Quantum algebras
Identifieur interne : 001F06 ( Main/Merge ); précédent : 001F05; suivant : 001F07Quantum algebras
Auteurs : H. Saller [Allemagne]Source :
- Il Nuovo Cimento [ 0369-3554 ] ; 1993-06-01.
English descriptors
- KwdEn :
- 02.10, 03.65, Abelian, Adjoint, Adjoint action, Algebra, Algebra elements, Algebraic, Algebraic element, Algebraic elements, Antisymmetric, Basic space, Basic vectors, Bilinear, Bilinear form, Bose, Bracket, Cimento, Clifford algebra, Clifford algebras, Complex algebra, Conjugation, Dual bases, Dual conjugation, Dual product, Endomorphism, Endomorphisms, Euclid conjugation, Exponential, Fermi, Fermi quantum algebra, Fock, Heisenberg, Heisenberg form, Hermann, Hilbert, Hilbert space, Hybrid, Hybrid bracket algebra, Inner derivations, Invariance, Invariance subalgebra, Invariant form, Invl, Isomorphic, Isomorphism, Iswv, Linear transformations, Logic, Minimal polynomial, Oscillator, Probability forms, Projector, Quantized, Quantized classes, Quantum, Quantum algebra, Quantum algebra invariants, Quantum algebras, Quantum theory, Quotient, Saller, Scalar, Simple algebra, Subalgebra, Symmetric algebra, Symmetric sesquilinear form, Tensor, Tensor algebra, Tensor product, Trace form, Trace forms, Traceless, Vector space, Vector spaces, Vector subspace, quantum mechanics, set theory and algebra.
- Teeft :
- Abelian, Adjoint, Adjoint action, Algebra, Algebra elements, Algebraic, Algebraic element, Algebraic elements, Antisymmetric, Basic space, Basic vectors, Bilinear, Bilinear form, Bose, Bracket, Cimento, Clifford algebra, Clifford algebras, Complex algebra, Conjugation, Dual bases, Dual conjugation, Dual product, Endomorphism, Endomorphisms, Euclid conjugation, Exponential, Fermi, Fermi quantum algebra, Fock, Heisenberg, Heisenberg form, Hermann, Hilbert, Hilbert space, Hybrid, Hybrid bracket algebra, Inner derivations, Invariance, Invariance subalgebra, Invariant form, Invl, Isomorphic, Isomorphism, Iswv, Linear transformations, Minimal polynomial, Oscillator, Probability forms, Projector, Quantized, Quantized classes, Quantum, Quantum algebra, Quantum algebra invariants, Quantum algebras, Quotient, Saller, Scalar, Simple algebra, Subalgebra, Symmetric algebra, Symmetric sesquilinear form, Tensor, Tensor algebra, Tensor product, Trace form, Trace forms, Traceless, Vector space, Vector spaces, Vector subspace.
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:CA8111EF1726395243138B08C7CA90BBA8EA7455Le document en format XML
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<idno type="ISSN">0369-3554</idno>
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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>02.10</term>
<term>03.65</term>
<term>Abelian</term>
<term>Adjoint</term>
<term>Adjoint action</term>
<term>Algebra</term>
<term>Algebra elements</term>
<term>Algebraic</term>
<term>Algebraic element</term>
<term>Algebraic elements</term>
<term>Antisymmetric</term>
<term>Basic space</term>
<term>Basic vectors</term>
<term>Bilinear</term>
<term>Bilinear form</term>
<term>Bose</term>
<term>Bracket</term>
<term>Cimento</term>
<term>Clifford algebra</term>
<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>
<term>Invariance</term>
<term>Invariance subalgebra</term>
<term>Invariant form</term>
<term>Invl</term>
<term>Isomorphic</term>
<term>Isomorphism</term>
<term>Iswv</term>
<term>Linear transformations</term>
<term>Logic</term>
<term>Minimal polynomial</term>
<term>Oscillator</term>
<term>Probability forms</term>
<term>Projector</term>
<term>Quantized</term>
<term>Quantized classes</term>
<term>Quantum</term>
<term>Quantum algebra</term>
<term>Quantum algebra invariants</term>
<term>Quantum algebras</term>
<term>Quantum theory</term>
<term>Quotient</term>
<term>Saller</term>
<term>Scalar</term>
<term>Simple algebra</term>
<term>Subalgebra</term>
<term>Symmetric algebra</term>
<term>Symmetric sesquilinear form</term>
<term>Tensor</term>
<term>Tensor algebra</term>
<term>Tensor product</term>
<term>Trace form</term>
<term>Trace forms</term>
<term>Traceless</term>
<term>Vector space</term>
<term>Vector spaces</term>
<term>Vector subspace</term>
<term>quantum mechanics</term>
<term>set theory and algebra</term>
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<keywords scheme="Teeft" xml:lang="en"><term>Abelian</term>
<term>Adjoint</term>
<term>Adjoint action</term>
<term>Algebra</term>
<term>Algebra elements</term>
<term>Algebraic</term>
<term>Algebraic element</term>
<term>Algebraic elements</term>
<term>Antisymmetric</term>
<term>Basic space</term>
<term>Basic vectors</term>
<term>Bilinear</term>
<term>Bilinear form</term>
<term>Bose</term>
<term>Bracket</term>
<term>Cimento</term>
<term>Clifford algebra</term>
<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>
<term>Invariance</term>
<term>Invariance subalgebra</term>
<term>Invariant form</term>
<term>Invl</term>
<term>Isomorphic</term>
<term>Isomorphism</term>
<term>Iswv</term>
<term>Linear transformations</term>
<term>Minimal polynomial</term>
<term>Oscillator</term>
<term>Probability forms</term>
<term>Projector</term>
<term>Quantized</term>
<term>Quantized classes</term>
<term>Quantum</term>
<term>Quantum algebra</term>
<term>Quantum algebra invariants</term>
<term>Quantum algebras</term>
<term>Quotient</term>
<term>Saller</term>
<term>Scalar</term>
<term>Simple algebra</term>
<term>Subalgebra</term>
<term>Symmetric algebra</term>
<term>Symmetric sesquilinear form</term>
<term>Tensor</term>
<term>Tensor algebra</term>
<term>Tensor product</term>
<term>Trace form</term>
<term>Trace forms</term>
<term>Traceless</term>
<term>Vector space</term>
<term>Vector spaces</term>
<term>Vector subspace</term>
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<front><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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