## Algebra & Number Theory

### Operad of formal homogeneous spaces and Bernoulli numbers

Sergei Merkulov

#### Abstract

It is shown that for any morphism, $ϕ:g→h$, of Lie algebras the vector space underlying the Lie algebra $h$ is canonically a $g$-homogeneous formal manifold with the action of $g$ being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2-coloured operad of formal homogeneous spaces whose minimal resolution gives a new conceptual explanation of both Ziv Ran’s Jacobi–Bernoulli complex and Fiorenza–Manetti’s $L∞$-algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary $L∞$-algebras.

#### Article information

Source
Algebra Number Theory, Volume 2, Number 4 (2008), 407-433.

Dates
Revised: 17 January 2008
Accepted: 9 March 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797268

Digital Object Identifier
doi:10.2140/ant.2008.2.407

Mathematical Reviews number (MathSciNet)
MR2411406

Zentralblatt MATH identifier
1162.18003

#### Citation

Merkulov, Sergei. Operad of formal homogeneous spaces and Bernoulli numbers. Algebra Number Theory 2 (2008), no. 4, 407--433. doi:10.2140/ant.2008.2.407. https://projecteuclid.org/euclid.ant/1513797268

#### References

• C. Berger and I. Moerdijk, “Resolution of coloured operads and rectification of homotopy algebras”, pp. 31–58 in Categories in algebra, geometry and mathematical physics, Contemp. Math. 431, Amer. Math. Soc., Providence, RI, 2007. http://www.emis.de/cgi-bin/MATH-item?0562.94001Zbl 0562.94001
• I. N. Bernšteĭn and B. I. Rosenfel'd, “Homogeneous spaces of infinite-dimensional Lie algebras and the characteristic classes of foliations”, Uspehi Mat. Nauk 28:4(172) (1973), 103–138.
• D. Fiorenza and M. Manetti, “$L\sb \infty$ structures on mapping cones”, Algebra Number Theory 1:3 (2007), 301–330.
• M. Kontsevich, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys. 66:3 (2003), 157–216.
• M. Kontsevich and Y. Soibelman, “Deformations of algebras over operads and the Deligne conjecture”, pp. 255–307 in Conférence Moshé Flato 1999 (Dijon, September 5–8, 1999), vol. I, edited by G. Dito and D. Sternheimer, Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht, 2000.
• P. van der Laan, “Operads up to homotopy and deformations of operad maps”, preprint, 2002.
• R. Longoni and T. Tradler, “Homotopy inner products for cyclic operads”, preprint, 2003.
• M. Manetti, “Lie description of higher obstructions to deforming submanifolds”, preprint, 2005.
• M. Markl, S. Shnider, and J. Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, American Mathematical Society, Providence, RI, 2002.
• S. A. Merkulov, “Strong homotopy algebras of a Kähler manifold”, Internat. Math. Res. Notices 3 (1999), 153–164.
• S. A. Merkulov, “Nijenhuis infinity and contractible differential graded manifolds”, Compos. Math. 141:5 (2005), 1238–1254. \codarefmath.DG/0403244
• S. A. Merkulov, “PROP profile of Poisson geometry”, Comm. Math. Phys. 262:1 (2006), 117–135.
• S. A. Merkulov, “Graph complexes with loops and wheels”, in Algebra, Arithmetic and Geometry: the Manin Festschrift, edited by Y. Tschinkel and Y. G. Zarhin, Birkhaüser, 2008.
• S. A. Merkulov, “Lectures on props, Poisson geometry and deformation quantization”, pp. 223–258 in Poisson Geometry in Mathematics and Physics, edited by G. Dito et al., Contemporary Mathematics 450, American Mathematical Society, Providence, RI, 2008. To appear.
• S. A. Merkulov and B. Vallette, “Deformation theory of representations of prop(erad)s”, preprint, 2007. To appear in J. Reine Angew. Math.
• Z. Ran, “Lie atoms and their deformations”, preprint, 2004. To appear in Geomet. Funct. Anal.
• Z. Ran, “Jacobi cohomology, local geometry of moduli spaces, and Hitchin connections”, Proc. London Math. Soc. $(3)$ 92:3 (2006), 545–580.
• B. Vallette, “A Koszul duality for PROPs”, Trans. Amer. Math. Soc. 359:10 (2007), 4865–4943.