Double and dual numbers

Hypercomplex numbers of the form , where and are real numbers, and where the double numbers satisfy the relation , while the dual numbers satisfy the relation (cf. Hypercomplex number). Addition of double and dual numbers is defined by

Multiplication of double numbers is defined by

and that of dual numbers by

Complex numbers, double numbers and dual numbers are also called complex numbers of hyperbolic, elliptic and parabolic types, respectively. These numbers are sometimes used to represent motions in the three-dimensional spaces of Lobachevskii, Riemann and Euclid (see, for instance, Helical calculus).

Both double and dual numbers form two-dimensional (with base 1 and ) associative-commutative algebras over the field of real numbers. As distinct from the field of complex numbers, these algebras comprise zero divisors, all these having the form in the algebra of double numbers. The algebra of double numbers may be split into a direct sum of two real number fields. Hence yet another name for double numbers — splitting complex numbers. Double numbers have yet another appellation — paracomplex numbers. The algebra of dual numbers is considered not only over the field of real numbers, but also over an arbitrary field or commutative ring. Let be a commutative ring and let be an -module. The direct sum of -modules equipped with the multiplication

is a commutative -algebra and is denoted by . It is known as the algebra of dual numbers with respect to the module . The -module is identical with the ideal of the algebra which is the kernel of the augmentation homomorphism

The square of this ideal is zero, while . If is a regular ring the converse is also true: If is an -algebra and is an ideal in such that and , then , where is regarded as an -module [4].

If , the algebra (then denoted by ) is isomorphic to the quotient algebra of the algebra of polynomials by the ideal . Many properties of an -module may be formulated as properties of the algebra ; as a result, many problems on -modules can be reduced to corresponding problems in the theory of rings [2].

Let be an arbitrary -algebra, let be a homomorphism and let be a derivation (cf. Derivation in a ring) of with values in the -module , regarded as a -module with respect to the homomorphism . The mapping () will then be a homomorphism of -algebras. Conversely, for any homomorphism of -algebras the composition , where is the projection of onto , is an -derivation of with values in , regarded as a -module with respect to the homomorphism . This property of double and dual numbers is utilized for the description of the tangent space to an arbitrary functor in the category of schemes [1], [3].

References

[1]	D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)
[2]	R. Fossum, P.A. Griffith, I. Reiten, "Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory" , Springer (1975)
[3]	M. Demazure, A. Grothendieck, "Schémas en groupes I" , Lect. notes in math. , 151–153 , Springer (1970)
[4]	S. Lichtenbaum, M. Schlessinger, "The cotangent complex of a morphism" Trans. Amer. Math. Soc. , 128 : 1 (1967) pp. 41–70

Comments

An old-fashioned term for an associative algebra with unit element over is system of hypercomplex numbers, and an element of is then called a hypercomplex number. There are (up to isomorphism) precisely three of these algebras of dimension 2: the complex numbers, the dual numbers and the double numbers.

References

[a1]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970)