Multiplicity, Invariants and Tensor Product Decomposition of Tame Representations of U(\infty)

The structure of r-fold tensor products of irreducible tame representations of the inductive limit U(\infty) of unitary groups U(n) are are described, versions of contragredient representations and invariants are realized on Bargmann-Segal-Fock spaces.Comment: 48 pages, LaTeX file, to appear in J. Math. Phy

New mechanisms and novel cation channels in cardiac and skeletal sarcoplasmic reticulum

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Multiplicatively closed bases for C(A)C (A)

A canonical basis for the centralizer $C(M)$ of a $n � n$ complex matrix M is obtained from the one $C(A)$ constructed for a Jordan canonical form A of M. These bases are closed under nonzero products and we obtain simple multiplication tables for their products. Two matrices have the same centralizer invariants iff their Jordan canonical forms have the same block structure. For square matrices M and N we prove that $C(M)$ and $C(N)$ are isomorphic as algebras over C iff they have the same size and the same centralizer invariants. We produce a similar canonical basis for the centralizer algebra of a real matrix over the real numbers which has analogous multiplicative properties in addition to conjugation properties. We also give some additional consequences of these results

Poincaré’s proof of the so-called BirkhoffWitt theorem

In honor of the 100th birthday of the article, “Sur les groupes continus” Abstract: A methodical analysis of the research related to the article, “Sur les groupes continus”, of Henri Poincaré reveals many historical misconceptions and inaccuracies regarding his contribution to Lie theory. A thorough reading of this article confirms the precedence of his discovery of many important concepts, especially that of the universal enveloping algebra of a Lie algebra over the real or complex field, and the canonical map (symmetrization) of the symmetric algebra onto the universal enveloping algebra. The essential part of this article consists of a detailed discussion of his rigorous, complete, and enlightening proof of the so-called Birkhoff-Witt theorem. TITRE: La démonstration de Poincaré du théorème de Birkhoff-Witt. RÉSUMÉ: Une analyse méthodique des travaux faits en connexion avec l’article, “Sur les groupes continus”, de Henri Poincaré révèle des erreurs historiques et des jugements injustes en ce qui concerne sa contribution à la théorie de Lie. Une étude approfondie de cet article confirme la précédence de sa découverte de plusieurs concepts importants; notamment de l’algèbre enveloppante universelle d’une algèbre de Lie sur le corps réel ou le corps complexe, et de l’application canonique (la symétrisation) de l’algèbre symétrique sur l’algèbre enveloppante universelle. L’essentiel de cet article consiste en un examen approfondi de sa démonstration rigoureuse et complète du théorème de Birkhoff-Witt. Liste de quelques mots-clés: L’algèbre enveloppante universelle, l’application canonique (symétrisation), le théorème de Birkhoff-Witt