nLab Albert Nijenhuis

Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a mathematician at the University of Pennsylvania.

He got his PhD degree in 1952 from the University of Amsterdam, advised by Jan Schouten.

He introduced the notion of a natural bundles and natural operation in differential geometry, which was subsequently developed by others.

Selected writings

On the Schouten-Nijenhuis bracket:

• Albert Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indagationes Mathematicae 17 (1955) 390–403 [doi:10.1016/S1385-7258(55)50054-050054-0)]

Introducing the notion of natural operations in differential geometry:

• Albert Nijenhuis, Geometric aspects of formal differential operations on tensor fields, Proceedings of the International Congress of Mathematicians 1958, 463–469.

• Albert Nijenhuis, Natural bundles and their general properties, in: Differential Geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 317–334.

Introducing the Frölicher-Nijenhuis bracket:

• Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part I. Derivations in the graded ring of differential forms, Indagationes Mathematicae (Proceedings) 59 (1956) 338–350 and 351–359 [doi:10.1016/s1385-7258(56)50046-7, doI;10.1016/s1385-7258(56)50047-9]

and its refinement for almost complex structures:

• Alfred Frölicher, Albert Nijenhuis, Theory of vector-valued differential forms. Part II. Almost-complex structures, Indagationes Mathematicae (Proceedings) 61 (1958) 414–421 and 422-429 [doi:10.1016/S1385-7258(58)50058-4, doi:10.1016/S1385-7258(58)50057-2]

Giving an explicit formula for what came to be known as the Nijenhuis–Richardson bracket:

• Albert Nijenhuis, Roger W. Richardson, Cohomology and deformations of algebraic structures, Bulletin of the American Mathematical Society 70 3 (1964) 406–412 [doi:10.1090/s0002-9904-1964-11117-4]

• Albert Nijenhuis, R. W. Richardson, Deformations of Lie Algebra Structures, Journal of Mathematics and Mechanics 17 1 (1967) 89-105 [jstor:24902154]

Beware that this other article has a similar title to Nijenhuis & Richardson 1964, but does not seem to mention the Nijenhuis–Richardson bracket:

• Albert Nijenhuis, R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966) 1-29 [doi:10.1090/S0002-9904-1966-11401-5]

Related entries

• Nijenhuis tensor

• Schouten–Nijenhuis bracket

• Frölicher–Nijenhuis bracket

• Nijenhuis–Richardson bracket

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