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The Shuffle Hopf Algebra and Noncommutative Full Completeness
, 1999
"... We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations b ..."
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Cited by 8 (3 self)
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We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cutfree proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *autonomous category, canonically enriched over vector spaces. This paper
On Hopf algebras and their generalizations
 Comm. Algebra
, 2007
"... Abstract. We survey Hopf algebras and their generalizations. In particular, we compare and contrast three wellstudied generalizations (quasiHopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (hopfish algebras and Hopf monads). Each of these notions was originally introduce ..."
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Cited by 2 (0 self)
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Abstract. We survey Hopf algebras and their generalizations. In particular, we compare and contrast three wellstudied generalizations (quasiHopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (hopfish algebras and Hopf monads). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.
A Noncommutative Full Completeness Theorem (Extended Abstract)
 Elsevier Science B.V
, 1996
"... ) R.F. Blute 1 P.J. Scott 1 Dept. of Mathematics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA E. N. T. C. S. Elsevier Science B. V. Abstract We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (Cy ..."
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Cited by 1 (1 self)
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) R.F. Blute 1 P.J. Scott 1 Dept. of Mathematics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA E. N. T. C. S. Elsevier Science B. V. Abstract We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (CyLL), first defined by Yetter. The semantics is obtained by considering dinatural transformations on a category of topological vector spaces which are invariant under certain actions of a noncocommutative Hopf algebra, called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that the space has the denotations of cutfree proofs in CyLL+MIX as a basis. This work is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic. In that paper, we consider dinaturals which are invariant under certain actions of the additive group of integers. The passage from group...
Enhancing an Rmatrix.
, 1996
"... In order to construct a representation of the tangle category one needs an enhanced Rmatrix. In this paper we define a sufficient and necessary condition for enhancement that can be checked easily for any Rmatrix. If the Rmatrix can be enhanced, we also show how to construct the additional data t ..."
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In order to construct a representation of the tangle category one needs an enhanced Rmatrix. In this paper we define a sufficient and necessary condition for enhancement that can be checked easily for any Rmatrix. If the Rmatrix can be enhanced, we also show how to construct the additional data that define the enhancement. As a direct consequence we find a sufficient condition for the construction of a knot invariant.