PhD thesis:
Affine braid group, affine Temperley-Lieb algebra and Markov trace
In this thesis we define a tower of affine Temperley-Lieb algebras of type à on which we define a Markov trace and we show that there is a unique such trace. In order to do so, we work on four levels of type à : affine braid groups, affine Coxeter groups, affine Hecke algebras and affine Temperley-Lieb algebras.
On the braid level, we show that the Ã-type affine braid group with n+1 generators surjects onto the $A$-type affine braid group with $n$ generators, we prove that this surjection comes from a quotient by a certain subgroup and we define a closure of an element of this group which is to be called an affine link.
On the Coxeter level, we study the Ã-type affine Coxeter group with n+1 generators, we give a full set of representatives of left cosets and double cosets of the Ã-type affine Coxeter group with n generators, then we classify fully commutative elements and we give a normal form for such elements.
On the Hecke level, we define a tower of Ã-type affine Hecke algebras, we show that this tower is a tower of inclusions, and we show that this tower surjects onto the tower of A-type Hecke algebras.
On the Temperley-Lieb level, we define a tower of Ã-type affine Temperley-Lieb algebras, we define a Markov trace as a collection of traces on this tower in its most general form (compatibility with affine links). We get the existence of such trace by showing that the mentioned tower surjects onto the tower of $A$-type Temperley-Lieb algebras and finally we show that this trace is unique by making use of the normal form of the fully commutative elements in the Ã-type affine Coxeter group.
In this thesis we define a tower of affine Temperley-Lieb algebras of type à on which we define a Markov trace and we show that there is a unique such trace. In order to do so, we work on four levels of type à : affine braid groups, affine Coxeter groups, affine Hecke algebras and affine Temperley-Lieb algebras.
On the braid level, we show that the Ã-type affine braid group with n+1 generators surjects onto the $A$-type affine braid group with $n$ generators, we prove that this surjection comes from a quotient by a certain subgroup and we define a closure of an element of this group which is to be called an affine link.
On the Coxeter level, we study the Ã-type affine Coxeter group with n+1 generators, we give a full set of representatives of left cosets and double cosets of the Ã-type affine Coxeter group with n generators, then we classify fully commutative elements and we give a normal form for such elements.
On the Hecke level, we define a tower of Ã-type affine Hecke algebras, we show that this tower is a tower of inclusions, and we show that this tower surjects onto the tower of A-type Hecke algebras.
On the Temperley-Lieb level, we define a tower of Ã-type affine Temperley-Lieb algebras, we define a Markov trace as a collection of traces on this tower in its most general form (compatibility with affine links). We get the existence of such trace by showing that the mentioned tower surjects onto the tower of $A$-type Temperley-Lieb algebras and finally we show that this trace is unique by making use of the normal form of the fully commutative elements in the Ã-type affine Coxeter group.