Groups With Triality

Abstract

Groups with triality, which arose in the papers of Glauberman and Doro, are naturally connected with Moufang loops. In this paper, we describe all possible, in a sense, groups with triality associated with a given Moufang loop. We also introduce several universal groups with triality and discuss their properties.

Hopf algebras in non-associative Lie theory

“…Moufang loops provide an example of how one can pass from loop identities to Sabinin algebra identities via the identities in the bialgebra of distributions. Given a Moufang loop M , the Moufang-Hopf identity (16), which holds in D e M , implies that the tangent space It is a Malcev algebra with the bracket given by [a, b] = ab − ba. The rest of the Sabinin algebra operations in T e M can be expressed via the Malcev bracket [44]; as a consequence, the Malcev algebra structure is sufficient to reconstruct a local Moufang loop.…”

Abelian-by-Cyclic Moufang Loops

“…Conversely, every Moufang loops arises so from a suitable group with triality. For more information on the relation between groups with triality and Moufang loops, see [7]. Every group with triality G possesses a (necessarily unique) maximal normal subgroup contained in C G (S) which we will denote by Z S (G).…”

“…Proof. Let g = (t, v 1 , v 2 , u) ∈ G. Then g −1 g σ is given by (7), where = (r 1 r 2 , 1), (x 2 , −r −1…”

Co-Moufang Deformations of the Universal Enveloping Algebra of the Algebra of Traceless Octonions

“…Another approach to the construction of the product of U (m) was carried out in [1]. This approach unified the connections between groups with triality and Moufang loops [8,10,13,20], and Lie algebras with triality and Malcev algebras [12,19] by means of the notion of Hopf algebra with triality. For an account of universal enveloping algebras of generalizations of Lie and Malcev algebras see [21][22][23].…”