Antibracket algebra

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In theoretical physics, the antibracket algebra is used in the Batalin-Vilkovisky formalism. A is an antibracket algebra if there are two bilinear products on it, . and ((,)) such that it is a Z×Z₂ graded associative algebra under . and satisfies the following:

for pure elements x, y, z, Let ghost[x] denote the Z grading of x and |x| be 0 or 1 depending on whether its Z₂ grading is even or odd.

Then, ghost[((x,y))]=ghost[x]+ghost[y]+1 and |((x,y))|=|x|+|y|+1 (mod 2). i.e. . and ((,)) do not satisfy the same grading relations.

Also,

((y,x))=-(-1)^{(|x|+1)(|y|+1)}((x,y))

((((x,y)),z))+(-1)^{(|x|+1)(|y|+|z|)}((((y,z)),x))+(-1)^{(|z|+1)(|x|+|y|)}((((z,x)),y))=0

((x,yz))=((x,y))z+(-1)^|y||z|((x,z))y

((xy,z))=x((y,z))+(-1)^|x||y|y((x,z))