On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture

For a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K\supset \mathbb C$, we deduce from the compatibility between cup products Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an alternative way of re-writing Kontsevich product $\star$ on $\mathrm S(\mathfrak g)$ by means of the Alekseev--Torossian flat connection (Alekseev and Torossian, 2010). We deduce a similar formula directly from the Kashiwara--Vergne conjecture (Kashiwara and Vergne, 1978).Comment: 8 pages, 1 figure; notation changed; corrected many other misprints recently noticed; comments are very welcome

The explicit equivalence between the standard and the logarithmic star product for Lie algebras

The purpose of this short note is to establish an explicit equivalence between the two star products $\star$ and $\star_{\log}$ on the symmetric algebra $\mathrm S(\mathfrak g)$ of a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K\supset\mathbb C$ of characteristic 0 associated with the standard angular propagator and the logarithmic one: the differential operator of infinite order with constant coefficients realizing the equivalence is related to the incarnation of the Grothendieck-Teichm\"uller group considered by Kontsevich.Comment: 2 figures; corrected and completed the formulation of Theorem 3.7. Comments are very welcome

Property, Legitimacy, Ideology: A Reality Check

Drawing on empirical evidence from history and anthropology, we aim to demonstrate that there is room for genealogical ideology critique within normative political theory. The test case is some libertarians’ use of folk notions of private property rights in defence of the legitimacy of capitalist states. Our genealogy of the notion of private property shows that asking whether a capitalist state can emerge without violations of self-ownership cannot help settling the question of its legitimacy, because the notion of private property presupposed by that question is a product of the entity it is supposed to help legitimise: the state. We anchor our genealogical critique in recent work on ideology in epistemology and philosophy of language, and in current debates on the methodology of political theory. But, unlike more traditional approaches that aim to debunk whole concepts or even belief systems, we propose a more targeted, argument-specific form of ideology critique

Deformation quantization with generators and relations

In this paper we prove a conjecture of B. Shoikhet which claims that two quantization procedures arising from Fourier dual constructions actually coincide.Comment: 9 pages; 4 figures; many typos have been corrected; the introduction has been considerably extended and a more detailed exposition of the Koszul theory behind the main idea has been added; the proof of Proposition 2.4 (iii) has been also extended; Subsection 3.2 has been enlarged, and a more detailed exposition of how the Duflo element arises has been adde

Joint Optimal Pricing and Electrical Efficiency Enforcement for Rational Agents in Micro Grids

In electrical distribution grids, the constantly increasing number of power generation devices based on renewables demands a transition from a centralized to a distributed generation paradigm. In fact, power injection from Distributed Energy Resources (DERs) can be selectively controlled to achieve other objectives beyond supporting loads, such as the minimization of the power losses along the distribution lines and the subsequent increase of the grid hosting capacity. However, these technical achievements are only possible if alongside electrical optimization schemes, a suitable market model is set up to promote cooperation from the end users. In contrast with the existing literature, where energy trading and electrical optimization of the grid are often treated separately or the trading strategy is tailored to a specific electrical optimization objective, in this work we consider their joint optimization. Specifically, we present a multi-objective optimization problem accounting for energy trading, where: 1) DERs try to maximize their profit, resulting from selling their surplus energy, 2) the loads try to minimize their expense, and 3) the main power supplier aims at maximizing the electrical grid efficiency through a suitable discount policy. This optimization problem is proved to be non convex, and an equivalent convex formulation is derived. Centralized solutions are discussed first, and are subsequently distributed. Numerical results to demonstrate the effectiveness of the so obtained optimal policies are then presented

Loop observables for BF theories in any dimension and the cohomology of knots

A generalization of Wilson loop observables for BF theories in any dimension is introduced in the Batalin-Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the paper has only trivalent interactions and, irrespective of the actual dimension, looks like a 3-dimensional Chern-Simons theory.Comment: 13 page

A gerbe for the elliptic gamma function

The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three (it is a stack). Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve.Comment: 54 page