Small non-associative division algebras up to isotopy

We classify small, non-associative division algebras up to isotopy. We reduce the classification problem to an involved case distinction that a computer program can solve. As a result, we classify algebras with 4, 8, 16, and 9 elements. In particular, we show that non-associative division algebras...

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Published in:	Algebra and Discrete Mathematics
Date:	2010
Main Author:	Schwarz, T.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2010
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/154498
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Cite this:	Small non-associative division algebras up to isotopy / T. Schwarz // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 103–108. — Бібліогр.: 4 назв. — англ.

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spelling	Schwarz, T. 2019-06-15T16:10:47Z 2019-06-15T16:10:47Z 2010 Small non-associative division algebras up to isotopy / T. Schwarz // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 103–108. — Бібліогр.: 4 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:17D99. https://nasplib.isofts.kiev.ua/handle/123456789/154498 We classify small, non-associative division algebras up to isotopy. We reduce the classification problem to an involved case distinction that a computer program can solve. As a result, we classify algebras with 4, 8, 16, and 9 elements. In particular, we show that non-associative division algebras of size 4, 8, and 9 are isotopes of a Galois field, whereas there are three isotopy classes of division algebras with 16 elements. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Small non-associative division algebras up to isotopy Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Small non-associative division algebras up to isotopy
spellingShingle	Small non-associative division algebras up to isotopy Schwarz, T.
title_short	Small non-associative division algebras up to isotopy
title_full	Small non-associative division algebras up to isotopy
title_fullStr	Small non-associative division algebras up to isotopy
title_full_unstemmed	Small non-associative division algebras up to isotopy
title_sort	small non-associative division algebras up to isotopy
author	Schwarz, T.
author_facet	Schwarz, T.
publishDate	2010
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	We classify small, non-associative division algebras up to isotopy. We reduce the classification problem to an involved case distinction that a computer program can solve. As a result, we classify algebras with 4, 8, 16, and 9 elements. In particular, we show that non-associative division algebras of size 4, 8, and 9 are isotopes of a Galois field, whereas there are three isotopy classes of division algebras with 16 elements.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/154498
citation_txt	Small non-associative division algebras up to isotopy / T. Schwarz // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 103–108. — Бібліогр.: 4 назв. — англ.
work_keys_str_mv	AT schwarzt smallnonassociativedivisionalgebrasuptoisotopy
first_indexed	2025-11-25T07:17:53Z
last_indexed	2025-11-25T07:17:53Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 9 (2010). Number 1. pp. 103 – 108 c© Journal “Algebra and Discrete Mathematics” Small non-associative division algebras up to isotopy Thomas Schwarz, S.J. Communicated by Robert Wisbauer Abstract. We classify small, non-associative division alge- bras up to isotopy. We reduce the classification problem to an involved case distinction that a computer program can solve. As a result, we classify algebras with 4, 8, 16, and 9 elements. In par- ticular, we show that non-associative division algebras of size 4, 8, and 9 are isotopes of a Galois field, whereas there are three isotopy classes of division algebras with 16 elements. Introduction Assume a distributed database consisting of a large number of objects that can be created, deleted, and modified by manipulations. We want to capture the state of the system in a very small bit-string (a signature as in [2]). We now hash (map) the set of all possible manipulations into a finite, non-associative algebra. We can capture the state of an object also in a signature, an element in that algebra, initially 1. Whenever a manipulation changes an object, we multiply the current signature of the object with the signature of the manipulation. Since there are frequent manipuliations that change a large number of objects in a given set (such as increasing the salary of all objects representing faculty), the distribu- tive law becomes useful. The state of the database is given as the sum of the signatures of all objects. Since manipulations do not commute, the finite algebra made up of all possible (non-zero) signatures needs to be non-communative, thus, it also needs to be non-associative. Furthermore, 2000 Mathematics Subject Classification: 17D99. Key words and phrases: Non-associative Division Algebras, Isotopy. Jo u rn al A lg eb ra D is cr et e M at h .104 Non-associative division algebras the algebra should be a division algebra, i.e. a vector space over a finite field with a multiplication such that the left multiplication of non-zero elements is invertible. A simple way to come up with candidate algebras is to use the isotope of a Galois field GF(2k). The concept of isotopy goes back to A. A. Albert [1]. Our application leads to the mathematical question on the classification of finite algebras up to isotopy. Finite algebras of dimension two are isotopes of Galois fields. Few work seems to have been done in this area, an exception being [3, 4]. In this article, we present a classification for algebras with 8, 16, and 27 elements. To do so, we reduce the number of possibilities using elementary mathematical arguments and then use software for a final, brute force calculation. The vast majority of the work presented here is spent verifying the software, while the actual programs run fast. Due to a combinatorial explosion, obtaining similar results for larger algebras is currently computationally infeasible. The limiting factor is the size of main memory and the much slower performance of hard drives which leads to runtimes of months and years regardless of parallelization. Further mathematical insight is needed here. Because of the bit-based nature of current computing, results in characteristic 2 are especially valuable. 1. Definition and basic properties Definition 1. A non-associative division algebra A over a field Φ is a non-associative division algebra such that for every element a 6= 0, the left-multiplication La : A → A, x 7→ a · x is a bijection. Definition 2. Let f, g, and h be vector space automorphisms of a non- associate division algebra A. The (f, g, h) isotope of a (non-associative) algebra (A, ·) is the same algebra (A, ⋆) with a new multiplication defined by x ⋆ y := h−1(f(x) · g(x)). Isotopy is an equivalence relation. Obviously, an isotope of a division algebra is also a division algebra. A finite-dimensional algebra is a di- vision algebra if and only if the equation x · y = 0 implies that x or y are zero. Thus, we can equivalently identify finite dimensional division algebras through the right multiplication. 1.1. Existence of unity If A is a division algebra with left multiplication L and a ∈ A is not zero, then a is a left one in the isotope with left multiplication L̂x = L−1 a ◦Lx. With a little bit more work, we can find an isotope with a (left and right) one. Assume that a ∈ A is a left one. Form the isotope with left Jo u rn al A lg eb ra D is cr et e M at h .T. Schwarz 105 multiplication L̂x = R−1 a ◦ Lx ◦ Ra. Clearly, a is still a left one in this isotope, but it is also a right one because x ⋆ a = R−1 a (x(aa)) = x. This important observation is due to Kaplansky, but seems to be unpublished (according to H. Petersson, Hagen). 1.2. Isotopes of fields Assume that (A, ·) is a field with one 1. Assume that its (f, g, h)-isotope (A, ⋆) has a one e. This implies h = Lg(e) ◦ f and g = Lg(e)/f(e) ◦ f. In particular, we have the identity f(a ⋆ b) = h−1 ( g(e)f(a)f(b) f(e) ) . From this, it follows that (A, ⋆) is associative. Thus: Proposition 1. An isotope of a field that has a one is also a field. 2. GF(3)3 We start out with a base B consisting of a one 1 and two elements x, y of GF(3)3. Obviously MatB(Lx) =   0 a b 1 c d 0 e f   with scalars a, b, c, d, e, f ∈ GF(3). In addition, we know that all linear combinations of the identity matrix and Lx are invertible. Replacing x′ by α · 1 + β · x in B changes this matrix to MatB(Lx′) =   0 β2a− α2 − αβc βb− αd 1 2α+ βc d 0 β2e α+ βf   To form a division algebra, all non-trivial linear combinations of the iden- tity matrix and Lx have to be invertible. A computerized brute force calculation reveals that the set of possible left multiplications with the x has 24 equivalence classes. A · Lx · A−1 is the left multiplication matrix with respect to the same base in the (A, id, A−1)-isotope of the original algebra. Accordingly, we introduce a further equivalence relation on the set of all possible left multiplications of the above form that now only has 5 equivalence classes. That is, we can assume without loss of generality and up to isotopy that Lx is one of the following matrices:   0 1 1 1 0 0 0 1 0     0 1 1 1 1 0 0 1 0     0 0 2 1 1 0 0 1 0     0 2 2 1 1 0 0 1 0     0 0 1 1 0 1 0 1 0   Jo u rn al A lg eb ra D is cr et e M at h .106 Non-associative division algebras The choice of these particular matrices is an artifact of the enumeration of matrices in our program. Similarly, MatB(Ly) =   0 a b 0 c d 1 e f   with (different) scalars a, b, c, d, e, f ∈ GF(3). Replacing y by y′ = α · 1 + β · y in B changes this matrix to MatB(Ly′) =   0 βa− αe −α2 + β2b− αβf 0 α+ βc β2d 1 e 2α+ βf   This gives of course again 24 equivalence classes of matrices. Taking the identity matrix, a matrix from the first list, and a matrix from the second list defines a non-associative algebra. However, it turns out that only 10 of them are division algebras. A final brute force calculation reveals that all these algebras are isotopes. 3. Algebras of size 4, 8, 16 In the case of four elements, we can use elementary case distinctions: Proposition 2. GF(4) is the only division algebra with unity 1 over GF(2) such that every sub-algebra generated by 1 and an arbitrary element x has dimension at most 2. Proof: Since Lx is invertible for x 6= 0, x2 = x implies either that x = 1 or that x = 0. If every sub-algebra generated by 1 and x has dimension at most 2, and x 6= 0, 1, then x2 = x + 1. If x and y are linearly independent and not equal to 1, we can apply this equation to x + y and obtain xy = yx + 1. Finally, if we apply this last equation to three linearly independent elements x, y, and z, neither of which equal 1, we obtain (x+ y+ z)2 = (x+ y+ z), which is a contradiction. Hence, the dimension of such an algebra cannot exceed 3. However, as we will see, it can also not have dimension 3. For assume a basis (1, x, y). Since x already has inverse 1 + x, the product xy cannot equal 1. xy = x implies y = 1, a contradiction, and xy = 1 + x means yx = x, also a contradiction. Hence we remain with xy = x + y or yx = x + y. In the first case, (1+x)(1+x+ y) = 0 and in the second (1+x+ y)(1+x) = 0, a contradiction. Hence, such an algebra can only have dimension 2, must have a base (1, x) with x2 = 1 + x, i.e. must be GF(4). Jo u rn al A lg eb ra D is cr et e M at h .T. Schwarz 107 Proposition 3. The only division algebras over GF(2) of dimension 3 are isotopes of GF(8). Proof: According to Proposition 2, there exists an element x such that 1, x, x2 are linearly independent. With respect to this base, the matrix representation of the left multiplications L1, Lx, and Lx2 are given by   1 0 0 0 1 0 0 0 1   ,   0 0 ∗ 1 0 ∗ 0 1 ∗   ,   0 ∗ ∗ 0 ∗ ∗ 1 ∗ ∗   . This gives us 29 possible algebras. Many are not division algebras and we can apply Proposition 1 to ascertain that those that are are indeed fields. Alternatively, we can find an explicit isotopy relationship to GF(8). In both cases, a brute force calculation proves the theorem. Recall that the opposite algebra Aop of an algebra A (with multipli- cation ·) is the same vectorspace, but a new multiplication defined by x ·op y = y · x. Theorem 1. There are three isotopy classes among the division algebras of dimension 4 over GF(2). One class contains GF(24), and any of the other two classes contains the opposite algebras of the remaining class. contains the opposite algebras of the other. Proof: We classify these division algebras using the matrix representa- tion of the left multiplications with respect to a chosen basis. For con- venience of presentation, we encode a column vector t(b1, b2, b3, b4) with coefficients in {0, 1} as the hexadecimal digit b1 ∗8+ b2 ∗4+ b3 ∗2+ b4 ∗1 ∈ {0, 1, . . . , 9, a, . . . f}. Because of Proposition 2, any four-dimensional division algebra over GF(2) contains an element x such that 1, x, x2 are linearly independent. Assume that x3 := x · x2 is in the linear span < 1, x, x2 > of 1, x, x2. We expand 1, x, x2 to a basis of the algebra and have with respect to this base L(x) =     0 0 ∗ ∗ 1 0 ∗ ∗ 0 1 ∗ ∗ 0 0 0 ∗     and either L(x) or L(x + 1) is not invertible. Therefore, 1, x, x2, x3 is a base. With respect to this base, the matrix for L(x), L(x2), and L(x3) respectively must have the form     0 0 0 1 1 0 0 ∗ 0 1 0 ∗ 0 0 1 ∗     ,     0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 1 ∗ ∗ ∗ 0 ∗ ∗ ∗     ,     0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 1 ∗ ∗ ∗     . Jo u rn al A lg eb ra D is cr et e M at h .108 Non-associative division algebras The small number of possibilities (228) allows us to use computer software to determine the isotopy classes. To keep the runtime under control, we first generate a list of all invertibe 4 by 4 matrices. We use this list to decide for each possible assignments of the free variables (the stars in the preceeding equation) whether the resulting algebra is a division algebra. This gives us 178 division algebras. We then determine isotopy classes by identifying in a first pass whether for X ∈ GL(4) and algebras A and A′ we have X{L(a)\|a ∈ A}X−1 = {XL(a)X−1\|a ∈ A} = {L′(a′)\|a′ ∈ A′} This leaves us with 6 equivalence classes under this relation with finer equivalency classes. Our second, much more compute-intensive pass then uses a brute force enumeration of all pairs of invertible matrices X and Y whether X{L(a)\|a ∈ A}Y = {L′(a′)\|a′ ∈ A′} This leaves exactly three equivalence classes. One has representative GF(16), the other ones are given by L(1), L(x), L(x2), and L(x3) equal to (8421, 4219, 21f5, 1a87) and (8421, 4219, 2945, 15f3). It turns out that the latter two algebras are opposite algebras of each other. I have not succeeded in proving or disproving that the only division algebras of dimension 3 over a field GF(p), p a prime, are isotopes of fields and leave it as a conjecture. References [1] A. A. Albert, Nonassociative Algebras. I, Annals of Mathematics, 43 (1942), p. 685-707. [2] W. Litwin and T. Schwarz. Algebraic Signatures for Scalable Distributed Data Structures. Proceedings of the 20th International Conference on Data Engineering (ICDE), Boston, 2004. [3] R. H. Oehmke and R. Sandler. The collineation group of division ring planes. I. Jordan algebras. Journal f. Reine und Angewandte Mathematik, 216 (1964), p. 67-87. [4] H. P. Petersson. Isotopism of Jordan Algebras. Proceedings of the American Math- ematical Society. 20(2), (1969), p. 477 - 482. Contact information Th. Schwarz, S.J. Facultad de Ingenieŕıa y Tecnoloǵıas Uni- versidad Católica del Uruguay, Av. 8 de Oc- tubre 2738, Montevideo, Uruguay E-Mail: TSchwarz@calprov.org URL: http://www.cse.scu.edu/ tschwarz Received by the editors: 25.11.2006 and in final form 14.05.2010.

Small non-associative division algebras up to isotopy

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