
In theoretical physics, BatalinVilkovisky (BV) formalism was developed as a method for determining the ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra action. The formalism, based on a Lagrangian that contains both fields and "antifields", can be thought of as a very complicated generalization of the BRST formalism. BatalinVilkovisky algebras A BatalinVilkovisky algebra is a graded supercommutative algebra (with identity 1) with a secondorder differential operator Δ of degree 1, with Δ^{2}=0 and Δ(1)=0. More precisely it satisfies the identities * ab = a + b (The product has degree 0) * Δ(a) = 1+a (Δ has degree 1) * (ab)c=a(bc), ab=(−1)^{ab}ba (the product is associative and (super) commutative) * Δ^{2}=0 * Δ(1)=0 (Normalization) * Δ is second order, in other words for any a, the supercommutator [Δ,a] is a derivation. A BatalinVilkovisky algebra becomes a Gerstenhaber algebra if one defines the Poisson bracket by [a,b] = ( − 1) ^{ a } Δ(ab) − ( − 1) ^{ a } Δ(a)b − aΔ(b). Master equation The (classical) master equation for an odd degree element S of a BatalinVilkovisky algebra (or more generally a Lie superalgebra) is the equation [S,S] = 0 The quantum master equation for an odd degree element S of a BatalinVilkovisky algebra (or more generally a Lie superalgebra with an odd derivation Δ) is the equation [S,S] = 2Δ(S) or equivalently [S − Δ,S − Δ] = 0. Examples * If L is a Lie superalgebra, and Π is the operator exchanging the even and odd parts of a super space, then the symmetric algebra of Π(L) (the "exterior algebra" of L) is a BatalinVilkovisky algebra with Δ given by the usual differential used to compute Lie algebra cohomology. * If A is a BatalinVilkovisky algebra, and S a solution of the quantum master equation, then changing Δ to Δ + [S, ] gives a new BatalinVilkovisky algebra. See also * analysis of flows. References * E. Getzler BatalinVilkovisky algebras and twodimensional topological field theories, Communications in Mathematical Physics, Volume 159, Number 2 / January, 1994, Pages 265285. doi:10.1007/BF02102639 * Steven Weinberg The Quantum Theory of Fields Vol. II ISBN 0521670543 Retrieved from "http://en.wikipedia.org/" 
