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The Exceptional Jordan Eigenvalue Problem
"... We discuss the eigenvalue problem for 3 × 3 octonionic Hermitian matrices which is relevant to the Jordan formulation of quantum mechanics. In contrast to the eigenvalue problems considered in our previous work, all eigenvalues are real and solve the usual characteristic equation. We give an element ..."
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We discuss the eigenvalue problem for 3 × 3 octonionic Hermitian matrices which is relevant to the Jordan formulation of quantum mechanics. In contrast to the eigenvalue problems considered in our previous work, all eigenvalues are real and solve the usual characteristic equation. We give an elementary construction of the corresponding eigenmatrices, and we further speculate on a possible application to particle physics. 1
The Kac Jordan superalgebra: automorphisms and maximal subalgebras. Preprint arXiv:mat.RA/0509040
"... Abstract. In this note the group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras. 1. Introduction. Finite dimensional simple Jordan superalgebras over an algebraically closed field of characteristic zero were classified by V. Kac in 1977 [9] ..."
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Abstract. In this note the group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras. 1. Introduction. Finite dimensional simple Jordan superalgebras over an algebraically closed field of characteristic zero were classified by V. Kac in 1977 [9] (see also Kantor [10], where a missing case is added). Among these superalgebras we find the ten dimensional Kac Jordan superalgebra, K10, which is exceptional (see
2.2 Geometric Properties....................... 5
, 2008
"... We investigate the spectral geometry of the exceptional Jordan algebra and its extensions. We examine the spectrum of the exceptional Jordan algebra over the sixteendimensional space of primitive idempotents, where it exhibits three real eigenvalues. We interpret the spectrum as coordinates for a c ..."
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We investigate the spectral geometry of the exceptional Jordan algebra and its extensions. We examine the spectrum of the exceptional Jordan algebra over the sixteendimensional space of primitive idempotents, where it exhibits three real eigenvalues. We interpret the spectrum as coordinates for a coincident Dbrane system where the real eigenvalues correspond to positions of three D0branes on a line in the octonionic projective plane. The F4 gauge symmetry arises from the isometries of the octonionic projective plane. An argument is also given for the exceptional Jordan C*algebra, where E6 symmetry arises. We conclude that Mtheory, in Jordan matrix models, is inherently a sixteendimensional theory originating in octonionic matrix space. This matrix space demonstrates the existence of a Jordan algebraic Gel’fandNaimark theorem, where a nonassociative geometry is
GRADED LIE ALGEBRAS DEFINED BY JORDAN ALGEBRAS AND THEIR REPRESENTATIONS
, 2004
"... Abstract. In this talk we introduce the notion of a generalized representation of a Jordan algebra with unit which has the following properties: 1) Usual representations and Jacobson representations correspond to special cases of generalized representations. 2) Every simple Jordan algebra has infini ..."
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Abstract. In this talk we introduce the notion of a generalized representation of a Jordan algebra with unit which has the following properties: 1) Usual representations and Jacobson representations correspond to special cases of generalized representations. 2) Every simple Jordan algebra has infinitely many nonequivalent generalized representations. 3) There is a onetoone correspondence between irreducible generalized representations of a Jordan algebra A and irreducible representations of a graded Lie algebra L(A) = U−1⊕U0⊕U1 corresponding to A (the Lie algebra L(A) coincides with the TKK construction when A has a unit). The latter correspondence allows to use the theory of representations of Lie algebras to study generalized representations of Jordan algebras. In particular, one can classify irreducible generalized representations of semisimple Jordan algebras and also obtain classical results about usual representations and Jacobson representations in a simple way.
Locality, Weak or Strong Anticipation and Quantum Computing. I. Nonlocality in Quantum Theory
"... Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Categ ..."
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Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the ChurchTuring hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinatefree mathematical language which is both constructive and nonlocal to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of ChurchTuring and quantum theories. It has constructively to bridge the nonlocal chasm between the weak anticipation of mathematics and the strong anticipation of physics.