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summary:We look at two examples of homotopy Lie algebras (also known as $L_{\infty }$ 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 $\Delta $ to verify the homotopy Lie data is shown to produce the same results

Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket