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Quantum algebras

Identifieur interne : 001F06 ( Main/Merge ); précédent : 001F05; suivant : 001F07

Quantum algebras

Auteurs : H. Saller [Allemagne]

Source :

RBID : ISTEX:CA8111EF1726395243138B08C7CA90BBA8EA7455

English descriptors

Abstract

Summary: Quantum algebras are the universal algebras, associated to a finite-dimensional vector space and its endomorphisms. The Fermi or Bose statistics of a quantum algebra reflects the (anti-)symmetry of the basic-space dual product. The relation to the universal enveloping algebra of the basic endomorphisms and to the dual-product Clifford algebra is discussed together with its invariants. According to the Abelian or non-Abelian basic-space endomorphism algebra, it carries two different trace-induced linear forms-the Fock and the Heisenberg form, respectively. With a quantum algebra conjugation,e.g. connected with a-not necessarily Euclidean unitary-time representation, quantum algebras have an inner product and, in the case of a positive conjugation, a Hilbert space of Fock type for the Abelian and of Heisenberg type for the non-Abelian case.

Url:
DOI: 10.1007/BF02826997

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ISTEX:CA8111EF1726395243138B08C7CA90BBA8EA7455

Le document en format XML

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<term>Algebra</term>
<term>Algebra elements</term>
<term>Algebraic</term>
<term>Algebraic element</term>
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<term>Antisymmetric</term>
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<term>Bose</term>
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<term>Exponential</term>
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<term>Fock</term>
<term>Heisenberg</term>
<term>Heisenberg form</term>
<term>Hermann</term>
<term>Hilbert</term>
<term>Hilbert space</term>
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<term>Hybrid bracket algebra</term>
<term>Inner derivations</term>
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<term>Quantized</term>
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<term>Algebraic</term>
<term>Algebraic element</term>
<term>Algebraic elements</term>
<term>Antisymmetric</term>
<term>Basic space</term>
<term>Basic vectors</term>
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<term>Bilinear form</term>
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<term>Bracket</term>
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<term>Clifford algebras</term>
<term>Complex algebra</term>
<term>Conjugation</term>
<term>Dual bases</term>
<term>Dual conjugation</term>
<term>Dual product</term>
<term>Endomorphism</term>
<term>Endomorphisms</term>
<term>Euclid conjugation</term>
<term>Exponential</term>
<term>Fermi</term>
<term>Fermi quantum algebra</term>
<term>Fock</term>
<term>Heisenberg</term>
<term>Heisenberg form</term>
<term>Hermann</term>
<term>Hilbert</term>
<term>Hilbert space</term>
<term>Hybrid</term>
<term>Hybrid bracket algebra</term>
<term>Inner derivations</term>
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<term>Invariance subalgebra</term>
<term>Invariant form</term>
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<term>Isomorphism</term>
<term>Iswv</term>
<term>Linear transformations</term>
<term>Minimal polynomial</term>
<term>Oscillator</term>
<term>Probability forms</term>
<term>Projector</term>
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<term>Quantized classes</term>
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<term>Quantum algebra</term>
<term>Quantum algebra invariants</term>
<term>Quantum algebras</term>
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<term>Saller</term>
<term>Scalar</term>
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<term>Symmetric sesquilinear form</term>
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<term>Tensor algebra</term>
<term>Tensor product</term>
<term>Trace form</term>
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<div type="abstract" xml:lang="en">Summary: Quantum algebras are the universal algebras, associated to a finite-dimensional vector space and its endomorphisms. The Fermi or Bose statistics of a quantum algebra reflects the (anti-)symmetry of the basic-space dual product. The relation to the universal enveloping algebra of the basic endomorphisms and to the dual-product Clifford algebra is discussed together with its invariants. According to the Abelian or non-Abelian basic-space endomorphism algebra, it carries two different trace-induced linear forms-the Fock and the Heisenberg form, respectively. With a quantum algebra conjugation,e.g. connected with a-not necessarily Euclidean unitary-time representation, quantum algebras have an inner product and, in the case of a positive conjugation, a Hilbert space of Fock type for the Abelian and of Heisenberg type for the non-Abelian case.</div>
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