Vector Fields and Their Duals
Identifieur interne : 009E03 ( Main/Exploration ); précédent : 009E02; suivant : 009E04Vector Fields and Their Duals
Auteurs : Philip Feinsilver ; René Schott [France]Source :
- Advances in Mathematics [ 0001-8708 ] ; 2000.
English descriptors
- Teeft :
- Academic press, Algebra, Algebraic, Boson, Calculus, Canonical, Canonical variables, Commutation, Duals, Feinsilver, Fundamental theorems, Group elements, Heisenberg algebra, Holomorphic, Holomorphic canonical calculus, Initial conditions, Jacobian, Matrix, Momentum variables, Other hand, Partial differentiation, Same result, Schott, Standard notations, Umbral, Umbral calculus, Vacuum state, Vector field, Vector fields.
Abstract
Abstract: A standard way of realizing a Lie algebra is as a family of vector fields closed under commutation. Using the action on the universal enveloping algebra, one finds a realization in a dual form—the double dual. This is an algebraic Fourier transform of a “vector fields realization” of the Lie algebra. On the other hand, in the subject of umbral calculus (canonical boson calculus) the duals-to-vector fields play a primary role. It is shown that the double dual realizations of Lie algebras provide a rich source of examples for the umbral calculus, which, complementarily, provides a canonical construction of polynomial systems associated to the Lie algebra. For any finite-dimensional Lie algebra, take an element in the local Lie group it generates. Then there is an abelian family of operators such that acting on a canonical vacuum state, the abelian group gives the same result as the group element constructed via the given Lie algebra. In other words, they yield the same coherent states.
Url:
DOI: 10.1006/aima.1999.1850
Affiliations:
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Cartan and Loria, Université Henri Poincaré-Nancy 1, 54506, Vandoeuvre-lès-Nancy</wicri:regionArea><placeName><region type="region" nuts="2">Grand Est</region><region type="old region" nuts="2">Lorraine (région)</region><settlement type="city">Vandœuvre-lès-Nancy</settlement></placeName></affiliation></author></analytic><monogr></monogr><series><title level="j">Advances in Mathematics</title><title level="j" type="abbrev">YAIMA</title><idno type="ISSN">0001-8708</idno><imprint><publisher>ELSEVIER</publisher><date type="published" when="2000">2000</date><biblScope unit="volume">149</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="182">182</biblScope><biblScope unit="page" to="192">192</biblScope></imprint><idno type="ISSN">0001-8708</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0001-8708</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="Teeft" xml:lang="en"><term>Academic press</term><term>Algebra</term><term>Algebraic</term><term>Boson</term><term>Calculus</term><term>Canonical</term><term>Canonical variables</term><term>Commutation</term><term>Duals</term><term>Feinsilver</term><term>Fundamental theorems</term><term>Group elements</term><term>Heisenberg algebra</term><term>Holomorphic</term><term>Holomorphic canonical calculus</term><term>Initial conditions</term><term>Jacobian</term><term>Matrix</term><term>Momentum variables</term><term>Other hand</term><term>Partial differentiation</term><term>Same result</term><term>Schott</term><term>Standard notations</term><term>Umbral</term><term>Umbral calculus</term><term>Vacuum state</term><term>Vector field</term><term>Vector fields</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: A standard way of realizing a Lie algebra is as a family of vector fields closed under commutation. Using the action on the universal enveloping algebra, one finds a realization in a dual form—the double dual. This is an algebraic Fourier transform of a “vector fields realization” of the Lie algebra. On the other hand, in the subject of umbral calculus (canonical boson calculus) the duals-to-vector fields play a primary role. It is shown that the double dual realizations of Lie algebras provide a rich source of examples for the umbral calculus, which, complementarily, provides a canonical construction of polynomial systems associated to the Lie algebra. For any finite-dimensional Lie algebra, take an element in the local Lie group it generates. Then there is an abelian family of operators such that acting on a canonical vacuum state, the abelian group gives the same result as the group element constructed via the given Lie algebra. In other words, they yield the same coherent states.</div></front></TEI><affiliations><list><country><li>France</li></country><region><li>Grand Est</li><li>Lorraine (région)</li></region><settlement><li>Vandœuvre-lès-Nancy</li></settlement></list><tree><noCountry><name sortKey="Feinsilver, Philip" sort="Feinsilver, Philip" uniqKey="Feinsilver P" first="Philip" last="Feinsilver">Philip Feinsilver</name></noCountry><country name="France"><region name="Grand Est"><name sortKey="Schott, Rene" sort="Schott, Rene" uniqKey="Schott R" first="René" last="Schott">René Schott</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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