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TemperleyLieb Algebra: From Knot Theory to . . .
"... Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics. ..."
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Our aim in this paper is to trace some of the surprising and beautiful connections which are beginning to emerge between a number of apparently disparate topics.
Symmetric SelfAdjunctions: A Justification of Brauer’s Representation of Brauer’s Algebras
, 2005
"... A classic result of representation theory is Brauer’s construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this pa ..."
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A classic result of representation theory is Brauer’s construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this paper is to provide a motivation for this result through the categorial notion of symmetric selfadjunction. Mathematics Subject Classification (2000): 14L24, 57M99, 20C99, 18A40 Keywords: Brauer’s centralizer algebras, matrix representation, orthogonal group, adjoint functor
Symmetric SelfAdjunctions and Matrices
, 2007
"... It is shown that the multiplicative monoids of Brauer’s centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the selfadjoint functors is satisfied. As in ..."
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It is shown that the multiplicative monoids of Brauer’s centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the selfadjoint functors is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric selfadjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.
unknown title
, 2007
"... The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of s ..."
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The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in TemperleyLieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a somewhat different related presentation is given, with completeness proved in a new and selfcontained manner.
unknown title
, 2007
"... The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of s ..."
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The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in TemperleyLieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a closely related presentation is given, with completeness proved in a new and selfcontained manner.
unknown title
, 2007
"... The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of s ..."
Abstract
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The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in TemperleyLieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a closely related presentation is given, with completeness proved in a new and selfcontained manner.
unknown title
, 2007
"... The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of s ..."
Abstract
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The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in TemperleyLieb algebras, and as the monoids of TemperleyLieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in TemperleyLieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a closely related presentation is given, with completeness proved in a new and selfcontained manner.
Belgrade, Serbia
"... A classic result of representation theory is Brauer’s construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this pa ..."
Abstract
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A classic result of representation theory is Brauer’s construction of a diagrammatical (geometrical) algebra whose matrix representation is a certain given matrix algebra, which is the commutating algebra of the enveloping algebra of the representation of the orthogonal group. The purpose of this paper is to provide a motivation for this result through the categorial notion of symmetric selfadjunction. 1.
Negation and Involutive Adjunctions
, 2005
"... This note analyzes in terms of categorial proof theory some standard assumptions about negation in the absence of any other connective. It is shown that the assumptions for an involutive negation, like classical negation, make a kind of adjoint situation, which is named involutive adjunction. The no ..."
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This note analyzes in terms of categorial proof theory some standard assumptions about negation in the absence of any other connective. It is shown that the assumptions for an involutive negation, like classical negation, make a kind of adjoint situation, which is named involutive adjunction. The notion of involutive adjunction amounts in a precise sense to adjunction where an endofunctor is adjoint to itself.
Ordinals in Frobenius Monads
, 809
"... This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for selfadjunctions (a ..."
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This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for selfadjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in ε0 play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in ε0, and by raising the question of the exact categorial abstraction of the notion of Frobenius algebra.