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‘Sur le concept de nombre en mathématique’ Cours inédit de Leopold Kronecker à Berlin (1891)

Jacqueline Boniface, Norbert Schappacher (2001)

Revue d'histoire des mathématiques

Le texte que nous présentons est le dernier cours du mathématicien berlinois Leopold Kronecker (1823–1891). Ce cours, publié ici pour la première fois, nous donne des informations importantes sur la philosophie des mathématiques de Kronecker, en particulier sur sa conception du nombre. Il précise, en outre, la position que Kronecker occupa dans le mouvement d’‘arithmétisation’ des mathématiques et permet de mieux comprendre comment, et pourquoi, il se situe à contre-courant de la tendance dominante...

Symmetric algebras and Yang-Baxter equation

Beidar, K., Fong, Y., Stolin, A. (1997)

Proceedings of the 16th Winter School "Geometry and Physics"

Let U be an open subset of the complex plane, and let L denote a finite-dimensional complex simple Lie algebra. A. A. Belavin and V. G. Drinfel’d investigated non-degenerate meromorphic functions from U × U into L ⊗ L which are solutions of the classical Yang-Baxter equation [Funct. Anal. Appl. 16, 159-180 (1983; Zbl 0504.22016)]. They found that (up to equivalence) the solutions depend only on the difference of the two variables and that their set of poles forms a discrete (additive) subgroup Γ of the...

Symmetrization of brace algebra

Daily, Marilyn, Lada, Tom (2006)

Proceedings of the 25th Winter School "Geometry and Physics"

Summary: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on ⨁ k ≥ 1 Hom ( V ⊗ k , V ) coincides with the natural symmetric brace structure on ⨁ k ≥ 1 Hom ( V ⊗ k , V ) a s , the direct sum of spaces of antisymmetric maps V ⊗ k → V .

Symplectic solution supermanifolds in field theory

Schmitt, T. (1997)

Proceedings of the 16th Winter School "Geometry and Physics"

Summary: For a large class of classical field models used for realistic quantum field theoretic models, an infinite-dimensional supermanifold of classical solutions in Minkowski space can be constructed. This solution supermanifold carries a natural symplectic structure; the resulting Poisson brackets between the field strengths are the classical prototypes of the canonical (anti-) commutation relations. Moreover, we discuss symmetries and the Noether theorem in this context.

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