Octonion multiplication and Heawood’s map
In this note, the octonion multiplication table is recovered from a regular tesselation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map.
Octonionic Cayley spinors and
Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group , and of its subgroups. We are therefore led to a description of in terms of octonionic matrices, generalizing previous results in the case. Our treatment naturally includes a description of several important subgroups of , notably , , and (the double cover of) . An interpretation of the actions...
On a class of non-associative rings
On Algebraic Power-Associative and Jordan Algebras with Semi-Normal Functionals.
On Algebras all of Whose Subalgebras are Simple; Some Solutions of Plonka's Problem
On automorphism groups of non-associative Boolean rings.
On free subgroups of units in quaternion algebras
It is well known that for the ring H(ℤ) of integral quaternions the unit group U(H(ℤ) is finite. On the other hand, for the rational quaternion algebra H(ℚ), its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ℤ ⊂ A ⊆ ℚ such that the group of units U(H(A)) of quaternions over A contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments...
On free subgroups of units in quaternion algebras II
Let A ⊆ ℚ be any subring. We extend our earlier results on unit groups of the standard quaternion algebra H(A) to units of certain rings of generalized quaternions H(A,a,b) = ((-a,-b)/A), where a,b ∈ A. Next we show that there is an algebra embedding of the ring H(A,a,b) into the algebra of standard Cayley numbers over A. Using this embedding we answer a question asked in the first part of this paper.
On generalized formal power series
On generalized Jordan derivations of Lie triple systems
Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple -derivation on a Lie triple system is a -derivation.
On Herstein's theorems relating modularity in A and A(+).
In this paper we will examine the relationship between modularity in the lattices of subalgebras of A and A(+), for A an associative algebra over an algebraically closed field. To this aim we will construct an ideal which measures the modularity of an algebra (not necessarily associative) in paragraph 1, examine modular associative algebras in paragraph 2, and prove in paragraph 3 that the ideal constructed in paragraph 1 coincides for A and A(+). We will also examine some properties of the ideal...
On homotopes of Novikov algebras.
On Leibniz algebras with maximal cyclic subalgebras
We begin to study the structure of Leibniz algebras having maximal cyclic subalgebras.
On Lie algebras in braided categories
The category of group-graded modules over an abelian group is a monoidal category. For any bicharacter of this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have -ary multiplications between various graded components. They possess universal enveloping algebras that are Hopf algebras in the given category. Their biproducts with the group ring are noncommutative...
On Linear and Quadratic Jordan Division Algebras.
On maximal subalgebras of central simple Malcev algebras.
In this paper the structure of the maximal elements of the lattice of subalgebras of central simple non-Lie Malcev algebras is considered. Such maximal subalgebras are studied in two ways: first by using theoretical results concerning Malcev algebras, and second by using the close connection between these simple non-Lie Malcev algebras and the Cayley-Dickson algebras, which have been extensively studied (see [4]).
On Nambu bracket representations of 3-algebras.
On one-sided division infinite-dimensional normed real algebras.
In this note we introduce the concept of Cayley homomorphism which is closely related with those of composition algebra and normalized orthogonal multiplication. The key result shows the existence of certain types of Cayley homomorphisms for infinite dimension. As an application we prove the existence of left division infinite-dimensional complete normed real algebras with left unity.
On positive and critical theories of some classes of rings.
On regular and extremely regular algebras