What is a planar algebra? Connection with subfactor A canonical MASA (maximal abelian self-adjoint subalgebra) Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Planar calculus Planar Algebras and Subfactors Tangle Planar algebra Connection with subfactor Subfactor Arnaud Brothier An invariant Planar algebra gives subfactor A canonical Université de Paris 7 MASA (maximal abelian self-adjoint 31 août 2009 subalgebra) Description Properties Free group factor Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Connection with subfactor A canonical MASA (maximal abelian self-adjoint subalgebra) Plan Planar Algebras and Subfactors Arnaud Brothier 1 What is a planar algebra? What is a planar Planar calculus algebra? Planar calculus Tangle Tangle Planar algebra Planar algebra Connection with subfactor 2 Connection with subfactor Subfactor An invariant Subfactor Planar algebra gives subfactor An invariant A canonical MASA (maximal Planar algebra gives subfactor abelian self-adjoint 3 A canonical MASA (maximal abelian self-adjoint subalgebra) Description subalgebra) Properties Free group factor Description Properties Free group factor Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Planar calculus Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Let ( E , +) be a set with an associative law. Planar calculus Tangle We can compute elements on a line as follow: Planar algebra Connection with a subfactor a + b Subfactor An invariant Planar algebra gives c + a + b subfactor We want to do the same but in the plane to give a sense to A canonical MASA (maximal b abelian self-adjoint something like that: c subalgebra) Description a Properties Free group factor For this, I have to define what is the "+" in the plane. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Planar calculus Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Let ( E , +) be a set with an associative law. Planar calculus Tangle We can compute elements on a line as follow: Planar algebra Connection with a subfactor a + b Subfactor An invariant Planar algebra gives c + a + b subfactor We want to do the same but in the plane to give a sense to A canonical MASA (maximal b abelian self-adjoint something like that: c subalgebra) Description a Properties Free group factor For this, I have to define what is the "+" in the plane. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Planar calculus Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Let ( E , +) be a set with an associative law. Planar calculus Tangle We can compute elements on a line as follow: Planar algebra Connection with a subfactor a + b Subfactor An invariant Planar algebra gives c + a + b subfactor We want to do the same but in the plane to give a sense to A canonical MASA (maximal b abelian self-adjoint something like that: c subalgebra) Description a Properties Free group factor For this, I have to define what is the "+" in the plane. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Planar calculus Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Let ( E , +) be a set with an associative law. Planar calculus Tangle We can compute elements on a line as follow: Planar algebra Connection with a subfactor a + b Subfactor An invariant Planar algebra gives c + a + b subfactor We want to do the same but in the plane to give a sense to A canonical MASA (maximal b abelian self-adjoint something like that: c subalgebra) Description a Properties Free group factor For this, I have to define what is the "+" in the plane. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Planar calculus Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Let ( E , +) be a set with an associative law. Planar calculus Tangle We can compute elements on a line as follow: Planar algebra Connection with a subfactor a + b Subfactor An invariant Planar algebra gives c + a + b subfactor We want to do the same but in the plane to give a sense to A canonical MASA (maximal b abelian self-adjoint something like that: c subalgebra) Description a Properties Free group factor For this, I have to define what is the "+" in the plane. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Planar calculus Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? Let ( E , +) be a set with an associative law. Planar calculus Tangle We can compute elements on a line as follow: Planar algebra Connection with a subfactor a + b Subfactor An invariant Planar algebra gives c + a + b subfactor We want to do the same but in the plane to give a sense to A canonical MASA (maximal b abelian self-adjoint something like that: c subalgebra) Description a Properties Free group factor For this, I have to define what is the "+" in the plane. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Tangle Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? * Planar calculus * Tangle Planar algebra * Connection with subfactor * Subfactor An invariant Planar algebra gives subfactor Exterior disc A canonical MASA (maximal Disjoints subdiscs D j abelian Strings self-adjoint subalgebra) First segment for each disc denote by a star Description Properties Each disc cross an even number of times the strings, we Free group factor denote by 2 k j the number of point of the boundary of the disc D j which touch the strings and by 2 k for the exterior disc. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Tangle Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? * Planar calculus * Tangle Planar algebra * Connection with subfactor * Subfactor An invariant Planar algebra gives subfactor Exterior disc A canonical MASA (maximal Disjoints subdiscs D j abelian Strings self-adjoint subalgebra) First segment for each disc denote by a star Description Properties Each disc cross an even number of times the strings, we Free group factor denote by 2 k j the number of point of the boundary of the disc D j which touch the strings and by 2 k for the exterior disc. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Tangle Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? * Planar calculus * Tangle Planar algebra * Connection with subfactor * Subfactor An invariant Planar algebra gives subfactor Exterior disc A canonical MASA (maximal Disjoints subdiscs D j abelian Strings self-adjoint subalgebra) First segment for each disc denote by a star Description Properties Each disc cross an even number of times the strings, we Free group factor denote by 2 k j the number of point of the boundary of the disc D j which touch the strings and by 2 k for the exterior disc. Arnaud Brothier Planar Algebras and Subfactors
What is a planar algebra? Planar calculus Connection with subfactor Tangle A canonical MASA (maximal abelian self-adjoint subalgebra) Planar algebra Tangle Planar Algebras and Subfactors Arnaud Brothier What is a planar algebra? * Planar calculus * Tangle Planar algebra * Connection with subfactor * Subfactor An invariant Planar algebra gives subfactor Exterior disc A canonical MASA (maximal Disjoints subdiscs D j abelian Strings self-adjoint subalgebra) First segment for each disc denote by a star Description Properties Each disc cross an even number of times the strings, we Free group factor denote by 2 k j the number of point of the boundary of the disc D j which touch the strings and by 2 k for the exterior disc. Arnaud Brothier Planar Algebras and Subfactors
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