SPECIAL n-FORMS ON A 2n-DIMENSIONAL VECTOR SPACE

The configuration of regular 3-forms in dimension 6 is generalized to n-forms in dimension 2n. The algebras of complex, paracomplex, and dual numbers are systematically used. The automorphism groups of all forms are determined. In the last decade there has arisen interest in exterior forms of higher degree, first of all in forms of degree 3 (see [2] and [3]). It is known (see [4] and [1]) that on a 6-dimensional real vector space there are exactly three types (= orbits) of regular 3-forms, and that these types are closely related with 2-dimensional unital algebras. In this note we show that these forms can be generalized to higher dimensions. Namely, using 2-dimensional unital algebras we construct n-forms on a 2n-dimensional real vector space, and investigate their properties. We shall consider all three 2-dimensional unital, associative and commutative real algebras, namely C = [1, i], i = −1 algebra of complex numbers, D = [1, d], d = 1 algebra of paracomplex numbers, E = [1, e], e = 0 algebra of dual numbers. Let V be a 2n-dimensional real vector space, n ≥ 3. On this vector space we shall consider consider three endomorphisms J , D, and E, respectively. We shall assume that they satisfy J = −I (complex structure) D = I, dimker(D − I) = dimker(D + I) = n (product structure) E = 0, dim imE = dimkerE = n (tangent structure). If V is endowed with a complex structure J (resp. product structure D, resp. tangent structure E), we can introduce on V a structure of a C-module (i. e. complex vector space) (resp. D-module, resp. E-module) in the following way (a+ bi)v = av + bJv (resp. (a+ bd)v = av + bDv, resp. (a+ be)v = av + bEv). On the other hand if V carries a structure of a C-module (resp. D-module, resp. E-module), we can introduce on V a complex structure J (resp. product structure