Gerstenhaber algebra

Definition

• |ab| = |a| + |b|
• || = |a| + |b| - 1
• c = a
• ab = ^|a||b|ba
• = c + ^|b|b
• = −
• ,c] + b,[a,c

Examples

• Gerstenhaber showed that the Hochschild cohomology H^* of an algebra A is a Gerstenhaber algebra.
• A Batalin–Vilkovisky algebra has an underlying Gerstenhaber algebra if one forgets its second order Δ operator.
• The exterior algebra of a Lie algebra is a Gerstenhaber algebra.
• The differential forms on a Poisson manifold form a Gerstenhaber algebra.
• The multivector fields on a manifold form a Gerstenhaber algebra using the Schouten–Nijenhuis bracket