Weil algebra

Motivated by algebraic geometry, A. Weil [a3] suggested the treatment of infinitesimal objects as homomorphisms from algebras of smooth functions into some real finite-dimensional commutative algebra with unit. The points in correspond to the choice , while the algebra , , of dual numbers (also called Study numbers) leads to the tangent vectors at points in (viewed as derivations on functions). At the same time, Ch. Ehresmann established similar objects, jets (cf. also Jet), in the realm of differential geometry, cf. [a1].

Since is formally real (i.e. is invertible for all ), the values of the homomorphisms in are in formally real subalgebras. Now, for each finite-dimensional real commutative unital algebra which is formally real, there is a decomposition of the unit into all minimal idempotent elements. Thus, , where , and are nilpotent ideals in . A real unital finite-dimensional commutative algebra is called a Weil algebra if it is of the form

where is the ideal of all nilpotent elements in . The smallest with the property is called the depth, or order, of .

In other words, one may also characterize the Weil algebras as the formally real and local (i.e. the ring structure is local, cf. also Local ring) finite-dimensional commutative real unital algebras. See [a2], 35.1, for details.

As a consequence of the Nakayama lemma, the Weil algebras can be also characterized as the local finite-dimensional quotients of the algebras of real polynomials . Consequently, the Weil algebras correspond to choices of ideals in of finite codimension. The algebra of Study numbers is given by , for example. Equivalently, one may consider the algebras of formal power series or the algebras of germs of smooth functions at the origin (cf. also Germ) instead of the polynomials.

The width of a Weil algebra is defined as the dimension of the vector space . If is an ideal of finite codimension in , , then the width of equals . For example, the Weil algebra

has width and order , and it coincides with the algebra of -jets of smooth functions at the origin in . Moreover, each Weil algebra of width and order is a quotient of .

Tensor products of Weil algebras are Weil algebras again. For instance, .

The infinitesimal objects of type attached to points in are simply . All smooth functions extend to by the evaluation of the Taylor series (cf. also Whitney extension theorem)

where , , are multi-indices, . Applying this formula to all components of a mapping , one obtains an assignment functorial in both and . Of course, this definition extends to a functor on all locally defined smooth mappings and so each Weil algebra gives rise to a Weil functor . (See Weil bundle for more details.)

The automorphism group of a Weil algebra is a Lie subgroup (cf. also Lie group) in and its Lie algebra coincides with the space of all derivations (cf. also Derivation in a ring) on , , i.e. all mappings satisfying , cf. [a2], 42.9.

References

[a1]	Ch. Ehresmann, "Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal. II. L'espace des jets d'ordre de dans . III. Transitivité des prolongements" C.R. Acad. Sci. Paris , 233 (1951) pp. 598–600; 777–779; 1081–1083 MR0045436 MR0045435 MR0044198
[a2]	I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993) MR1202431 Zbl 1084.53001
[a3]	A. Weil, "Théorie des points proches sur les variétés differentielles" Colloq. Internat. Centre Nat. Rech. Sci. , 52 (1953) pp. 111–117