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By De Simone A., Mundici D.

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The fusion of algebra, research and geometry, and their program to genuine global difficulties, were dominant topics underlying arithmetic for over a century. Geometric algebras, brought and categorized by way of Clifford within the past due nineteenth century, have performed a renowned function during this attempt, as visible within the mathematical paintings of Cartan, Brauer, Weyl, Chevelley, Atiyah, and Bott, and in functions to physics within the paintings of Pauli, Dirac and others.

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Let J = Hn (C) be the formally real Jordan algebra of dimension n2 over the reals consisting of all Z ∈ Mn (C) with Z ∗ = Z. Then the positive cone Cone(J) consists precisely of the positivedefinite matrices (the hermitian matrices whose Jordan form has only positive real eigenvalues). The structure group is generated by the two involutory transformations Z → −Z and Z → Z = Z tr together with the connected subgroup G = Strg(J)0 of all Z → AZA∗ for A ∈ GLn (C). The connected component K = Aut (J)0 of the automorphism group consists of all Z → U ZU ∗ = U ZU −1 for unitary U ∈ Un (C).

An isomorphism of Hermitian manifolds is a biholomorphic map of analytic manifolds whose differential is isometric on each tangent space. A hermitian symmetric space is a (connected) hermitian manifold having at each point p a symmetry sp [an involutive global isomorphism of the manifold having p as isolated fixed point]. We henceforth assume that all our Hermitian manifolds are connected. These are abstract manifolds, but every Hermitian symmetric space of “noncompact type” [having negative holomorphic sectional curvature] is a bounded symmetric domain, a down-to-earth bounded domain in Cn each point of which is an isolated fixed point of an involutive biholomorphic map of the domain.

4 Links with Differential Geometry Though mathematical physics gave birth to Jordan algebras and superalgebras, and Lie algebras gave birth to Jordan triples and pairs, differential geometry has had a more pronounced influence on the algebraic development of Jordan theory than any other mathematical discipline. Investigations of the role played by Jordan systems in differential geometry have revealed new perspectives on purely algebraic features of the subject. We now indicate what Jordan algebras were doing in such a strange landscape.

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A Cantor-Bernstein Theorem for Complete MV-Algebras by De Simone A., Mundici D.

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