An envelope for Malcev algebras

Pérez-Izquierdo, José M.; Shestakov, I. P.

doi:10.1016/s0021-8693(03)00389-2

Cited by 77 publications

(101 citation statements)

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“…The analogue of our second result above was established for Malcev algebras framed by Pérez-Izqquierdo and Shestakov [14]. The collection of all representations of Bol algebra and the morphisms between them form a category, named the category of representations of Bol algebras Rep(B).…”

Section: Theorem 12 Let B Be a Finite-dimensional Right Bol Algebramentioning

confidence: 59%

“…In the case that L is finite dimensional, the Ado-Iwasawa theorem says that A can be taken finite dimensional too. This extension of Ado-Iwasawa theorem was established for the Malcev algebras by Pérez-Izqquierdo and Shestakov [14]. There is a version of the Poincaré-Birkhoff-Witt theorem for Bol algebra proved by Kuz'min and Zaidi [4].…”

Section: An Extension Of Ado-iwasawa Theorem To Bol Algebrasmentioning

confidence: 76%

“…There is a version of the Poincaré-Birkhoff-Witt theorem for Bol algebra proved by Kuz'min and Zaidi [4]. Now let B be a Bol algebra [14] that there is an alternative algebra A and an injective map…”

Section: An Extension Of Ado-iwasawa Theorem To Bol Algebrasmentioning

confidence: 98%

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On Representations of Bol Algebras

N¹,

T²

2016

J Generalized Lie Theory Appl

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In this paper, we introduce the notion of representation of Bol algebra. We prove an analogue of the classical Engel's theorem and the extension of Ado-Iwasawa theorem for Bol Algebras. We study the category of representations of Bol algebras and show that it is a tensor category. In the case of regular representations of Bol algebras, we give a complete classification of them for all two-dimensional Bol algebras.

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Section: Theorem 12 Let B Be a Finite-dimensional Right Bol Algebramentioning

confidence: 59%

Section: An Extension Of Ado-iwasawa Theorem To Bol Algebrasmentioning

confidence: 76%

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On Representations of Bol Algebras

N¹,

T²

2016

J Generalized Lie Theory Appl

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“…The Jacobi identity can be extended to Malcev identity [11]. 2)(uv−vu) are the only vector product algebra and there is no infinite dimensional vector product algebra.…”

Section: Theorem Of Number Systemsmentioning

confidence: 99%

Untitled

2015

JMSS

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Einstein claimed that one cannot define global time, and in order to formulate physical dynamics, it is useful to adopt fiber bundle structure. We define topological space E which consists of base space X and fibers F = Π −1 (X), where Π is a projection of an event on the base space. Relations between initial data and final data are defined by group G and a Fiber bundle is defined as as set (E, Π, F, G, X).Tangent bundle TX of real linear space X is defined by the projection π TX = TX → X; (x,a) → a for any a ∈ X and a sphere S n any non negative integer n may be thought to be a smooth submanifold of R n+1 and TS n is identified asConnes proposed that when one adopts non-commutative geometry, one can put two fibers at each point of X and on top of the two fibers define the initial input event and the final detection event. When one considers dynamics of leptons defined by Dirac equation, group G is given by quaternions H, and the base space X is usually taken to be S 3 .E. Cartan studied dynamics of spinors which are described by octonions or Cayley numbers which is an ordered product of two quaternions. The asymptotic form Y of this system is S 7 . Cayley numbers of S 7 are defined as a 3-sphere bundle over S 4 with group S 3 . Therefore in T X there are two manifolds S 3 × R 4 and S 3 ' × R 4 and the direction of propagation of time on S 3 and S' 3 are not necessarily same. We apply this formulation to experimentally observed violation of time reversal symmetry in p  t process and for understanding the result of time reversal based nonlinear elastic wave spectroscopy (TR-NEWS) in memoducers.

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“…That was surprising since the construction of those bialgebras [PIS04] had no relation with Moufang loops. Distributions provide a natural connection between identities in loops and identities in bialgebras.…”

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confidence: 99%

Formal multiplications, bialgebras of distributions and nonassociative Lie theory

Mostovoy

Pérez-Izquierdo

2010

Transformation Groups

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Abstract. We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras.Starting with a formal multiplication we construct a non-associative bialgebra, namely, the bialgebra of distributions with the convolution product. Considering the primitive elements in this bialgebra gives a functor from formal loops to Sabinin algebras. We compare this functor to that of Mikheev and Sabinin and show that although the brackets given by both constructions coincide, the multioperator does not. We also show how identities in loops produce identities in bialgebras. While associativity in loops translates into associativity in algebras, other loop identities (such as the Moufang identity) produce new algebra identities. Finally, we define a class of unital formal multiplications for which Ado's theorem holds and give examples of formal loops outside this class.A by-product of the constructions of this paper is a new identity on Bernoulli numbers. We give two proofs: one coming from the formula for the non-associative logarithm, and the other (due to D. Zagier) using generating functions.

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An envelope for Malcev algebras

Cited by 77 publications

References 9 publications

On Representations of Bol Algebras

On Representations of Bol Algebras

Untitled

Formal multiplications, bialgebras of distributions and nonassociative Lie theory

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