Schouten BracketEdit
The Schouten bracket is a foundational construct in differential geometry that generalizes the familiar Lie bracket of vector fields to the whole exterior algebra of a manifold’s tangent bundle. Named after Jan Arnold Schouten, it provides a natural graded Lie algebra structure on the space of polyvector fields, turning the direct sum of exterior powers of the tangent bundle into a rich algebraic object that interacts nicely with the wedge product and with notions of derivations. In many texts it is referred to as the Schouten–Nijenhuis bracket, acknowledging a parallel development by Nijenhuis and others; the term is now standard in both pure geometry and its applications to physics.
The bracket acts on sections of the exterior powers of the tangent bundle, that is, on polyvector fields. If P is a p-vector field and Q is a q-vector field, then [P, Q] is a (p+q−1)-vector field. This bracket extends the ordinary Lie bracket of vector fields in the sense that for two vector fields X and Y (i.e., 1-vectors), [X, Y] is the usual Lie bracket. It also interacts with functions and the wedge product in a way that makes the whole collection of polyvector fields into a graded Lie algebra.
Definition and basic properties
Setup and basic objects
- Let M be a smooth manifold, and denote by Γ(Λ^k TM) the space of smooth sections of the k-th exterior power of the tangent bundle. The direct sum Γ(Λ^• TM) = ⊕_{k≥0} Γ(Λ^k TM) is the space of all polyvector fields.
- The Schouten bracket [·,·] is a bilinear operation [·,·] : Γ(Λ^p TM) × Γ(Λ^q TM) → Γ(Λ^{p+q−1} TM) defined for homogeneous P ∈ Γ(Λ^p TM) and Q ∈ Γ(Λ^q TM).
Defining rules (coordinate-free description)
- If P = X_1 ∧ … ∧ X_p and Q = Y_1 ∧ … ∧ Y_q are simple p- and q-vectors made from vector fields X_i, Y_j, then [P, Q] = ∑{i=1}^p ∑{j=1}^q (-1)^{i+j} [X_i, Y_j] ∧ X_1 ∧ … ∧ X_{i−1} ∧ X_{i+1} ∧ … ∧ X_p ∧ Y_1 ∧ … ∧ Y_{j−1} ∧ Y_{j+1} ∧ … ∧ Y_q.
- The bracket is extended to arbitrary P, Q by bilinearity and by requiring the above formula to hold on decomposable elements and to satisfy linearity.
- For a polyvector P ∈ Γ(Λ^p TM) and a function f ∈ C∞(M) (a 0-vector), the action reduces to [P, f] = i_P df, where i_P is the interior product (contraction) of P with the one-form df. In particular, for p = 1 (P is a vector field X), [X, f] = X(f).
- The bracket satisfies the graded antisymmetry [P, Q] = −(−1)^{(p−1)(q−1)} [Q, P], and the graded Jacobi identity [P, [Q, R]] = [[P, Q], R] + (−1)^{(p−1)(q−1)} [Q, [P, R]], for homogeneous P, Q, R.
Compatibility with the wedge product
- The Schouten bracket turns ⊕_k Γ(Λ^k TM) into a graded Lie algebra in the sense above, and it satisfies a Leibniz-type rule with respect to the wedge product: [P, Q ∧ R] = [P, Q] ∧ R + (−1)^{(p−1) q} Q ∧ [P, R].
- This makes the space of polyvector fields a graded Lie algebra with a compatible graded-commutative algebra structure given by the wedge product.
Relation to the ordinary Lie bracket
- When p = q = 1, P and Q are vector fields, and [P, Q] recovers the usual Lie bracket of vector fields. The Schouten bracket therefore can be viewed as an extension of the Lie algebra of vector fields to higher-degree objects.
Graded structure and special cases
Degrees and conventions
- The degree of the Schouten bracket lowers the total degree by one: if P has degree p and Q has degree q, then [P, Q] has degree p+q−1. This degree convention is part of what makes the bracket a graded Lie bracket.
- Different authors may adopt slightly different sign conventions or notational preferences (for instance, when relating to the Gerstenhaber bracket or to the Nijenhuis bracket). The essential algebraic structure—graded antisymmetry and the graded Jacobi identity—persists across these conventions.
Extensions to supergeometry
- On supermanifolds and in graded geometry, the Schouten bracket generalizes to act on graded exterior algebras, preserving the same essential properties but with signs adjusted for parity. This is particularly relevant in the study of Poisson supermanifolds and in the mathematical formulation of certain physical theories.
Examples and applications
Basic examples
- If P and Q are functions (0-vectors), their Schouten bracket is zero: [f, g] = 0.
- If P is a vector field X (a 1-vector) and f is a function, [X, f] = X(f), i.e., the directional derivative of f along X.
- If P and Q are both vector fields, [X, Y] is the ordinary Lie bracket of vector fields.
Poisson geometry
- A bivector field π ∈ Γ(Λ^2 TM) defines a bracket on functions by {f, g} = π(df, dg). This bracket is a genuine Poisson bracket precisely when the Schouten bracket vanishes on the bivector with itself: [π, π] = 0.
- The condition [π, π] = 0 encodes the Jacobi identity for the induced Poisson bracket, linking the Schouten bracket directly to the core criterion for a Poisson structure on M.
- In this context the Schouten bracket sits at the interface between differential geometry and classical mechanics, providing a coordinate-free way to express Poisson equations and Hamiltonian flows.
Lie algebroids and higher structures
- The Schouten bracket generalizes to Lie algebroids, where A is a vector bundle with a bracket on sections and an anchor map to TM. The exterior algebra of A’s sections carries a Schouten-type bracket that mirrors the geometric structure of the base manifold and the algebroid bracket.
- This framework underpins the study of deformations, cohomology theories, and the algebraic aspects of foliation theory.
Physics and the BV formalism
- In theoretical physics, an odd version of the Poisson bracket—often called the antibracket or BV bracket—arises in the Batalin–Vilkovisky formalism. Mathematically, this is closely related to a Schouten-type bracket on a graded ring of functionals. The master equation (S, S) = 0 is the BV version of a consistency condition for gauge theories.
- The Schouten bracket thus provides a rigorous mathematical language for structures that are central to the quantization of gauge theories.
Controversies and conventions
Sign and degree conventions
- A practical point of contention in the literature is the precise sign convention and degree assignment for the Schouten bracket in different contexts (pure geometry vs. physics; ordinary geometry vs. supergeometry). While the core properties are invariant, the explicit signs in formulas can vary between authors, leading to confusion if one switches between sources. This is a technical issue rather than a substantive disagreement about the mathematics.
Nomenclature and scope
- Some authors emphasize the term Schouten bracket, others prefer Schouten–Nijenhuis bracket, and a few discuss related brackets (e.g., the Nijenhuis bracket) in adjacent contexts. The choice often reflects historical lineage or emphasis on related structures. The exact scope—whether one uses the term for the full graded Lie structure on polyvector fields or reserves it for a particular degree range—can differ across texts.
Conventions in physics vs. geometry
- The use of the Schouten bracket in physics, especially in the BV formalism and in the language of graded manifolds, sometimes adopts conventions that differ from pure differential geometry. For practitioners, awareness of the chosen grading, parity, and sign rules is essential to avoid misinterpretations when translating results across disciplines.
Role in deformation theory
- In deformation theory, the Schouten bracket appears as part of the Gerstenhaber algebra structure on polyvector fields, encoding infinitesimal deformations of structures like Poisson brackets. Debates here focus on the proper handling of higher-order terms, the interpretation of cohomology classes, and the compatibility with additional algebraic structures (e.g., L-infinity algebras) that may arise in specific contexts.