Nijenhuis-Richardson bracket

In mathematics, the algebraic bracket or Nijenhuis-Richardson bracket is a graded Lie algebra structure on the space of alternating multilinear forms of a vector space to itself, introduced by A. Nijenhuis and R. Richardson (1966, 1967). It is related to but not the same as the Frölicher-Nijenhuis bracket and the Schouten-Nijenhuis bracket.

Definition

The primary motivation for introducing the bracket was to develop a uniform framework for discussing all possible Lie algebra structures on a vector space, and subsequently the deformations of these structures. If "V" is a vector space and "p" ≥ -1 is an integer, let :$Alt^p(V)\; =\; (wedge^\{p+1\}\; V^*)otimes\; V$be the space of all skew-symmetric ("p"+1)-multilinear mappings of "V" to itself. The direct sum Alt("V") is a graded vector space. A Lie algebra structure on "V" is determined by a skew-symmetric bilinear map μ : "V" × "V" → "V". That is to say, μ is an element of Alt¹(V). Furthermore, μ must obey the Jacobi identity. The Nijenhuis-Richardson bracket supplies a systematic manner for expressing this identity in the form [μ,μ] =0.

In detail, the bracket is a bilinear bracket operation defined on Alt("V") as follows. On homogeneous elements "P" ∈ Alt^p("V") and "Q" ∈ Alt^q("V"), the Nijenhuis-Richardson bracket ["P","Q"] ^∧ ∈ Alt^p+q(V) is given by:$[P,Q]\; ^and\; =\; i\_P\; Q\; -\; (-1)^\{pq\}i\_Q\; P.,$Here the interior product "i"_"P" is defined by:$(i\_P\; Q)(X\_0,X\_1,ldots,X\_\{p+q\})\; =\; sum\_\{sigmain\; Sh\_\{p,qmathrm\{sgn\}(sigma)\; P(Q(X\_0,X\_1,ldots,X\_q),X\_\{q+1\},ldots,X\_\{q+p\})$where the sum is over all (p,q) shuffles of the indices. On non-homogeneous elements, the bracket is extended by bilinearity.

Derivations of the ring of forms

The Nijenhuis-Richardson bracket can be defined on the vector valued forms Ω^*("M", "T"("M")) on a smooth manifold "M"in a similar way. Vector valued forms act as derivations on the supercommutative ring Ω^*("M") of forms on "M"by taking "K" to the derivation "i_K", and the Nijenhuis-Richardson bracket then corresponds to the commutator of two derivations. This identifies Ω^*("M", "T"("M")) with the algebra of derivations that vanish on smooth functions. Not all derivations are of this form; for the structure of the full ring of all derivations see the article Frölicher-Nijenhuis bracket.

The Nijenhuis-Richardson bracket and the Frölicher-Nijenhuis bracket both make Ω^*("M", "T"("M")) into a graded superalgebra, but have different degrees.

References

*Pierre Lecomte, Peter W. Michor, Hubert Schicketanz, "The multigraded Nijenhuis–Richardson algebra, its universal property and application" J. Pure Appl. Algebra, 77 (1992) 87–102
*springer|id=F/f120230|author=P. W. Michor|title=Frölicher–Nijenhuis bracket
*P. W. Michor, H. Schicketanz, [http://arxiv.org/math.DG/9201255 "A cohomology for vector valued differential forms"] Ann. Global Anal. Geom. 7 (1989), 163-169
*A. Nijenhuis, R. Richardson, "Cohomology and deformations in graded Lie algebras" Bull. Amer. Math. Soc. , 72 (1966) pp. 1–29
*A. Nijenhuis, R. Richardson, "Deformation of Lie algebra structures", J. Math. Mech. 17 (1967), 89–105.