Frölicher–Nijenhuis Bracket
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Frölicher–Nijenhuis bracket is an extension of the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
of
vector fields In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and direct ...
to
vector-valued differential form In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differe ...
s on a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
. It is useful in the study of
connections Connections may refer to: * Connection (disambiguation), plural form Television * '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series * ''Connections'' (British TV series), a 1978 documentary tele ...
, notably the
Ehresmann connection In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does ...
, as well as in the more general study of projections in the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
. It was introduced by
Alfred Frölicher Alfred Frölicher (often misspelled Fröhlicher) was a Swiss mathematician (8 October 8 1927 – 1 July 2010). He was a full professor at the Université de Fribourg (1962–1965), and then at the Université de Genève (1966–1993). He introdu ...
and
Albert Nijenhuis Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a Dutch-American mathematician who specialized in differential geometry and the theory of deformations in algebra and geometry, and later worked in combinatorics. His high school st ...
(1956) and is related to the work of Schouten (1940). It is related to but not the same as the Nijenhuis–Richardson bracket and the Schouten–Nijenhuis bracket.


Definition

Let Ω*(''M'') be the
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
of
exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s on a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
''M''. This is a
graded algebra In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
in which forms are graded by degree: :\Omega^*(M) = \bigoplus_^\infty \Omega^k(M). A
graded derivation In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geom ...
of degree ℓ is a mapping :D:\Omega^*(M)\to\Omega^(M) which is linear with respect to constants and satisfies :D(\alpha\wedge\beta) = D(\alpha)\wedge\beta + (-1)^\alpha\wedge D(\beta). Thus, in particular, the
interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterio ...
with a vector defines a graded derivation of degree ℓ = −1, whereas the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
is a graded derivation of degree ℓ = 1. The vector space of all derivations of degree ℓ is denoted by DerΩ*(''M''). The direct sum of these spaces is a
graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a ''grading'' or ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For ...
whose homogeneous components consist of all graded derivations of a given degree; it is denoted :\mathrm\, \Omega^*(M) = \bigoplus_^\infty \mathrm_k\, \Omega^*(M). This forms a
graded Lie superalgebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...
under the anticommutator of derivations defined on homogeneous derivations ''D''1 and ''D''2 of degrees ''d''1 and ''d''2, respectively, by : _1,D_2= D_1\circ D_2 - (-1)^D_2\circ D_1. Any
vector-valued differential form In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differe ...
''K'' in Ω''k''(''M'', T''M'') with values in the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of ''M'' defines a graded derivation of degree ''k'' − 1, denoted by ''i''''K'', and called the insertion operator. For ω ∈ Ω(''M''), :i_K\,\omega(X_1,\dots,X_)=\frac\sum_\textrm\,\sigma \cdot \omega(K(X_,\dots,X_),X_,\dots,X_) The Nijenhuis–Lie derivative along ''K'' ∈ Ωk(''M'', T''M'') is defined by :\mathcal_K = _K, d=i_K\,\,d-(-1)^ d\,\, i_K where ''d'' is the exterior derivative and ''i''K is the insertion operator. The Frölicher–Nijenhuis bracket is defined to be the unique vector-valued differential form : cdot, \cdot : \Omega^k(M,\mathrmM) \times \Omega^\ell(M,\mathrmM) \to \Omega^(M,\mathrmM) : (K, L) \mapsto , L such that :\mathcal_ = mathcal_K, \mathcal_L Hence, : , L=-(-1)^ ,K. If ''k'' = 0, so that ''K'' ∈ Ω0(''M'', T''M'') is a vector field, the usual homotopy formula for the Lie derivative is recovered :\mathcal_K = _K,d=i_K \,\, d+d \,\, i_K. If ''k''=''ℓ''=1, so that ''K,L'' ∈ Ω1(''M'', T''M''), one has for any vector fields ''X'' and ''Y'' : , L(X,Y) = X, LY X, KY(KL+LK) ,YK( X,Y
, LY The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
-L( X,Y
, KY The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
. If ''k''=0 and ''ℓ''=1, so that ''K=Z''∈ Ω0(''M'', T''M'') is a vector field and ''L'' ∈ Ω1(''M'', T''M''), one has for any vector field ''X'' : , L(X) =
, LX The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
L ,X An explicit formula for the Frölicher–Nijenhuis bracket of \phi\otimes X and \psi\otimes Y (for forms φ and ψ and vector fields ''X'' and ''Y'') is given by :\left.\right. phi \otimes X,\psi \otimes Y = \phi\wedge\psi\otimes ,Y+ \phi\wedge\mathcal_X \psi\otimes Y - \mathcal_Y \phi\wedge\psi \otimes X +(-1)^(d\phi \wedge i_X(\psi)\otimes Y +i_Y(\phi) \wedge d\psi \otimes X).


Derivations of the ring of forms

Every derivation of Ω*(''M'') can be written as :i_L + \mathcal_K for unique elements ''K'' and ''L'' of Ω*(''M'', T''M''). The Lie bracket of these derivations is given as follows. *The derivations of the form \mathcal_K form the Lie superalgebra of all derivations commuting with ''d''. The bracket is given by :: mathcal_,\mathcal_ \mathcal_ :where the bracket on the right is the Frölicher–Nijenhuis bracket. In particular the Frölicher–Nijenhuis bracket defines a
graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...
structure on \Omega(M,\mathrmM), which extends the
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
of
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s. *The derivations of the form i_L form the Lie superalgebra of all derivations vanishing on functions Ω0(''M''). The bracket is given by :: _,i_ i_ :where the bracket on the right is the Nijenhuis–Richardson bracket. *The bracket of derivations of different types is given by :: mathcal_, i_L i_ - (-1)^\mathcal_ : for ''K'' in Ωk(''M'', T''M''), ''L'' in Ωl+1(''M'', T''M'').


Applications

The
Nijenhuis tensor In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
of an
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
''J'', is half the Frölicher–Nijenhuis bracket of ''J'' with itself. An almost complex structure is a complex structure if and only if the Nijenhuis tensor is zero. With the Frölicher–Nijenhuis bracket it is possible to define the
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
and cocurvature of a vector-valued 1-form which is a
projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...
. This generalizes the concept of the curvature of a
connection Connection may refer to: Mathematics *Connection (algebraic framework) *Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold * Connection (affine bundle) *Connection (composite bun ...
. There is a common generalization of the Schouten–Nijenhuis bracket and the Frölicher–Nijenhuis bracket; for details see the article on the Schouten–Nijenhuis bracket.


References

*. *. * *. {{DEFAULTSORT:Frolicher-Nijenhuis Bracket Bilinear maps Differential geometry