split semisimple algebras are cellular for

any involution xing simples Conversely let A be cellular with resp ect

to some involution say i Then i xes simple mo dules hence for any two



simple mo dules L and L there are extensions in one direction if and only

there are extensions in the other direction By the denition of directed

ness extensions in one direction are zero Hence there are no nontrivial

extensions at all and A is semisimple Thus A is a direct sum of its cell

ideals hence it is split semisimple

As an application we consider the following question When is a Brauer

tree algebra cellular For denition and prop erties of Brauer tree algebras

we refer to chapter ve

Prop osition ABrauer tree algebra with an arbitrary number of excep

tional vertices is cel lular if and only if the Brauer treeisastraight line that

is at each vertex thereare at most two edges with arbitrary multiplicities

Pro of If there is a vertex with more than two edges meeting there

then by denition of Brauer tree algebra there are simple mo dules L

L L corresp onding to these edges for some m such that

 m

Ext L L to b e read cyclically and all other rst extensions b e

j j 

tween two of these simples v anish On the other hand b eing cellular implies

the existence of an involution i xing isomorphism classes of simple mo dules

and hence providing vector space isomorphisms b etween Ext L L and

j j 

Ext L L whichisa contradiction

j  j

Thus it remains to show that in the case of a straight line the algebra

is cellular This can be done easily using the wellknown description by

quiver and relations We describ e the pro cedure in the basic case The

involution i xes the vertices and reverses the arrows The cell chain can

be chosen in such a way that J is one dimensional the unique so cle of

a pro jective mo dule P which b elongs to an edge with one neighbour

only Then J J is fourdimensional and intersects nontrivially with P



and its neighbour P If the vertex has multiplicity one then J J is



a heredity ideal Otherwise one continues to factor out fourdimensional

ideals containing comp osition factors L and L until the idemp otentat

has b een factored out Then one continues inductively

It follows that for example Brauer trees of ArikiKoike Hecke algebras

are straight lines this has b een proved already by Geck also using an

involution argumenthowever his result is more general

In we will see that for the examples we are interested in that is

the integral cellular algebras the involution i has a rather sp ecial shap e In

general however it seems to b e an op en problem to nd all involutions on

agiven nite dimensional algebra

Homological prop erties

In prop osition we have seen that over a eld for a cell ideal J there

are precisely two p ossibilities Either it is a heredity ideal or it has square

the rst case the homological prop erties are well understo o d for zero In

example there is a homological epimorphism A AJ andAcohomology

can to a large extent be read o from AJ see In the second

case this is not true cohomology of A and of AJ can b e rather dierentas



one can see already from the example k xx In this section we give a

list of homological prop erties of J in the second case in particular weshow

how the cell basis can b e read o naturally from certain Ext or Tor groups

We restrict to algebras over a eld in this section in order to have the ab ove

dichotomyavailable

We start by recalling an exercise of CartanEilenb erg VI If

A is a ring and I is a right ideal and J is a left ideal then the following

isomorphisms are easily shown by using long exact sequences

A

Tor AI AJ I J I J

mul t

A

Tor AI AJ kernelI J IJ

A



A A

Tor AI AJ Tor I J n

n n

Prop osition For any ideal J in a k algebra A the fol lowing two as

sertions are equivalent



I J

A

II Tor AJ AJ J J

A



Pro of Applying J to the exact sequence J A AJ

A

pro duces the exact sequence

A

Tor J AJ J J J A J AJ

A A A

 

Now the last term equals JJ Hence if J the rst two terms must

be isomorphic Dimension shift then proves that I I is valid Conversely

again by dimension shift condition I I implies that the rst two terms

hence also the last two terms are isomorphic thus I is valid

Now assume A has a nilp otent cell ideal say J which is isomorphic to

i where i is an involution on A and is a left ideal inside J

k

Since J is isomorphic to i we get an isomorphism of k vector

k

spaces J J i i thus the Tor space in the previous

A A

k k

prop osition will b e quite large provided i is not zero But the latter

A

space is the k dual of Hom Hom ik which is nonzero since it

A

k

contains the map top socl eHom ik Hom ik

k k

Corollary Let J be anilpotent cel l ideal in the k algebra A Then the

A

space Tor AJ AJ is not zero



This has the following consequence If A has a cell chain whichcontains

nilp otent cell ideals then this chain of ideals do es not give a recollement

of derived mo dule categories and the chain of ideals do es not make A into

a stratied algebra in the sense of Cline Parshall and Scott Hence

their notion of stratied algebra is quite dierent from the notion of cellu

lar algebra although b oth classes of algebras contain the quasihereditary

algebras

Another consequence of the exercise in CartanEilenb erg is the following

here we do not need that k is a eld it is enough to assume that k is a

no etherian integral domain

If J can b e written as i and m is the k rank of then there are

k

m linearly indep endentemb eddings of in to J and similarly for i So

J can b e written as a direct sum J i where j and l run from

j

l

A

to m each But i is isomorphic to Tor AiA whichin

particular must have rank one over k Thus the choice of the cell basis

can be seen as cho osing a decomp osition of the ab ove Torspace

The involution i on A induces an antiequivalence A mod mod A

Its eect on standard mo dules can b e describ ed explicitly

Prop osition Let A be cel lular with respect to an involution i and de

note by the antiequivalence A mod mod A induced by i Then

for a standard module of minimal index there is an isomorphism of left

Amodules A Hom Aik

k

Pro of We rst recall the denition of the mo dule structures on the

two mo dules On Hom Aik an element a A acts by sending a

k

linear form to the form whichmapsx to xa On A whichas a

vector space coincides with Hom Ak the action is by sending f to

k

af x f iax

Now the isomorphism is straightforward Send f to x f ix

f

An inductive construction

For a quasihereditary algebra there are two inductive constructions due

to Parshall and Scott and to Dlab and Ringel One should note

that there are to o many quasihereditary algebras so one cannot use these

constructions in order to classify them all but one can inductively check

prop erties or pro duce examples in this way

The rst of these constructions takes as input an algebra B with

n simples and pro duces as output an algebra A with n simples and a

heredity ideal J of A such that AJ B One can use this construction as

well in case A of our situation ie for heredity ideals one just has to

add the existence of an involution as an additional condition

The aim of this section is to lo ok at an inductive construction similar

to that of whichworks in the second case of a nilp otent cell ideal In

particular we will use this construction for pro ducing examples which give

negative answers to some of the questions asked in the third section As

positive results we get a statement ab out global dimensions and b ounds for

the Lo ewy length of a cellular algebra

Prop osition Let B be a cel lular algebra with an involution i Let

be any B module Let be the image of under the antiequivalence

B mod mod B denedbyi thus as vector space equals Denote

Dene i on J by sending x y to y x Pick by J the B bimodule

R



a Hochschild cocycle H B J which satises in addition the equation

ixiy iy x for al l x y B

   

Then A J B with multiplication denedbyj bj b jb j b

 

b b bb is a cel lular algebra with nilpotent cel l ideal J

Conversely any cel lular algebrawithanilpotent cel l ideal can be written

in this form

Pro of First we assume that A is a cellular algebra with a nilp otent



cell ideal J Then J implies that the Amo dule structure of J factors

over the quotient algebra B AJ Hence Ho chschild cohomology can be

applied It is easy to check that the involution i on A imp oses the ab ove

condition on the Ho chschild co cycle

Conversely it is wellknown that the ab ove data dene an asso ciative

R algebra structure on A and that J is an ideal Since i is dened both

on B and on J it is dened on A as well and the condition on the co cycle

implies that i actually is an in volutory antiautomorphism of A Clearly J is

an ideal of A Its Abimo dule structure is given byitsB mo dule structure

thus it is a cell ideal

An easy sp ecial case of Ho chschild extensions are trivial extensions where

one cho oses the co cycle to be zero One should not exp ect our examples

to be trivial extensions in general But trivial extensions can be used to

construct cellular algebras whichprovide negative answers to the questions

mentioned after denition In the theorem can be chosen arbitrary

As a B mo dule it is not distinguished byany prop erty Since its Amo dule

structure coincides with its B mo dule structure this implies that do es

not have to be indecomp osable Moreover there is no restriction on its

endomorphism ring on its comp osition factors and so on

We will see in that the situation is much b etter if we restrict to

a smaller class of algebras still containing all the examples provided by

Graham and Lehrer

Nowwe apply the inductive construction to get more information on the

ring structure of cellular algebras For an algebra A we denote by LLA

its Lo ewy length

Prop osition Let A be a cel lular algebra with a nilpotent cel l ideal J

and B AJ the quotient algebra Then the fol lowing assertions hold true

a radAradB J

b there is a commutative diagram

 

X rad A rad B

 



J radA radB

 



 

Y radAr ad A radB r ad B

   

where X J rad AradB J JradB fb b bb radB bb



g and Y rad B J radB radB radB J JradB with

A

indicating a nondirect sum and indicating multiplication in A In

A

particular the quiver of A c ontains the quiver of B as a subquiver and the

additional arrows correspond to a basis of Y

c there is an inequality LLA LLB

Pro of Since J is nilp otent it is contained in radA this implies a

 

Multiplying two elements in radAgives a pro duct of the form j b bj

 

b b bb which implies exactness of the rst row in the diagram in b

The rest of the diagram is then clear Statement c follows since J is a

B mo dule

In the case of heredity ideals there is a similar b ound as in c see

thus we get

Corollary Let A be a cel lular algebra with a cel l chain consisting of n

n

ideals Then the Loewy length of A is bounded above by

Quasihereditary algebras alway have nite global dimension This is

not true for cellular algebras for example there are many lo cal cellular

algebras which are not simple

Corollary If A is a trivial extension of an algebra B by a nilpotent cel l

ideal J then the global dimension of A is innite

the decomp osition A J B is even a Pro of For a trivial extension

decomp osition of B bimo dules which induces B bimo dule decomp ositions

 

radA J radB and rad A rad B JradB radB J

 

Thus radAr ad A decomp oses into radB r ad B JradB J

JradB Since J is xed by the involution i the spaceJradB J

JradB is xed by i as well In particular there is an idemp otent e in A

and a nonzero element say x in this space such that ex xe Hence

the quiver of A contains a lo op and by the global dimension of A is

innite

The examples we have checked seem to indicate that there is a much

larger class of cellular algebras having innite global dimension although

not necessarily containing lo ops in their quiver moreover the results of

seem to be applicable in a more general context We note that the

ab ove result also follows from a general criterion deciding when a trivial

extension has nite global dimension We also note that typical examples

of cellular alegbras like the group algebras of symmetric groups are self

injective hence either semisimple or of innite global dimension

Problem When do es an extension of an algebra B byanilp otent cell

ideal J have innite global dimension In particular When do es a cellular

algebra have innite global dimension

A dierent inductive construction of cellular algebras not distinguishing

between the two sorts of cell ideals and valid over any ground ring will b e

given in

References

AAAlbert Structure of algebras AMS

JLAlperin Lo cal representation theoryCambridge University Press

SAriki On the semisimplicity of the Hecke algebra of Zr Z o J Alg

n

SAriki On the decomp osition numb ers of the Hecke algebra of Gm n Preprint

SAriki and KKoike AHecke algebra of Zr Z o and construction of its irre

n

ducible representations Adv Math

RBrauer On algebras which are connected with the semisimple continous groups

Annals of Math

M Br oue and G Malle Zyklotomische Heckealgebren Asterisque

HCartan and SEilenberg Homological algebra Princeton University Press

ECline BParshall and LScott Stratifying endomorphism algebras Memoir

AMS

CWCurtis Representations of Hecke algebras Asterisque

BMDeng and CCXi Quasihereditary algebras whicharetwisted double inci

dence algebras of p osets Contrib Alg Geom

VDlab and CMRingel Quasihereditary algebras Illinois JMath

VDlab and CMRingel A construction of quasihereditary algebras Comp os

Math

JDu and HRui Based algebras and standard bases for quasihereditary algebras

Preprint

MGeck Brauer trees of Hecke algebras Comm Alg

JJGraham Mo dular representations of Hecke algebras and related algebras PhD

Thesis Sydney

JJGraham and GILehrer Cellular algebras InventMath

RMGreen On representations of ane Temp erleyLieb algebras Preprint

RMGreen Completions of cellular algebras Preprint

PHanlon and DWales On the decomp osition of Brauers centralizer algebras J

Alg

PHanlon and DWales Eigenvalues connected with Brauers centralizer algebras

J Alg

PHanlon and DWales Atower construction for the radical in Brauers centralizer

algebras J Alg

DKazhdan and GLusztig Representations of Coxeter groups and Hecke algebras

InventMath

SKonig A criterion for quasihereditary and an abstract straightening formula

Invent Math

SKonig and CCXi Integral cellular algebras In preparation

SKonig and CCXi Cellular algebras inations and Morita equivalences Preprint

ALascoux BLeclerc and JYThibon Hecke algebras at ro ots of unityand

crystal bases of quantum ane algebras Comm Math Phys

HLenzing Nilp otente Elemente in Ringen von endlicher globaler Dimension Math

Z

CLofwall The global homological dimensions of trivial extension rings J Alg

GLusztig Intro duction to quantum groups Progress in Math Birkhauser

SOehms Symplectic q Schur algebras In preparation

BParshall and LScott Derived categories quasihereditary algebras and al

gebraic groups Pro c of the OttawaMo osonee Workshop in Algebra Math

Lecture Note Series Carleton University and Universite dOttawa

HWenzl On the structure of Brauers centralizer algebras Annals of Math

HWenzl Quantum groups and subfactors of typ e B C and D Comm Math

Physics

Steffen Konig Fakultat fur Mathematik Universitat Bielefeld Post

fach D Bielefeld Germany ko enigmathematikunibielefeldde

Changchang Xi Department of Mathematics Beijing Normal University

Beijing PR China xiccbnueducn