Free Homotopy Algebras

An explicit description of free strongly homotopy associative and free strongly homotopy Lie algebras is given. A variant of the Poincaré-Birkhoff-Witt theorem for the universal enveloping A(m)-algebra of a strongly homotopy Lie algebra is formulated. Introduction This note was originated many years ago as my reaction to questions of several people how free strongly homotopy algebras can be described and what can be said about the structure of the universal enveloping A(m)-algebra of an L(m)algebra constructed in [3], and then circulated as a “personal communication.” I must honestly admit that it contains no really deep result and that everything I did was that I expanded definitions and formulated a couple of statements with more or less obvious proofs. A(m)-algebras and their strict homomorphisms [5, pages 147–148] form an equationally given algebraic category A(m). It follows from general theory that the forgetful functor to the category gVect of graded vector spaces, 2 : A(m) → gVect, has a left adjoint m A : gVect → A(m). Given a graded vector space X ∈ gVect, the object m A(X) ∈ A(m) is the free A(m)-algebra on the graded vector space X . We will also, for n < m, consider forgetful functors 2 : A(m) → A(n) and their left adjoints m:n A : A(n) → A(m); here the case n = 1 is particularly important, because m:1 A : dgVect → A(m) describes the free A(m)-algebra generated by a differential graded vector space. We believe there is no need to emphasize the important rôle of free objects in mathematics. Each A(m)-algebra is a quotient of a free one and free A(m)algebras were used in our definition of the universal enveloping algebra of a strongly homotopy Lie algebra [3, page 2154]. It could be useful to have a concrete description of these free algebras, as explicit as, for example, the description of free associative algebras by tensor algebras. This brief note gives such a description in terms of planar trees. Replacing planar trees by non-planar ones, one can equally easy represent also free strongly homotopy Lie algebras. Surprisingly, these free algebras are simpler objects that their strict counterparts and admit a nice linear basis (Section 3), while free (strict) Lie algebras are very complicated objects (see, for example, [9, Chapter IV]). This might be This work was supported by the grant GA ČR 201/96/0310 and MŠMT ME603. . 2000 Mathematics Subject Classification: 08B20