36,704 research outputs found
Finitely generated free Heyting algebras via Birkhoff duality and coalgebra
Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and
thus the free algebras can be obtained by a direct limit process. Dually, the
final coalgebras can be obtained by an inverse limit process. In order to
explore the limits of this method we look at Heyting algebras which have mixed
rank 0-1 axiomatizations. We will see that Heyting algebras are special in that
they are almost rank 1 axiomatized and can be handled by a slight variant of
the rank 1 coalgebraic methods
Duality for Convexity
This paper studies convex sets categorically, namely as algebras of a
distribution monad. It is shown that convex sets occur in two dual adjunctions,
namely one with preframes via the Boolean truth values {0,1} as dualising
object, and one with effect algebras via the (real) unit interval [0,1] as
dualising object. These effect algebras are of interest in the foundations of
quantum mechanics
Target Space Duality II: Applications
We apply the framework developed in Target Space Duality I: General Theory.
We show that both nonabelian duality and Poisson-Lie duality are examples of
the general theory. We propose how the formalism leads to a systematic study of
duality by studying few scenarios that lead to open questions in the theory of
Lie algebras. We present evidence that there are probably new examples of
irreducible target space duality.Comment: LaTeX, 28 pages. Companion to Target Space Duality I: General Theory.
Added a couple of references and corrected a couple of typos. An FAQ that
discusses some subtle points is at
http://www.physics.miami.edu/~alvarez/papers/duality
Frobenius-Schur indicator for categories with duality
We introduce the Frobenius-Schur indicator for categories with duality to
give a category-theoretical understanding of various generalizations of the
Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras,
weak Hopf C*-algebras and association schemes. Our framework also clarifies a
mechanism how the `twisted' theory arises from the ordinary case. As a
demonstration, we give a twisted Frobenius-Schur theorem for semisimple
quasi-Hopf algebras. We also give several applications to the quantum SL_2.Comment: 38 pages; final version published in the Special Issue on "Hopf
Algebras, Quantum Groups and Yang-Baxter Equations" of Axiom
Bi-algebras, generalised geometry and T-duality
A study of sigma models whose target space is a group G that admits a
compatible Poisson structure is presented. The natural action of O(D,D;Z) on
the generalised tangent bundle TG+T*G and a generalisation of the Courant
bracket that appears are reviewed. This background provides a concrete example
where the generalised geometry and doubled geometry descriptions are both well
understood. Connections between the two formalisms are discussed and the
world-sheet theory from Hamiltonian and Lagrangian perspectives is
investigated. The comparisons between the approaches given by generalised
geometry and doubled geometry suggest possible ways of generalising the
analysis beyond the known examples.Comment: 43 page
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