The concept
Prague Astronomical Clock
The word was introduced on the previous page; but what is it really? Anticommutativity is a totally strange concept that makes you wonder if it has any reality the first time you encounter it. This feeling stems from the fact that, if it could apply to whole numbers - but fortunately it cannot, it would translate for example into an incomprehensible relation like:
2.3 + 3.2 = 0; absurd, unthinkable!
Yet square matrices give it substance and physics does too whenever certain quantum operators are involved. In this context it is no longer senseless to write:
[A].[B] + [B].[A] = [0]
Various dictionaries (Larousse, Wikipedia, etc.) agree that the word "anticommutativity" describes the mathematical property of binary functions (synonym: involving two arguments) of reversing the sign of the result whenever the positions of the arguments they act upon are swapped. For example, for a set of square matrices equipped with the binary operation "product of two matrices":
[A].[B] → [C]
[B].[A] → - [C]
Because the purpose of these dictionaries is primarily literary, they often do not detail this capability.
For example, they do not specify the structure that must be possessed by the set of elements constituting the pairs of arguments upon which these functions act.
A logical analysis of the definition of anticommutativity — "yielding a result of opposite sign" — highlights the fact that the set of results obtained through this binary operation, which could generically be written as "*", is implicitly endowed with an addition and an additive identity, that is to say, a zero.
Thus, if the set of results is still located within the set of arguments interacting via the operation *, this set of arguments contains a subset tacitly endowed with an additive group structure.
Discussions on anticommutativity concern sets symbolically denoted as {E, +, *} which are such that at least a portion of their elements satisfies the relation:
∃ (a, b) ∈ {E, +, *} : a * b + b * a = 0E
It seems that this concept emerged as a natural by-product of the work of mathematician Hermann Grassmann (1809 - 1877) in his 1844 treatise [01].
There, he invents the exterior (vector) product which happens to be both associative and anticommutative. The existence and action of this product on the elements of certain sets endow them with a structure called an exterior algebra. The initial intention of this approach was to describe geometric characteristics using vectors and operations on these vectors. A concern shared by O. Rodrigues and W.R. Hamilton which led them to invent quaternions in order to describe rotations in a three-dimensional space mathematically.
William Kingdon Clifford (1845 - 1879) unified the work of the previous two in 1878 [02] and founded geometric algebras (now known as Clifford algebras).
Furthermore, it turns out that all associative algebras can be equipped with a (Sophus) Lie bracket (1842 - 1899). Every Lie bracket defines an anticommutative operation on any set {F, +, ₼} even if the operation ₼ is not anticommutative. Indeed, since (i) by definition:
[a, b] = a ₼ b - b ₼ a
And (ii) addition being a commutative binary operation, it is easy to verify that:
[a, b] + [b, a] = (a ₼ b – b ₼ a) + (b ₼ a – a ₼ b) = (a ₼ b + b ₼ a) - (b ₼ a + a ₼ b) = 0(E)
Undoubtedly as an indirect consequence of the advent of algebraic geometry in particular, and of the development of mathematics in general, several fundamental discussions arise.
A) On symbolic mathematical notations; more generally on the problem of the representation of mathematical objects and concepts.
The entire introduction to Heaviside's historical work [03] summarizing J. C. Maxwell's theory in four vector equations consists, for example, of explaining why Cartesian writings (due to Descartes) and vector notations seem more appropriate than an exposition making use of quaternions.
In the same spirit, it is certainly appropriate to recall the efforts of the mathematician Giuseppe Peano (1858 - 1932) aimed at expressing theorems in a unified symbolic language [04]. He is one of the founders of mathematical logic and set theory.
These examples amply illustrate the fact that mathematical language, much like living languages, evolves. Its stabilization and normalization constitute a prerequisite for the understanding of documents produced or published by a broader public. It is true that the early schools of mathematicians had made a habit of keeping their reflections secret. The use of symbols to translate the ideas and early results of articulated thought ultimately coincides with the birth of writing.
B) On the notion of mathematical structure [05].
C) On the possible existence of various geometries [06].
It turns out that the set H of quaternions contains a subset of elements whose pairs are anticommutative. I will denote this subset of H x H as Anticom(H). It is defined as follows:
Anticom(H) : { (q,q') ∈ H2 | q.q' + q'.q = 0H }
It would of course be interesting to specify the morphology of the quaternions belonging to this set. But, before studying the anticommutativity of quaternions, I will examine further how it is possible to link them with complex numbers.
[01] Grassmann, H. : Die Wissenschaft der extensiven Größe oder die Ausdehnungslehre, eine neue mathematische Disziplin; Verlag Otto Wigand, Leipzig 1844.
[02] Clifford, W. K.: Applications of Grassmann's Extensive Algebra. American Journal of Mathematics. 1, 350-358. doi.org/10.2307/2369379. 1878.
[03] Heaviside, O.: XI. On the Forces, Stresses, and Fluxes of Energy in Electromagnetic Field (June 1891 - Mai 1892), pp. 423-480 ; [online (pdf)] : Royal Society Publishing: doi/10.1098/rsta.1892.0011/1272264/rsta.1892.0011 [seen: 8 June 2026].
[04] Peano, G.: Formulario Mathematico (1908), Torino, Fratres Bocca Editores.
[05] Cartan, É. : Sur la structure des groupes infinis de transformation ; Annales scientifique de l'É.N.S. 3ème série, tome 21 (1904), p.153-206.
[06] Hilbert, D. : Grundlage der Geometrie, Festschrift 1899; © Springer Verlag Berlin Heidelberg, 2015, ISBN 978-3-662-45568-5, 268 pages au total (texte original et commentaires).