2,115 research outputs found

    On several varieties of cacti and their relations

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    Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasi-operads by showing that both types of cacti are semi-direct products of the quasi-operad of their normalized versions with a re-scaling operad based on R>0. Furthermore, we introduce the notion of bi-crossed products of quasi-operads and show that the cacti proper are a bi-crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S^1. We also prove that this particular bi-crossed operad product is homotopy equivalent to the semi-direct product of the spineless cacti with the group S^1. This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm

    Twisting algebras using non-commutative torsors

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    Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori.Comment: 27 pages. Masuoka is a new coauthor. Introduction was revised. Sections 1 and 2 were thoroughly restructured. The presentation theorem in Section 3 is now put in a more general framework and has a more general formulation. Section 4 was shortened. All examples (quantum affine spaces and tori, twisting of SL(2), twisting of the enveloping algebra of sl(2)) are left unchange

    Graded q-pseudo-differential Operators and Supersymmetric Algebras

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    We give a supersymmetric generalization of the sine algebra and the quantum algebra Ut(sl(2))U_{t}(sl(2)). Making use of the qq-pseudo-differential operators graded with a fermionic algebra, we obtain a supersymmetric extension of sine algebra. With this scheme we also get a quantum superalgebra Ut(sl(2/1)U_{t}(sl(2/1).Comment: 10 pages, Late

    Cleft Extensions and Quotients of Twisted Quantum Doubles

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    Given a pair of finite groups F,GF, G and a normalized 3-cocycle ω\omega of GG, where FF acts on GG as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension kωG#c kF\Bbbk^G_\omega\#_c\,\Bbbk F where cc denotes some suitable cohomological data. When F→F‟:=F/AF\rightarrow \overline{F}:=F/A is a quotient of FF by a central subgroup AA acting trivially on GG, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of the type kωG#c kF→kωG#c‟ kF‟\Bbbk^G_\omega\#_c\, \Bbbk F\rightarrow \Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{F}. Our construction is particularly natural when F=GF=G acts on GG by conjugation, and kωG#ckG\Bbbk^G_\omega\#_c \Bbbk G is a twisted quantum double Dω(G)D^{\omega}(G). In this case, we give necessary and sufficient conditions that Rep(kωG#c‟ kG‟\Bbbk^G_\omega\#_{\overline{c}} \, \Bbbk \overline{G}) is a modular tensor category.Comment: LaTex; 14 page

    Some open questions on anti-de Sitter geometry

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    We present a list of open questions on various aspects of AdS geometry, that is, the geometry of Lorentz spaces of constant curvature -1. When possible we point out relations with homogeneous spaces and discrete subgroups of Lie groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.Comment: Not a research article in the usual sense but rather a list of open questions. 19 page

    Summer Field Day Scheduled for ISU Northern Iowa Research Farm

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    Crop management, nitrogen management, and crops grown for an energy source will be highlighted topics of the summer field day at the Iowa State University Northern Research and Demonstration Farm. The field day will take place June 25, at the farm’s northern location at 310 S. Main St., Kanawha, immediately south of town on county road R-35

    Ambiskew Hopf algebras

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    Necessary and sufficient conditions are obtained for an ambiskew polynomial algebra A over a Hopf k-algebra R to possess the structure of a Hopf algebra extending that of R, in which the added variables X+ and X- are skew primitive. The coradical filtration is calculated, many examples are described, and properties determined

    String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy

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    We consider scalar-Gauss-Bonnet and modified Gauss-Bonnet gravities and reconstruct these theories from the universe expansion history. In particular, we are able to construct versions of those theories (with and without ordinary matter), in which the matter dominated era makes a transition to the cosmic acceleration epoch. It is remarkable that, in several of the cases under consideration, matter dominance and the deceleration-acceleration transition occur in the presence of matter only. The late-time acceleration epoch is described asymptotically by de Sitter space but may also correspond to an exact Λ\LambdaCDM cosmology, having in both cases an effective equation of state parameter ww close to -1. The one-loop effective action of modified Gauss-Bonnet gravity on the de Sitter background is evaluated and it is used to derive stability criteria for the ensuing de Sitter universe.Comment: LaTeX20 pages, 4 figures, version to apear in PR

    Cotensor Coalgebras in Monoidal Categories

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    We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an abelian monoidal category. Under some further conditions we show that such a cotensor coalgebra exists and satisfies a meaningful universal property. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth

    Braidings of Tensor Spaces

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    Let VV be a braided vector space, that is, a vector space together with a solution R^∈End(V⊗V)\hat{R}\in {\text{End}}(V\otimes V) of the Yang--Baxter equation. Denote T(V):=⹁kV⊗kT(V):=\bigoplus_k V^{\otimes k}. We associate to R^\hat{R} a solution T(R^)∈End(T(V)⊗T(V))T(\hat{R})\in {\text{End}}(T(V)\otimes T(V)) of the Yang--Baxter equation on the tensor space T(V)T(V). The correspondence R^⇝T(R^)\hat{R}\rightsquigarrow T(\hat{R}) is functorial with respect to VV.Comment: 10 pages, no figure
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