arXiv-1205
June-20-12
8:00 AM
Mathematics > Quantum Algebra
The explicit equivalence between the standard and the logarithmic star product for Lie algebras
Carlo A. Rossi
(Submitted on 24 May 2012)
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. 
Comments: 
2 figures;this paper replaces the previous one, in that we changed the title radically, we also expanded the abstract and added some comments at the end; comments are very welcome
Subjects: 
Quantum Algebra (math.QA)
MSC classes: 
17B35, 81R60
arXiv:1205.5512v1 [math.QA]
Submission history
From: Carlo Antonio Rossi [view email] 
[v1] Thu, 24 May 2012 17:44:05 GMT (14kb)
Pasted from <[link]> 
Mathematics > Quantum Algebra
On KV, AT and compatibility between cup products
Carlo A. Rossi
(Submitted on 31 May 2012)
For a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K$ of characteristic 0 which contains $\mathbb C$, we deduce from the compatibility between cup products [Kontsevich, Section 8] and the vanishing of the weights associated to wheel graphs with spokes pointing inwards Shoikhet an alternative way of re-writing the Kontsevich $\star$-product on $\mathrm S(\mathfrak g)$ using the Alekseev-Torossian flat connection (Alekseev and Torossian). Further, we deduce directly from the Kashiwara-Vergne conjecture a similar way of re-writing $\star$. 
Comments: 
8 pages, 1 figure; the paper presents a proof different from the original one of Kontsevich of a special case of compatibility between cup products for the tangent formality morphism, which makes the link to the Alekseev-Torossian connection, compatibility between cup products and the Kashiwara-Vergne conjecture more transparent
Subjects: 
Quantum Algebra (math.QA)
MSC classes: 
17B35, 17B40, 81R60
arXiv:1205.7029v1 [math.QA]
Submission history
From: Carlo Antonio Rossi [view email] 
[v1] Thu, 31 May 2012 16:13:27 GMT (14kb)
Pasted from <[link]> 