User:AndrewReeves/Cellular algebra (draft article)

In algebraic representation theory, a cellular algebra is an algebra which has a basis with certain combinatorial properties. The existence of such a basis leads to a natural characterisation of the simple modules of the algebra. Cellular algebras are often quasi-hereditary, though this is not always the case.

The first definition of a cellular algebra was given in 1996 by J.J. Graham and G. I. Lehrer [1]. Later, S. König and C.C. Xi gave an alternative (but equivalent) definition[2]. In different applications, either one of the two definitions may turn out to be the more useful; the original definition is more combinatorial, the second is stated more in terms of abstract ring theory.

Examples of cellular algebras include the group algebra of the symmetric group, the Brauer algebra, all Hecke algebras of finite type[3], the Temperley-Lieb algebra and the Birman-Murakami-Wenzl algebra.

Definition

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The original definition given by Graham and Lehrer is as follow:.

Let   be a commutative ring and let   be an  -algebra free over  .   is cellular if there exists a (not necessarily unique) cell datum consisting of

  • A poset  ;
  • For each   a set  ;
  • An injection  ;

such that the image of   is an  -basis   of  , the map   extends  -linearly to an antiautomorphism of   and for any   and every   there exist   such that

 

where   denotes a linear combination of basis elements   with  

Representation Theory

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Cell Modules

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Suppose that   is a cellular  -algebra with cell datum ( , , ). Using the same notation as above, we can define for any   a cell module  . This is the module with  -basis   and algebra action defined by

 

for all   and  

For any cell module   there is an  -bilinear form   given by  . Graham and Lehrer[1] showed that this form is symmetric and invariant under the action of  . Based on this they also showed how to extract all the simple modules of   from the cell modules.

Simple Modules

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Suppose that   is a field and that   is finite (which implies that   is a finite dimensional algebra).

Define the radical  . This is a submodule of the cell module  , so the quotient   is well-defined. It is trivial only when   is identically zero.

Let   be the set of all   such that   is not identically zero. Then a theorem of Graham and Lehrer[1] states that

  •   is absolutely irreducible for any  ;
  •   is a complete set of representatives of the distinct isomorphism classes of simple  -modules;
  • The cell module   is simple if and only if   is non-degenerate on  ;
  • The following statements are equivalent:
    •   is a semisimple algebra;
    • Every cell module of   is absolutely irreducible;
    • The forms   are non-degenerate for every  ;

An Alternative Characterisation

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When Is A Cellular Algebra Quasi-Hereditary?

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Further Reading

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  • Deng, B.; Du, J.; Parshall, B; Wang, J. (2008), Finite Dimensional Algebras and Quantum Groups, American Mathematical Society, pp. 699–726, ISBN 978-0821841860
  • Mathas, Andrew (1999), Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group, University Lecture Series, American Mathematical Society, pp. 15–26, ISBN 978-0821819265

References

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  1. ^ a b c Graham, J.J; Lehrer, G.I. (1996), "Cellular algebras", Inventiones Mathematicae, 123: 1–34
  2. ^ König, S.; Xi, C.C. (1996), "On the structure of cellular algebras", Algebras and modules II. CMS Conference Proceedings: 365–386
  3. ^ Geck, Meinolf (2007), "Hecke algebras of finite type are cellular", Inventiones mathematicae, 169: 501–517