We recall the definition of strong homotopy derivations of algebras and introduce the corresponding definition for algebras. We define strong homotopy inner derivations for both algebras and exhibit explicit examples of both.
We look at two examples of homotopy Lie algebras (also known as algebras) in detail from two points of view. We will exhibit the algebraic point of view in which the generalized Jacobi expressions are verified by using degree arguments and combinatorics. A second approach using the nilpotency of Grassmann-odd differential operators to verify the homotopy Lie data is shown to produce the same results.
Summary: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on coincides with the natural symmetric brace structure on , the direct sum of spaces of antisymmetric maps .
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