Equivalences and iterations for weak crossed products Ramn Gonzlez - PowerPoint PPT Presentation

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Equivalences and iterations for weak crossed products Ramón González Rodríguez http://www.dma.uvigo.es/˜rgon/ Departamento de Matemática Aplicada II. Universidade de Vigo Based in a joint work with A.B. Rodríguez Raposo and J.M. Fernández Vilaboa Research supported by Ministerio de Economía y Competitividad of Spain (European Feder support included). Grant MTM2013-43687-P Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Outline Weak crossed products 1 Equivalences between weak crossed products 2 Iterations for weak crossed products 3 Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Some notation and conventions. From now on C denotes a strict monoidal category with tensor product denoted by ⊗ and unit object K . We assume that every idempotent morphism q : Y → Y splits, i.e., there exist an object Z and morphisms i : Z → Y and p : Y → Z such that q = i ◦ p and p ◦ i = id Z . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Some notation and conventions. From now on C denotes a strict monoidal category with tensor product denoted by ⊗ and unit object K . We assume that every idempotent morphism q : Y → Y splits, i.e., there exist an object Z and morphisms i : Z → Y and p : Y → Z such that q = i ◦ p and p ◦ i = id Z . ( A , η A , µ A ) is a monoid with product µ A and unit η A . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Some notation and conventions. From now on C denotes a strict monoidal category with tensor product denoted by ⊗ and unit object K . We assume that every idempotent morphism q : Y → Y splits, i.e., there exist an object Z and morphisms i : Z → Y and p : Y → Z such that q = i ◦ p and p ◦ i = id Z . ( A , η A , µ A ) is a monoid with product µ A and unit η A . For simplicity of notation, given three objects V , U , B in C and a morphism f : V → U , we write B ⊗ f for id B ⊗ f and f ⊗ B for f ⊗ id B . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Weak crossed products Weak crossed products 1 Equivalences between weak crossed products 2 Iterations for weak crossed products 3 Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Preunits and weak crossed products, J. Pure Appl. Algebra (2009). Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products J.M. Fernández Vilaboa, R. González Rodríguez, R., A.B. Rodríguez Raposo, Preunits and weak crossed products, J. Pure Appl. Algebra (2009). Let A be a monoid and let V be an object in C . Suppose that there exists a morphism ψ A V : V ⊗ A → A ⊗ V such that the following equality holds ( µ A ⊗ V ) ◦ ( A ⊗ ψ A V ) ◦ ( ψ A V ⊗ A ) = ψ A ( 1 ) V ◦ ( V ⊗ µ A ) . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products The morphism ∇ A ⊗ V = ( µ A ⊗ V ) ◦ ( A ⊗ ψ A V ) ◦ ( A ⊗ V ⊗ η A ) : A ⊗ V → A ⊗ V is idempotent. Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products The morphism ∇ A ⊗ V = ( µ A ⊗ V ) ◦ ( A ⊗ ψ A V ) ◦ ( A ⊗ V ⊗ η A ) : A ⊗ V → A ⊗ V is idempotent. For the idempotent morphism ∇ A ⊗ V we denote by A × V the object such that p A ⊗ V ◦ i A ⊗ V = id A × V , where i A ⊗ V : A × V → A ⊗ V , p A ⊗ V : A ⊗ V → A × V the injection and the projection associated to ∇ A ⊗ V . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products The morphism ∇ A ⊗ V = ( µ A ⊗ V ) ◦ ( A ⊗ ψ A V ) ◦ ( A ⊗ V ⊗ η A ) : A ⊗ V → A ⊗ V is idempotent. For the idempotent morphism ∇ A ⊗ V we denote by A × V the object such that p A ⊗ V ◦ i A ⊗ V = id A × V , where i A ⊗ V : A × V → A ⊗ V , p A ⊗ V : A ⊗ V → A × V the injection and the projection associated to ∇ A ⊗ V . From now on we consider quadruples ( A , V , ψ A V , σ A V ) where A is an algebra, V an object, ψ A V : V ⊗ A → A ⊗ V satisfies (1) and σ A V : V ⊗ V → A ⊗ V is a morphism in C . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products For a quadruple ( A , V , ψ A V , σ A V ) define the product µ A ⊗ V = ( µ A ⊗ V ) ◦ ( µ A ⊗ σ A V ) ◦ ( A ⊗ ψ A V ⊗ V ) : A ⊗ V ⊗ A ⊗ V → A ⊗ V and let µ A × V be the product µ A × V = p A ⊗ V ◦ µ A ⊗ V ◦ ( i A ⊗ V ⊗ i A ⊗ V ) : A × V ⊗ A × V → A × V . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products For a quadruple ( A , V , ψ A V , σ A V ) define the product µ A ⊗ V = ( µ A ⊗ V ) ◦ ( µ A ⊗ σ A V ) ◦ ( A ⊗ ψ A V ⊗ V ) : A ⊗ V ⊗ A ⊗ V → A ⊗ V and let µ A × V be the product µ A × V = p A ⊗ V ◦ µ A ⊗ V ◦ ( i A ⊗ V ⊗ i A ⊗ V ) : A × V ⊗ A × V → A × V . What conditions are needed to make µ A ⊗ V and µ A × V associative? Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Twisted condition We say that ( A , V , ψ A V , σ A V ) satisfies the twisted condition if ( µ A ⊗ V ) ◦ ( A ⊗ ψ A V ) ◦ ( σ A V ⊗ A ) = ( µ A ⊗ V ) ◦ ( A ⊗ σ A V ) ◦ ( ψ A V ⊗ V ) ◦ ( V ⊗ ψ A V ) Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Twisted condition We say that ( A , V , ψ A V , σ A V ) satisfies the twisted condition if ( µ A ⊗ V ) ◦ ( A ⊗ ψ A V ) ◦ ( σ A V ⊗ A ) = ( µ A ⊗ V ) ◦ ( A ⊗ σ A V ) ◦ ( ψ A V ⊗ V ) ◦ ( V ⊗ ψ A V ) Cocycle condition We say that ( A , V , ψ A V , σ A V ) satisfies the the cocycle condition if ( µ A ⊗ V ) ◦ ( A ⊗ σ A V ) ◦ ( σ A V ⊗ V ) = ( µ A ⊗ V ) ◦ ( A ⊗ σ A V ) ◦ ( ψ A V ⊗ V ) ◦ ( V ⊗ σ A V ) . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Theorem Let ( A , V , ψ A V , σ A V ) be a quadruple satisfying the twisted and the cocycle conditions. The product µ A ⊗ V is associative and normalized with respect to ∇ A ⊗ V . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Theorem Let ( A , V , ψ A V , σ A V ) be a quadruple satisfying the twisted and the cocycle conditions. The product µ A ⊗ V is associative and normalized with respect to ∇ A ⊗ V . Normal condition Normalized with respect to ∇ A ⊗ V : ∇ A ⊗ V ◦ µ A ⊗ V = µ A ⊗ V = µ A ⊗ V ◦ ( ∇ A ⊗ V ⊗ ∇ A ⊗ V ) . Ramón González Rodríguez Equivalences and iterations for weak crossed products

Weak crossed products Equivalences between weak crossed products Iterations for weak crossed products Theorem Let ( A , V , ψ A V , σ A V ) be a quadruple satisfying the twisted and the cocycle conditions. The product µ A ⊗ V is associative and normalized with respect to ∇ A ⊗ V . Normal condition Normalized with respect to ∇ A ⊗ V : ∇ A ⊗ V ◦ µ A ⊗ V = µ A ⊗ V = µ A ⊗ V ◦ ( ∇ A ⊗ V ⊗ ∇ A ⊗ V ) . Corollary Let ( A , V , ψ A V , σ A V ) be a quadruple satisfying the twisted and the cocycle conditions. The product µ A × V is associative. Ramón González Rodríguez Equivalences and iterations for weak crossed products