283 research outputs found
A magic pyramid of supergravities
By formulating N = 1, 2, 4, 8, D = 3, Yang-Mills with a single Lagrangian and
single set of transformation rules, but with fields valued respectively in
R,C,H,O, it was recently shown that tensoring left and right multiplets yields
a Freudenthal-Rosenfeld-Tits magic square of D = 3 supergravities. This was
subsequently tied in with the more familiar R,C,H,O description of spacetime to
give a unified division-algebraic description of extended super Yang-Mills in D
= 3, 4, 6, 10. Here, these constructions are brought together resulting in a
magic pyramid of supergravities. The base of the pyramid in D = 3 is the known
4x4 magic square, while the higher levels are comprised of a 3x3 square in D =
4, a 2x2 square in D = 6 and Type II supergravity at the apex in D = 10. The
corresponding U-duality groups are given by a new algebraic structure, the
magic pyramid formula, which may be regarded as being defined over three
division algebras, one for spacetime and each of the left/right Yang-Mills
multiplets. We also construct a conformal magic pyramid by tensoring conformal
supermultiplets in D = 3, 4, 6. The missing entry in D = 10 is suggestive of an
exotic theory with G/H duality structure F4(4)/Sp(3) x Sp(1).Comment: 30 pages, 6 figures. Updated to match published version. References
and comments adde
Derived equivalences for Rational Cherednik algebras
Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra
for depending on a parameter c. One can consider the category O for H_c(W).
We prove a conjecture of Rouquier that the categories O for H_c(W) and
H_{c'}(W) are derived equivalent provided the parameters c,c' have integral
difference. Two main ingredients of the proof are a connection between the
Ringel duality and Harish-Chandra bimodules and an analog of a deformation
technique developed by the author and Bezrukavnikov. We also show that some of
the derived equivalences we construct are perverse.Comment: 32 pages, preliminary version; v2 33 pages some proofs rewritten; v3
32 pages, proofs of the main results substantially simplified, other minor
changes; v4 some proofs were modified, other minor change
An Eilenberg-like theorem for algebras on a monad
An Eilenberg–like theorem is shown for algebras on a given monad. The main idea is to explore the
approach given by Bojan´czyk that defines, for a given monad T on a category D, pseudovarieties of
T–algebras as classes of finite T–algebras closed under homomorphic images, subalgebras, and finite
products. To define pseudovarieties of recognizable languages, which is the other main concept for
an Eilenberg–like theorem, we use a category C that is dual to D and a recent duality result
between Eilenberg–Moore categories of algebras and coalgebras by Salamanca, Bonsangue, and Rot.
Using this duality we define the concept of a pseudovariety o
On Affine Logic and {\L}ukasiewicz Logic
The multi-valued logic of {\L}ukasiewicz is a substructural logic that has
been widely studied and has many interesting properties. It is classical, in
the sense that it admits the axiom schema of double negation, [DNE]. However,
our understanding of {\L}ukasiewicz logic can be improved by separating its
classical and intuitionistic aspects. The intuitionistic aspect of
{\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the
commutativity of a weak form of conjunction. This is equivalent to a very
restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed
both as an extension of classical affine logic with [CWC], or as an extension
of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE],
intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic
affine logic by the schema [CWC]. At first glance, intuitionistic affine logic
seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results
such as intuitionistic analogues of De Morgan's laws. However the proofs can be
very intricate. We present these results using derived connectives to clarify
and motivate the proofs and give several applications. We give an analysis of
the applicability to these logics of the well-known methods that use negation
to translate classical logic into intuitionistic logic. The usual proofs of
correctness for these translations make much use of contraction. Nonetheless,
we show that all the usual negative translations are already correct for
intuitionistic {\L}ukasiewicz logic, where only the limited amount of
contraction given by [CWC] is allowed. This is in contrast with affine logic
for which we show, by appeal to results on semantics proved in a companion
paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page
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