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# Mathematical Research Letters

## Volume 15 (2008)

### Number 6

### A universal enveloping for $L_\infty$-algebras.

Pages: 1073 – 1089

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n6.a1

#### Author

#### Abstract

For any $L_\infty$-algebra $L$ we construct an $A_\infty$-algebra structure on the symmetric coalgebra $Sym^*_c(L)$ and prove that this structure satisfies properties generalizing those of the usual universal enveloping algebra. These properties follow from an invariant contracting homotopy one the cobar construction of an exterior coalgebra and its relation to combinatorics of permutahedra and semistandard Young tableaux.