Niveau: Supérieur, Doctorat, Bac+8
Jordan structures and non-associative geometry Wolfgang Bertram Institut Elie Cartan, Universite Nancy I, Faculte des Sciences, B.P. 239, 54506 Vandœuvre-les-Nancy, Cedex, France Abstract. We give an overview of constructions of geometries associated to Jor- dan structures (algebras, triple systems and pairs), featuring analogs of these con- structions with the Lie functor on the one hand and with the approach of non- commutative geometry on the other hand. Keywords: Jordan pair, Lie triple system, graded Lie algebra, filtered Lie alge- bra, generalized projective geometry, flag geometries, (non-) associative algebra and geometry AMS subject classification: 17C37, 17B70, 53C15 Introduction Let us compare two aspects of the vast mathematical topic “links between ge- ometry and algebra”: on the one hand, the Lie functor establishes a close rela- tion between Lie groups (geometric side) and Lie algebras (algebraic side); this is generalized by a correspondence between symmetric spaces and Lie triple systems (see [Lo69]). On the other hand, the philosophy of Non-Commutative Geometry generalizes the relation between usual, geometric point-spaces M (e.g., manifolds) and the commutative and associative algebra Reg(M,K) of “regular” (e.
- reg
- algebra
- associative geometry
- geometric space
- banach-jordan
- graded lie
- commutative geometry
- purely geometric
- jordan pairs
- lie algebras