Martin Markl

On the origin of higher braces

The 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.