An explicit construction of the Quillen homotopical category of dg Lie algebras
| dc.creator | Shoikhet, Boris | |
| dc.date | 2007-06-09 | |
| dc.date.accessioned | 2026-07-07T08:04:49Z | |
| dc.date.available | 2026-07-07T08:04:49Z | |
| dc.description | Let $\g_1$ and $\g_2$ be two dg Lie algebras, then it is well-known that the $L_\infty$ morphisms from $\g_1$ to $\g_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $\Bbbk(\g_1,\g_2)$. Then the gauge action by exponents of the zero degree component $\Bbbk(\g_1,\g_2)^0$ on $MC\subset\Bbbk(\g_1,\g_2)^1$ gives an explicit "homotopy relation" between two $L_\infty$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_\infty$ algebras and morphisms are $L_\infty$ morphisms modulo the gauge relation) is well-defined, and is a localization of the category of dg Lie algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to an equivalence, it is equivalent to the Quillen-Hinich homotopical category of dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillen's concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D]. | |
| dc.description | LaTeX, 17 pages | |
| dc.identifier | https://arxiv.org/abs/0706.1333 | |
| dc.identifier | http://arxiv.org/abs/0706.1333 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130101 | |
| dc.subject | K-Theory and Homology | |
| dc.subject | Quantum Algebra | |
| dc.title | An explicit construction of the Quillen homotopical category of dg Lie algebras | |
| dc.type | text |