On the group of automorphisms of the algebra of plural numbers
- DOI
- 10.5922/0321-4796-2023-54-2-6
- Pages
- 63-70
Abstract
The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element.
In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he established that higher-order tangent bundles over smooth manifolds have the structure of a smooth manifold over algebras of plural numbers. This allowed him in the 70s of the twentieth century to construct a theory of lifts of tensor fields and linear connections from a smooth manifold to its tangent bundles of arbitrary order.
In this paper, we study automorphisms of the algebra of plural numbers. It is proved that the set of all automorphisms of the algebra of plural numbers forms a group. The structure of this group is described. The groups of automorphisms of the algebra of plural numbers with small dimension are indicated as examples.
Reference
1. Vishnevskiy, V. V., Shirokov, А. P., Shurygin, V. V.: Spaces over algebras. Kazan (1984).
2. Vinberg, E. B.: Course of algebra. Moscow (2001).
3. Curtis, I., Reiner, I.: Theory of representations of finite groups and associative algebras. Moscow (1969).
4. Kostrikin, A. I.: Introduction to Algebra. Parts 1—3. Moscow (2000).
5. Kurosh, A. G.: Lectures on general algebra. Moscow (1962).
6. Nikitina, Ya. V., Sultanov, A. Ya.: Weil bundle over the tensor product of two algebras of dual numbers. Izvestia vuzov. Volga region. Phys.-Math. Nauki, 4:28, 17—28 (2013).
7. Sultanov, A. Ya.: On automorphism groups of special linear algebras. Izvestia Penza State Ped. Univ., 8:22, 70—74 (2010).
8. Sultanova, G. A.: On the dimensions of Lie algebras of automorphisms in tangent bundles with a complete lift connection over a projective Euclidean base. Dalnevost. Math. J., 16:1, 83—95 (2016).
9. Sultanova, G. A.: On an estimate for the dimensions of Lie algebras of infinitesimal automorphisms of tangent bundles with a complete lift connection over a nonprojective Euclidean base. DGMF, 47, 146—153 (2016).
10. Shirokov, A. P.: A note on structures in tangent bundles. Tr. Geom. Sem., 5, 259—309 (1973).