524 research outputs found
Homogeneous spaces and quasigroups
No full textThe paper under review is a survey on quasigroup methods in the theory of homogeneous spaces essentially based on the works of the author and his students. The main tool here is the so-called R. Baer-L. V. Sabinin construction establishing the equivalence of categories of left loops and left homogeneous transversalled spaces, which is displayed in the smooth case. Reductive, symmetric and generalized symmetric spaces are treated as smooth loops with identities. The unifying concept of transsymmetric space is also treated (as a smooth correct left F-quasigroup). par In this framework a new concept of hyporeductive homogeneous space is described in terms of an appropriate hyporeductive algebra. The quasigroup analysis of homogeneous spaces leads to new classes of such spaces as almost symmetric and antisymmetric spaces and generates new linear algebras that are close to Malʹtsev and Sagle algebras. par The bibliography (29 titles) contains some titles related to physical applications
Nonassociative geometry: Friedmann-Robertson-Walker spacetime
No full textIn (Phys. Rev. D 62 (2000) 081501(R)) we proposed a unified description of continuum and discrete spacetime based on nonassociative geometry. It follows from our approach that at distances comparable with Planck length the standard concept of spacetime might be replaced by the nonassociative discrete structure, and nonassociativity is the algebraic equivalent of the curvature. In the framework of the nonassociative geometry we introduce discrete Friedmann-Robertson-Walker (FRW) model and show that the standard FRW spacetime appears as its 'continuum limit'. � 2006 World Scientific Publishing Company
Entanglement generated by a Dicke phase transition
No full textIn the framework of nonassociative geometry a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called diodular discrete structure which at large spacetime scales "looks like" a differentiable manifold. After a brief review of foundations of nonassociative geometry, we discuss the nonassociative smooth and discrete de Sitter spacetimes. "2000 The American Physical Society.",,,,,,,,,"http://hdl.handle.net/20.500.12104/43170","http://www.scopus.com/inward/record.url?eid=2-s2.0-16644373096&partnerID=40&md5=b36e382541de3151bfc7cb5f6d8699b9",,,,,,"8",,"Physical Review D - Particles, Fields, Gravitation and Cosmology",,"
The theory of smooth hyporeductive and pseudoreductive loops
No full textFor Bol loops and reductive loops one can construct an infinitesimal theory similar to Lie group theory by associating with a loop a certain binary-ternary algebra with identities, namely the Bol algebra for a Bol loop and the triple Lie algebra for a reductive loop. It is also possible to construct a proper infinitesimal theory for hyporeductive loops that generalize Bol loops and reductive loops. This can be achieved by associating with these loops a hyporeductive algebra with two binary and one ternary operation and the system of identities [see L. V. Sabinin, in {it Variational methods in modern geometry (Russian)}, 50--69, Univ. Druzhby Narodov, Moscow, 1990; [msn] MR1130903 (92g:22008) [/msn]; Dokl. Akad. Nauk SSSR {bf 314} (1990), no.~3, 565--568; [msn] MR1094021 (92d:22002) [/msn]; in {it Webs and quasigroups (Moscow, 1989)}, 129--137, Tver. Gos. Univ., Tverʹ, 1991; see [msn] MR1140959 (92f:53003) [/msn]]. The latter algebra generalizes both the Bol algebra and the triple Lie algebra. In the paper under review the author constructs the infinitesimal theory for smooth hyporeductive and pseudoreductive loops
Nonassociative geometry: Towards discrete structure of spacetime
No full textIn the framework of nonassociative geometry a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called diodular discrete structure which at large spacetime scales "looks like" a differentiable manifold. After a brief review of foundations of nonassociative geometry, we discuss the nonassociative smooth and discrete de Sitter spacetimes. ©2000 The American Physical Society
Smooth loops, generalized coherent states, and geometric phases
No full textA description of generalized coherent states and geometric phases is given in the light of the general theory of smooth loops
Bol loop actions
No full textsummary:The notions of left Bol and Bol-Bruck actions are introduced. A purely algebraic analogue of a Nono family (Lie triple family), the so called Sabinin-Nono family, is given. It is shown that any Sabinin-Nono family is a left Bol-Bruck action. Finally it is proved that any local Nono family is a local left Bol-Bruck action. On general matters see [L.V. Sabinin 91, 99]
Smooth loops. I
No full textThe author presents the fundamentals of the theory of smooth loops generalizing Lie group theory. In the paper under review he deals with the infinitesimal theory of smooth loops
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