Matrix classification of two-dimensional algebras over the field Z2
https://doi.org/10.22405/2226-8383-2025-26-4-398-418
Abstract
The problem of finding orbits of the group 𝐺𝐿(𝑉 ) on the space 𝐴𝑙𝑔(𝑉 ) of all bilinear mappings 𝑉 ×𝑉 → 𝑉 is investigated. The work considers only two-dimensional algebras over the field Z2 without using methods of invariant theory. The classification of algebras is considered from new perspectives. To distinguish a large class of algebras, a mapping 𝑃 is used, which naturally arises as a mapping 𝐴𝑙𝑔(𝑉 ) → 𝑉 * × 𝑉 *, assigning to each algebra structure on the space 𝑉 a pair of linear forms 𝑇𝑟1 and 𝑇𝑟2, defined as traces of the left and right multiplication
operators in this algebra. The 𝜏 -action of the group 𝐺𝐿(2,Z2), consisting of non-degenerate square matrices 𝑔, on the set MSC(A) — matrices of structural constants is studied. This action is generally written as 𝜏 : (𝑔,MSC(A)) → 𝑔MSC(A)(𝑔−1)⊗2. The action 𝜏 defines equivalence between matrix operators of two-dimensional algebras and determines the structure for describing action orbits.
For this action, the 𝑃-mapping was used, which has the form 𝑃(𝜏 (𝑔,MSC(A))) == 𝑃(MSC(A))𝑔−1.
For matrix representatives of two-dimensional algebras, a complete matrix classification with various orbits over Z2 is proposed. The connection 𝑃(𝑔MSC(A)(𝑔−1)⊗2) = 𝑒 between MSC(A) and its defining linearly independent system {𝑇𝑟𝑘} = {𝑇𝑟1, 𝑇𝑟2} is presented. This connection of different matrix representatives of orbits equals 𝑞4. The work also addresses the fact that the matrix representatives of the orbits under the action of 𝜏 could potentially intersect; however, the study rigorously proves their non-intersection.
The interrelation in the form of a system of equalities between elements of equivalent orbits of 𝜏 -action by the group 𝐺𝐿(2,Z2) on MSC(A) for further study over fields of higher order is
shown. The results show that the number of different orbits of 𝜏 -action equals 52.
As a consequence in the group-theoretic sense, this problem is equivalent to describing
multiplication on a two-generated abelian group of the form Z2 ⊕ Z2 up to isomorphism.
In conclusion, some properties of the lattice Z𝑞×Z𝑞 (𝑇𝑟𝑘) using the system of vectors {𝑇𝑟𝑘} are given, from which, in particular, follows the description of equivalent matrices with respect
to number by five disjoint linear forms {𝑇𝑟𝑘} = {𝑇𝑟1(MSC(A)), 𝑇𝑟2(MSC(A))} = {𝑇𝑟1(𝐴), 𝑇𝑟2(𝐴)} of the dual space of the algebra A.
About the Author
Danila Alekseevich Sharipov
Financial University under the Government of the Russian Federation; National Research University Higher School of Economics
Russian Federation
Russian Federation
References
1. Petersson, H.P., Scherer, M. 2004, “The number of nonisomorphic two-dimensional algebras over a finite field”, Results Math. 42(1–2), pp. 137–152.
2. Petersson, H.P. 2000, “The classification of two-dimensional nonassociative algebras”, Results Math. 37(1–2), pp. 120–154.
3. Bekbaev, U. 2023, “Complete classification of two-dimensional algebras over any basic field”, AIP Conference Proceedings 2880, 030001.
4. Verhulst, N.D. 2020, “Counting finite-dimensional algebras over finite field”, Results Math. 75, pp. 153.
5. Althoen, S.C., Hansen, K.D. 1992, “Two-dimensional real algebras with zero divisors”, Acta Sci. Math (Szeged) 56, pp. 23–42.
6. Henderson, H.V., Searle, S.R. 1981, “The vec-permutation matrix, the vec operator and Kronecker products: a review”, Linear Multilinear Algebra, 9(4), pp. 271–288.
7. Rotman, J.J. 2012, “An Introduction to the Theory of Groups”, Grad. Texts in Math, vol. 148.
8. Springer, Berlin. ISBN: 9781461241768.
9. R¨ohrl, H.A. 1977, “A theorem on non-associative algebras and its applications to differential equations”, Manuscripta Math. 21, pp. 181–187.
10. R¨ohrl, H.A. 1979, “Finite dimensional algebras without nilpotent elements over algebraically closed fields”, Arch. Math. 32, pp. 10–12.
11. Fuchs, L. 1970, “Infinite abelian groups. vol. 1”,Academic Press.
12. Fuchs, L. 1973, “Infinite abelian groups. vol. 2”, Academic Press.
13. Kulikov, L. Ya. 1941, “On the theory of abelian groups of arbitrary power”, Mat. Sbornik, vol. 16, pp. 129–162 (Russian).
14. Arnold, D.M. 1982, “Finite rank torsion-free abelian groups and rings”, Lecture Notes in Math. Vol. 931. Springer, NY.
15. Szele, T. 1955, “Nilpotent Artinian rings”, Publ. Math. Debrecen, 4, pp. 71–78.
16. Feigelstock, S. 1983, “Additive groups of rings”, Vol. 1, 2. Pitman Advanced Publishing Program, 1986.
Review
For citations:
Sharipov D.A. Matrix classification of two-dimensional algebras over the field Z2. Chebyshevskii Sbornik. 2025;26(4):398-418. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-398-418
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