Cellularity and the Jones basic construction Cellularity definition - PowerPoint PPT Presentation

Cellularity and the Jones basic construction Frederick Goodman Introduction The Jones basic construction Cellularity and the Jones basic construction Cellularity definition Ponidicherry Conference, 2010 weaker, better definition basis free formulation The objects of our study Frederick Goodman Coherence of cellular structures The main theorem University of Iowa goodman@math.uiowa.edu Example – The BMW algebras Some idea of the proof

Cellularity and the Introduction Jones basic construction Frederick Goodman Introduction The Jones basic construction The goal of this work is to study certain finite dimensional Cellularity algebras that arise in invariant theory, knot theory, definition weaker, better definition subfactors, QFT, and statistical mechanics. basis free formulation The objects of our The algebras in question have parameters; for generic values study of the parameters, they are semisimple, but it is also Coherence of cellular structures interesting to study non–semisimple specializations. It turns The main theorem out that certain ideas from the semisimple world— Example – The specifically, the Jones basic construction — are still useful in BMW algebras Some idea of the the non–semisimple case. proof

Cellularity and the The Jones basic construction Jones basic construction Frederick Goodman There are two themes in this work: “Cellularity" and “the Introduction Jones basic construction." Cellularity means something The Jones basic specific, and it will be defined. construction Cellularity What the Jones basic construction means in the general definition weaker, better definition context we consider is not exactly clear. When it does make basis free formulation sense, the construction is a machine which, given a pair of The objects of our study algebras 1 ∈ A ⊆ B , will produce a third algebra J , with Coherence of A ⊆ B ⊆ J . cellular structures The main theorem If A and B are split semisimple over a field F , then it is clear Example – The BMW algebras what J should be, namely J = End ( B A ) . Some idea of the proof Suppose now that A and B are not only split s.s., but also that we have an F –valued trace on B , which is faithful on B and has faithful restriction to A . (Faithful means that the bilinear form ( x , y ) = tr ( xy ) is non–degenerate.)

Cellularity and the Basic construction, cont. Jones basic construction Frederick Goodman Introduction The Jones basic In the case just described, we have a unital A – A bimodule construction map ǫ : B → A determined by tr ( ba ) = tr ( ǫ ( b ) a ) for b ∈ B and Cellularity a ∈ A . Then we also have definition weaker, better definition basis free formulation End ( B A ) = B ǫ B ∼ = B ⊗ A B , The objects of our study Coherence of where the latter isomorphism is as B – B bimodules cellular structures The main theorem Now consider three successive BMW algebras Example – The A n − 1 ⊆ A n ⊆ A n + 1 . Then A n + 1 contains an essential BMW algebras Some idea of the idempotent e n , and we would like to understand the ideal proof J n + 1 = A n + 1 e n A n + 1 .

Cellularity and the Basic construction, slide 3 Jones basic construction Frederick Goodman Suppose (for the moment) that we work over � and the parameters of the BMW algebras are chosen generically. Introduction The Jones basic Then all the algebras in sight are s.s. and construction J n + 1 ∼ = End (( A n ) A n − 1 ) ∼ Cellularity = A n ⊗ A n − 1 A n . definition weaker, better definition basis free formulation That is, J n + 1 is the basic construction for A n − 1 ⊆ A n . The objects of our study Does some part of this persist in the non–s.s. case? Now, Coherence of cellular structures work over the generic integral ground ring for the BMW The main theorem algebras; i.e. work with the “integral form" of the BMW Example – The algebras. Now J n + 1 is no longer even a unital algebra and A n is BMW algebras not projective as an A n − 1 module. Nevertheless, it remains Some idea of the proof true that J n + 1 ∼ = A n ⊗ A n − 1 A n . This is what makes our treatment of cellularity for BMW algebras, and other similar algebras, work.

Cellularity and the Cellularity Jones basic construction Frederick Goodman Introduction The Jones basic construction Cellularity Cellularity is a concept due to Graham and Lehrer that is definition weaker, better definition useful in the study of a number of important algebras: Hecke basis free formulation algebras, Brauer and BMW algebras, Schur algebras, etc. The objects of our study I will give the definition, say what it is good for, and then Coherence of cellular structures make some general observations about cellularity, including The main theorem a suggested variant on the definition, and a new basis free Example – The formulation. BMW algebras Some idea of the proof

Cellularity and the What is cellularity? Jones basic construction Frederick Goodman Let A be an algebra with involution ∗ over an integral domain Introduction S . A is said to be cellular if there exists a finite partially The Jones basic ordered set (Λ , ≥ ) and for each λ ∈ Λ , a finite index set � ( λ ) , construction Cellularity such that definition ◮ A has an S –basis { c λ s , t : λ ∈ Λ ; s , t ∈ � ( λ ) } . weaker, better definition basis free formulation s , t ) ∗ = c λ ◮ ( c λ The objects of our t , s . study ◮ For a ∈ A , Coherence of � cellular structures a c λ s , t ≡ s r c λ r , t , The main theorem r Example – The modulo the span of basis elements c µ BMW algebras u , v , with µ > λ , Some idea of the where the coefficients in the expansion depend only on proof a and s and not on t . Such a basis is called a cellular basis, and the whole apparatus (Λ , ≥ , � ( λ ) , { c λ s , t } ) is called a cell datum.

Cellularity and the Cellularity – Example Jones basic construction Frederick Goodman Definition 1 Introduction The Hecke algebra H S n ( q 2 ) over a ring S is the quotient of the The Jones basic construction braid group algebra over S by the Hecke skein relation: Cellularity definition = ( q − q − 1 ) − weaker, better definition . basis free formulation The objects of our study Fact: Coherence of cellular structures The Hecke algebras H S n ( q 2 ) are cellular, with Λ = Y n , the set of The main theorem Young diagrams with n boxes, ordered by dominance , and Example – The � ( λ ) the set of standard tableaux of shape λ . The cell BMW algebras modules are known as Specht modules . The cellular structure Some idea of the proof is due to Murphy. See, for example, A. Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group , AMS University Lecture Series.

Cellularity and the What is cellularity?, cont. Jones basic construction Frederick Goodman Introduction It follows immediately from the definition of cellularity that The Jones basic ◮ For every order ideal Γ of Λ , construction Cellularity A (Γ) : = span { c λ s , t : λ ∈ Γ , s , t ∈ � ( λ ) } definition weaker, better definition basis free formulation is a ∗ –ideal of A . The objects of our study In particular, write A λ for A ( { µ : µ ≥ λ } ) and ˘ A λ for Coherence of cellular structures A ( { µ : µ > λ } ) . The main theorem ◮ For each λ ∈ Λ , there is an A –module ∆ λ , free as Example – The BMW algebras S –module, with basis { c λ t : t ∈ � ( λ ) } , such that the map Some idea of the A λ → ∆ λ ⊗ R (∆ λ ) ∗ defined by α λ : A λ / ˘ proof A λ �→ c s ⊗ ( c λ α λ : c λ t ) ∗ is an A – A bimodule s , t + ˘ isomorphism.

Cellularity and the What is this good for? Jones basic construction Frederick Goodman Introduction The Jones basic ◮ The modules ∆ λ are called cell modules . When the construction Cellularity ground ring is a field, and the algebra A is semisimple, definition weaker, better definition these are exactly the simple modules. basis free formulation ◮ In general, general, ∆ λ has a canonical bilinear form. The objects of our study With rad ( λ ) the radical of this form, and with the ground Coherence of ring a field, ∆ λ / rad ( λ ) is either zero or simple, and all cellular structures The main theorem simples are of this form. Example – The BMW algebras So cellularity gives an approach to finding all the simple Some idea of the modules. It’s also useful for investigating the block structure, proof etc.

Cellularity and the Proposed weakening of the definition Jones basic construction Frederick Goodman Introduction The Jones basic I propose weakening the definition of cellularity, replacing construction s , t ) ∗ = c λ ◮ ( c λ Cellularity t , s definition weaker, better definition by basis free formulation s , t ) ∗ ≡ c λ ◮ ( c λ t , s modulo ˘ A λ . The objects of our study Coherence of Disadvantages: none that I know of. cellular structures The main theorem All the results of Graham & Lehrer are still valid. Moreover, Example – The the weakened definition is equivalent to the original BMW algebras definition if 2 is invertible in the ground ring, so we haven’t Some idea of the proof lost much.