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On the Structure of Cellular Algebras 1 Introduction

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On the Structure of Cellular Algebras 1 Introduction 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 representation theory 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 symmetric group 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 commutative ring 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
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