Schouten–Nijenhuis Bracket - Definition and Properties

Definition and Properties

An alternating multivector field is a section of the exterior algebra ∧∗TM over the tangent bundle of a manifold M. The alternating multivector fields form a graded supercommutative ring with the product of a and b written as ab (some authors use a∧b). This is dual to the usual algebra of differential forms Ω∗M by the pairing on homogeneous elements:

The degree of a multivector A in ∧pTM is defined to be |A| = p.

The skew symmetric Schouten–Nijenhuis bracket is the unique extension of the Lie bracket of vector fields to a graded bracket on the space of alternating multivector fields that makes the alternating multivector fields into a Gerstenhaber algebra. It is given in terms of the Lie bracket of vector fields by

for vector fields a_i, b_j and

for vector fields a_i and smooth function f, where i_df is the common inner product operator. It has the following properties.

• |ab| = |a| + |b| (The product has degree 0)
• || = |a| + |b| − 1 (The Schouten–Nijenhuis bracket has degree −1)
• (ab)c = a(bc), ab = (−1)|a||b|ba (the product is associative and (super) commutative)
• = c + (−1)|b|(|a| − 1)b (Poisson identity)
• = −(−1)(|a| − 1)(|b| − 1) (Antisymmetry of Schouten–Nijenhuis bracket)
• ,c] = ] − (−1)(|a| − 1)(|b| − 1)] (Jacobi identity for Schouten–Nijenhuis bracket)
• If f and g are functions (multivectors homogeneous of degree 0), then = 0.
• If a is a vector field, then = L_ab is the usual Lie derivative of the multivector field b along a, and in particular if a and b are vector fields then the Schouten–Nijenhuis bracket is the usual Lie bracket of vector fields.

The Schouten–Nijenhuis bracket makes the multivector fields into a Lie superalgebra if the grading is changed to the one of opposite parity (so that the even and odd subspaces are switched), though with this new grading it is no longer a supercommutative ring. Accordingly, the Jacobi identity may also be expressed in the symmetrical form