An approach to finite-dimensional real division composition algebras through reflections

Alsaody, Seidon

doi:10.1016/j.bulsci.2014.10.001

Cited by 1 publication

(5 citation statements)

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“…We denote by X A = PGL(A) 2 /N the orbit set of the induced N -action on PGL(A) 2 . Then (1) induces an action of PG(A) ≃ T /N on X A , given by (2) ϕ · [α, β] = ϕ 1 αϕ −1 , ϕ 2 βϕ −1 for [α, β] ∈ X A and ϕ ∈ PG(A). Here ϕ 1 , ϕ 2 are any pair of triality components of ϕ, that is,…”

Section: Proposition 2 Let a Be A Hurwitz Algebra Any Isotope B Of mentioning

confidence: 99%

“…Let X (A) = PG(A) X A be the groupoid of the action (2). For any Hurwitz algebra A, let I (A) denote the category of principal isotopes of A, andǏ (A) the groupoid obtained from I (A) by removing all non-isomorphisms between the objects.…”

Section: Proposition 2 Let a Be A Hurwitz Algebra Any Isotope B Of mentioning

confidence: 99%

“…) . We denote by X A = PGL(A) 2 /N the orbit set of the induced N -action on PGL(A) 2 . Then (1) induces an action of PG(A) ≃ T /N on X A , given by…”

Section: Proposition 2 Let a Be A Hurwitz Algebra Any Isotope B Of A ...mentioning

confidence: 99%

“…(2) The set X A = PGL(A) 2 /N may also be viewed as the orbit set PGL(A) 2 /N (A) * of the N (A) * -action on PGL(A) 2 given by w…”

Section: And Hence the Elements ([α] [β]) And ([γ] [δ]) In Pgl(mentioning

confidence: 99%

“…Isotopes of Hurwitz algebras of dimension two is a special case of Petersson's [17] description of all 2-dimensional algebras, which is also based on the concept of isotopy. There is a large number of articles on finite-dimensional composition algebras and absolute valued algebras (some examples are [2,3,8,14,18]), most of which use the concept of isotopy as a central tool.…”

Section: Introductionmentioning

confidence: 99%

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Isotopes of Hurwitz Algebras

Darpö¹

2018

Mediterr. J. Math.

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We study the class of all algebras that are isotopic to a Hurwitz algebra. In the general case, isomorphism classes of such algebras are shown to correspond to orbits of a certain group action. Our main focus is the isotopes of Hamilton's quaternions, for which a more detailed, and geometrically intuitive description of the isomorphism classes is given. As an application, we also demonstrate how some results concerning the classification of finite-dimensional composition algebras can be deduced from our general results.Key words and phrases. Hurwitz algebra, isotope, quaternion algebra, octonion algebra. MSC 2010: 17A60 (17A35, 17A75). 1 Some authors have used a weaker notion of non-degeneracy for the norm n, requiring that n(x + y) = n(y) for all y implies x = 0. This definition gives rise to additional unital composition algebras over fields k of characteristic two, in form of purely inseparable field extensions of k [14]. If char k = 2, the two definitions are equivalent.

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Section: Proposition 2 Let a Be A Hurwitz Algebra Any Isotope B Of mentioning

confidence: 99%

Section: Proposition 2 Let a Be A Hurwitz Algebra Any Isotope B Of mentioning

confidence: 99%

“…) . We denote by X A = PGL(A) 2 /N the orbit set of the induced N -action on PGL(A) 2 . Then (1) induces an action of PG(A) ≃ T /N on X A , given by…”

Section: Proposition 2 Let a Be A Hurwitz Algebra Any Isotope B Of A ...mentioning

confidence: 99%

“…(2) The set X A = PGL(A) 2 /N may also be viewed as the orbit set PGL(A) 2 /N (A) * of the N (A) * -action on PGL(A) 2 given by w…”

Section: And Hence the Elements ([α] [β]) And ([γ] [δ]) In Pgl(mentioning

confidence: 99%