Brauer algebras of Dynkin type Arjeh Cohen research reported on is - PowerPoint PPT Presentation

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Brauer algebras of Dynkin type Arjeh Cohen research reported on is joint work with David Wales, Shona Yu, Di´ e Gijsbers, and Shoumin Liu 9 September 2013, Universit¨ at Stuttgart

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Outline Motivation 1 Definitions 2 Simply laced types 3 Non-simply laced Dynkin types 4 Conclusion 5

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Outline Motivation 1 Definitions 2 Simply laced types 3 Non-simply laced Dynkin types 4 Conclusion 5

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups; occurs as endomorphism algebra in tensor categories for the above groups;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups; occurs as endomorphism algebra in tensor categories for the above groups; is cellular and (generically) semisimple;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups; occurs as endomorphism algebra in tensor categories for the above groups; is cellular and (generically) semisimple; maps homomorphically onto the group algebra of Sym n ;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups; occurs as endomorphism algebra in tensor categories for the above groups; is cellular and (generically) semisimple; maps homomorphically onto the group algebra of Sym n ; contains the Temperley-Lieb algebra TL n as a subalgebra;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups; occurs as endomorphism algebra in tensor categories for the above groups; is cellular and (generically) semisimple; maps homomorphically onto the group algebra of Sym n ; contains the Temperley-Lieb algebra TL n as a subalgebra; is a specialization of the Birman-Wenzl-Murakami algebra BMW n ;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation Theorem The Brauer algebra Br n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic groups; occurs as endomorphism algebra in tensor categories for the above groups; is cellular and (generically) semisimple; maps homomorphically onto the group algebra of Sym n ; contains the Temperley-Lieb algebra TL n as a subalgebra; is a specialization of the Birman-Wenzl-Murakami algebra BMW n ; has a natural definition in terms of generators and relations.

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Motivation cont’d The relations can be summarized by use of the Dynkin diagram of type A n − 1 , whose Weyl group is Sym n . Similary for BMW n . To what extent are there similar algebras for other Dynkin types?

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Outline Motivation 1 Definitions 2 Simply laced types 3 Non-simply laced Dynkin types 4 Conclusion 5

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Br n by diagrams, example for n = 10 As a Z [ δ ]-module, Br n is spanned by Brauer diagrams on 2 n nodes.

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Br n by diagrams, example cont’d

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Br n by diagrams cont’d Known pictures for Sym n . Extended by cups and caps. For a diagram T and a circle C in T T = δ · ( T \ C ) .

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Main properties Theorem (Brauer, 1937) For δ ∈ N and V = C δ there is a surjective homomorphism Br n → End O ( V ) ( ⊗ n ( V )) . For δ ∈ − 2 N and V = C − δ there is a surjective homomorphism Br n → End Sp ( V ) ( ⊗ n ( V )) . This followed a result of Schur’s for End GL ( V ) ( ⊗ n ( V )). Theorem (Wenzl, Hanlon & Wales, Doran, Rui & Si) If δ �∈ Z or | δ | < n, then Br n is semisimple of dimension dim( Br n ) = n !! = 1 · 3 · · · (2 n − 1) .

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion TL n by diagrams Definition TL n is the subalgebra of Br n spanned by the diagrams without crossings. Lemma dim( TL n ) is the n-th Catalan number. Theorem TL n is the quotient of the Hecke algebra of type A n − 1 by the central elements of the parabolic subalgebras of rank two.

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion BMW n by diagrams Same diagrams as for Brauer, but with distinction of over and under crossings. Braid relations. Skein relations.

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion The Kauffman skein relation = + m + m g − 1 g i + m 1 = + m e i i

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Properties of BMW n Theorem The Birman-Wenzl-Murakami algebra BMW n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic quantum groups;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Properties of BMW n Theorem The Birman-Wenzl-Murakami algebra BMW n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic quantum groups; is cellular and (generically) semisimple;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Properties of BMW n Theorem The Birman-Wenzl-Murakami algebra BMW n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic quantum groups; is cellular and (generically) semisimple; maps homomorphically onto the Hecke algebra of type A n − 1 ;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Properties of BMW n Theorem The Birman-Wenzl-Murakami algebra BMW n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic quantum groups; is cellular and (generically) semisimple; maps homomorphically onto the Hecke algebra of type A n − 1 ; contains the Temperley-Lieb algebra TL n as a subalgebra;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Properties of BMW n Theorem The Birman-Wenzl-Murakami algebra BMW n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic quantum groups; is cellular and (generically) semisimple; maps homomorphically onto the Hecke algebra of type A n − 1 ; contains the Temperley-Lieb algebra TL n as a subalgebra; has a natural definition in terms of generators and relations;

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion Properties of BMW n Theorem The Birman-Wenzl-Murakami algebra BMW n maps homeomorphically onto the centralizer of n-fold tensors of the natural representations of the orthogonal and symplectic quantum groups; is cellular and (generically) semisimple; maps homomorphically onto the Hecke algebra of type A n − 1 ; contains the Temperley-Lieb algebra TL n as a subalgebra; has a natural definition in terms of generators and relations; has a Markov trace leading to a knot theory invariant.

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion BMW n by presentation (Wenzl) Diagram A n − 1 = ◦ ◦ 2 · · · · · · ◦ 1 n − 1 Coefficients δ , m , l such that m (1 − δ ) = l − l − 1 . Single node i : g 2 i = 1 − m ( g i − l − 1 e i ) e i g i = l − 1 e i g i e i = l − 1 e i e 2 i = δ e i

Motivation Definitions Simply laced types Non-simply laced Dynkin types Conclusion BMW n by presentation, cont’d Two nodes i and j of A n − 1 with i ∼ j : g i g j = g j g i e i g j = g j e i e i e j = e j e i