View source for Weil algebra

Motivated by [[Algebraic geometry|algebraic geometry]], A. Weil [[#References|[a3]]] suggested the treatment of infinitesimal objects as homomorphisms from algebras of smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200501.png" /> into some real finite-dimensional commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200502.png" /> with unit. The points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200503.png" /> correspond to the choice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200504.png" />, while the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200506.png" />, of dual numbers (also called Study numbers) leads to the tangent vectors at points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200507.png" /> (viewed as derivations on functions). At the same time, Ch. Ehresmann established similar objects, jets (cf. also [[Jet|Jet]]), in the realm of [[Differential geometry|differential geometry]], cf. [[#References|[a1]]].

Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200508.png" /> is formally real (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200509.png" /> is invertible for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005010.png" />), the values of the homomorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005011.png" /> are in formally real subalgebras. Now, for each finite-dimensional real commutative [[unital algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005012.png" /> which is formally real, there is a decomposition of the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005013.png" /> into all minimal idempotent elements. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005016.png" /> are nilpotent ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005017.png" />. A real unital finite-dimensional commutative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005018.png" /> is called a Weil algebra if it is of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005020.png" /> is the ideal of all nilpotent elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005021.png" />. The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005022.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005023.png" /> is called the depth, or order, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005024.png" />.

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|Local ring]]) finite-dimensional commutative real unital algebras. See [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005025.png" />. Consequently, the Weil algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005026.png" /> correspond to choices of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005028.png" /> of finite codimension. The algebra of Study numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005029.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005030.png" />, for example. Equivalently, one may consider the algebras of [[Formal power series|formal power series]] or the algebras of germs of smooth functions at the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005031.png" /> (cf. also [[Germ|Germ]]) instead of the polynomials.

The width of a Weil algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005032.png" /> is defined as the dimension of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005034.png" /> is an ideal of finite codimension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005036.png" />, then the width of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005037.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005038.png" />. For example, the Weil algebra

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005039.png" /></td> </tr></table>

has width <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005040.png" /> and order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005041.png" />, and it coincides with the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005043.png" />-jets of smooth functions at the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005044.png" />. Moreover, each Weil algebra of width <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005045.png" /> and order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005046.png" /> is a quotient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005047.png" />.

Tensor products of Weil algebras are Weil algebras again. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005048.png" />.

The infinitesimal objects of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005049.png" /> attached to points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005050.png" /> are simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005051.png" />. All smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005052.png" /> extend to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005053.png" /> by the evaluation of the [[Taylor series|Taylor series]] (cf. also [[Whitney extension theorem|Whitney extension theorem]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005054.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005057.png" /> are multi-indices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005058.png" />. Applying this formula to all components of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005059.png" />, one obtains an assignment functorial in both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005061.png" />. Of course, this definition extends to a functor on all locally defined smooth mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005062.png" /> and so each Weil algebra gives rise to a Weil functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005063.png" />. (See [[Weil bundle|Weil bundle]] for more details.)

The automorphism group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005064.png" /> of a Weil algebra is a Lie subgroup (cf. also [[Lie group|Lie group]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005065.png" /> and its [[Lie algebra|Lie algebra]] coincides with the space of all derivations (cf. also [[Derivation in a ring|Derivation in a ring]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005067.png" />, i.e. all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005068.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005069.png" />, cf. [[#References|[a2]]], 42.9.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Ch. Ehresmann, "Les prolongements d'une variété différentiable. I. Calcul des jets, prolongement principal. II. L'espace des jets d'ordre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005070.png" /> de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005071.png" /> dans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005072.png" />. III. Transitivité des prolongements" ''C.R. Acad. Sci. Paris'' , '''233''' (1951) pp. 598–600; 777–779; 1081–1083 {{MR|0045436}} {{MR|0045435}} {{MR|0044198}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Kolář, P.W. Michor, J. Slovák, "Natural operations in differential geometry" , Springer (1993) {{MR|1202431}} {{ZBL|1084.53001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Weil, "Théorie des points proches sur les variétés differentielles" ''Colloq. Internat. Centre Nat. Rech. Sci.'' , '''52''' (1953) pp. 111–117</TD></TR></table>