XXXIII Workshop on Geometric Methods in Physics | 29.06-5.07.2014 |

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## Martin Markl## On the origin of higher bracesThe Koszul hierarchy (aka higher braces or Koszul braces) is an explicit construction that, for any commutative associative algebra A with a differential Delta (which is, very crucially, not necessary a derivation), produces a sequence of multilinear maps Phi_n : A x ... x A ---> A (n copies of A) such that (1) the operations Phi_n form a strongly homotopy Lie algebra, and (2) Phi_n = 0 implies Phi_{n+1} = 0 (heredity property) Koszul braces are used for instance to define higher-order derivations: Delta is a degree n derivation if Phi_{n+1}(Delta) = 0. Higher order derivations play an important role e.g. in BRST approach to closed string field theory. Recently, a similar construction appeared also for associative (non-commutative) algebras. I will show that both braces are given by the twisting by a specific unique automorphism and that they are essentially unique. Consequently, the notion of higher-order derivations is God given, not human invention. |

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