Generalized Clifford algebras and hyperspin manifolds

Fleury, N.; Traubenberg, Michel Rausch de; Yamaleev, Robert M.

doi:10.1007/bf00673754

Cited by 19 publications

(56 citation statements)

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“…The matrix representations of such algebras [3] lead to natural algebraic extensions of the Clifford and Grassmann algebras [4,5]. These families endow extension of complex numbers [6,7] leading to the corresponding extension of trigonometric functions, dubbed multisine functions (see also [8]). …”

mentioning

confidence: 99%

“…The set of multicomplex (MC) numbers is generated by one element e which satisfies e n = −1 [6] (IMI l C n = x = n−1 i=0…”

mentioning

confidence: 99%

“…It is worth stressing that most of the results of usual complex mumbers analysis remains true for MC-numbers, were it be for algebraic [6] or analytic properties [7].…”

mentioning

confidence: 99%

“…Having defined an appropriate pseudo-norm ( x ) [6,7], it is possible to write any multicomplex number x ( x = 0) in an exponential representation x = n−1 i=0…”

mentioning

confidence: 99%

“…However, the parametrisation in terms of Θ given in [6,7] is inconvenient to derive explicit formulas for the mus-functions. From the properties of e in its matrix representation (see [6]) it is easy to see that e i + e n−i (resp. e i − e n−i ) are represented by pure imaginary (real) matrices, and, hence among the n − 1 directions in…”

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confidence: 99%

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Extension of Sine–gordon Field Theory From Generalized Clifford Algebras

1998

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Linearization of homogeneous polynomials of degree n and k variables leads to generalized Clifford algebras. Multicomplex numbers are then introduced in analogy to complex numbers with respect to usual Clifford algebra. In turn multicomplex extensions of trigonometric functions are constructed in terms of 'compact' and 'non-compact' variables. It gives rise to the natural extension of the d−dimensional sine-Gordon field theory in the n−dimensional multicomplex space. In dimension 2, the cases n = 1, 2, 3, 4 are identified as the quantum integrable Liouville, sineGordon and known deformed Toda models. The general case is discussed.Mod. Phys. Lett. A 13 (1998Lett. A 13 ( ) 2531 In most domain of research activities apparently disparate facts and data are first accumulated and later on classified and interpreted in terms of an underlying hidden order. There are numerous examples in physics ranging from Maxwell's early unification of electricity and magnetism to the actual understanding of particle physics and elementary interactions. The hidden structure takes the form of underlying algebras. A recent celebrated example relates to the interpretation of the universality of critical phenomena in two dimensions in term of Virasoro algebra. Among the possible algebras those induced by bilinear relations have a special status, probably due to the bilinear aspects of fundamental objects such as quadratic metric, commutators, anticommutators, etc· · · . However, algebras going beyond the quadratics ones have been constructed in the 70's from underlying polynomials of degree higher than two. There are dubbed by mathematicians as Clifford algebras of polynomials [1,2]. The matrix representations of such algebras [3] lead to natural algebraic extensions of the Clifford and Grassmann algebras [4,5]. These families endow extension of complex numbers [6,7] leading to the corresponding extension of trigonometric functions, dubbed multisine functions (see also [8]).The potential usefulness of the multisine functions has already been pointed out and explored only briefly so far. In the context of field theory it is natural to focus on one of the 'simplest' (apparently) famous form, the sine-Gordon (SG) model, known for its integrability and fermionization properties related to the massive Thirring model for a specific relation between the couplings [9, 10, 11]. By analogy, Toda and affine Toda field theories (TFT and ATFT) based on simply or non-simply laced algebras are the natural simple Lie group extension of the SU(2) and SU(2) Toda and affine Toda field theories, e.g. Liouville and SG models. The interest in such deformed Toda theories comes from their integrability and duality properties [12,13] which can be used to pave the way to the understanding of electric-magnetic duality in four-dimensional gauge theories, conjectured in [14] and developed in [15]. Thereby nonperturbative analysis of the spectrum and of phase structure in supersymmetric YangMills theory becomes possible.Although the filiation of these models i...

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confidence: 99%

“…The set of multicomplex (MC) numbers is generated by one element e which satisfies e n = −1 [6] (IMI l C n = x = n−1 i=0…”

mentioning

confidence: 99%

“…It is worth stressing that most of the results of usual complex mumbers analysis remains true for MC-numbers, were it be for algebraic [6] or analytic properties [7].…”

mentioning

confidence: 99%

“…Having defined an appropriate pseudo-norm ( x ) [6,7], it is possible to write any multicomplex number x ( x = 0) in an exponential representation x = n−1 i=0…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Extension of Sine–gordon Field Theory From Generalized Clifford Algebras

1998

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The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

Kinani¹,

Ouarab

1999

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From Generalized Clifford Algebras to nambu’s formulation of dynamics

Yamaleev¹

2000

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Abstract. We considera commutative part of the Generalized Clifford Algebras, denominated as algebra of multicomplex numbers. By using the multicomplex algebra as underlying algebraic structure we construct oscillator model for the Nambu's formulation of dyaamics. We propose a new dynamical principle which gives rise to two kinds of Hamilton-Nambu equations in D > 2-dimensional phase space. The first one is formulatcd with (D -1)-evolution parameter anda single Hamiltonian. The Haniltonian of the oscillator model in such dynamics is given by D-degree homogeneous forro, hi the ~econd formulation, vice versa, the evolution of the system along a single evolution paramcter is generated by (D -1) Hamiltonian.

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Generalized Clifford algebras and hyperspin manifolds

Cited by 19 publications

References 18 publications

Extension of Sine–gordon Field Theory From Generalized Clifford Algebras

Extension of Sine–gordon Field Theory From Generalized Clifford Algebras

The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

From Generalized Clifford Algebras to nambu’s formulation of dynamics

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