Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop.[1]
Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy − yx.
The 7-sphere may be given the structure of a smooth Moufang loop by identifying it with the unit octonions. The tangent space of the identity of this Moufang loop may be identified with the 7-dimensional space of imaginary octonions. The imaginary octonions form a Malcev algebra with the Malcev product xy − yx.
malcev, algebra, algebras, groups, malcev, algebra, mathematics, maltsev, algebra, moufang, algebra, over, field, nonassociative, algebra, that, antisymmetric, that, displaystyle, satisfies, malcev, identity, displaystyle, they, were, first, defined, anatoly, . For the Lie algebras or groups see Malcev Lie algebra In mathematics a Malcev algebra or Maltsev algebra or Moufang Lie algebra over a field is a nonassociative algebra that is antisymmetric so that x y y x displaystyle xy yx and satisfies the Malcev identity x y x z x y z x y z x x z x x y displaystyle xy xz xy z x yz x x zx x y They were first defined by Anatoly Maltsev 1955 Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups Namely just as the tangent space of the identity element of a Lie group forms a Lie algebra the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra Moreover just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold For example this is true for a connected simply connected real analytic Moufang loop 1 Contents 1 Examples 2 See also 3 Notes 4 ReferencesExamples EditAny Lie algebra is a Malcev algebra Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy yx The 7 sphere may be given the structure of a smooth Moufang loop by identifying it with the unit octonions The tangent space of the identity of this Moufang loop may be identified with the 7 dimensional space of imaginary octonions The imaginary octonions form a Malcev algebra with the Malcev product xy yx See also EditMalcev admissible algebraNotes Edit Nagy Peter T 1992 Moufang loops and Malcev algebras PDF Seminar Sophus Lie 3 65 68 CiteSeerX 10 1 1 231 8888 References EditElduque Alberto Myung Hyo C 1994 Mutations of alternative algebras Kluwer ISBN 0 7923 2735 7 Filippov V T 2001 1994 Mal tsev algebra Encyclopedia of Mathematics EMS Press Mal cev A I 1955 Analytic loops Mat Sb New Series in Russian 36 78 569 576 MR 0069190 This algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Malcev algebra amp oldid 1013838357, wikipedia, wiki, book, books, library,
