It was then discovered (Wong and Masaro (22) and Masaro and Wong (12)) that these conditions reflected the fact that the Wishartness of Y'WY depended on whether or not a certain linear transformation was a Jordan algebra homomorphism.
We shall exploit the connection between Jordan algebras and the Wishart distribution, and characterize the Wishartness of a quadratic form Q(Y) in terms of Jordan algebra homomorphisms or more precisely, Jordan algebra representations.
2 which links the concept of a Jordan algebra homomorphism with the moment generating function of the Wishart distribution.
These conditions are closely linked to the natural Jordan algebra structure on [H.
2 which connects the notion of a Jordan algebra homomorphism with the moment generating function of the Wishart distribution.
A commutative algebra X with product x [o] y is called a Jordan algebra if [x.
A complex Jordan algebra C with product x [o] y and involution x [?
Upmeier, Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, Regional Conference Series in Mathematics, Amer.
The seven chapters in the volume are dedicated to nonlinear elliptic equations, division algebras, exceptional lie groups, and calibrations, Jordan algebras
and the Cartan isoparametric cubics, solutions from trialities and isoparametric forms, cubic minimal cones, and singular solutions in calibrated geometrics.
They cover gradings on algebras, associative algebras, classical Lie algebras, composition algebras and type G2, Jordan algebras
with type F4, other simple Lie algebras in characteristic zero, and Lie algebras of Caran type in prime characteristic.
Contributions from a 2007 conference on Algebras, Representations, and Applications held in Brazil to commemorate mathematician Ivan Shestakov's 60th birthday are included in this volume for students and researchers working with the Theory of Lie, Jordan algebras
, superalgebras and quantum rings.
Ten chapters discuss the skew field of quaternions; elements of the geometry of S3, Hopf bundles, and spin representations; internal variables of singularity free vector fields in a Euclidean space; isomorphism classes, Chern classes, and homotopy classes of singularity free vector fields in 3-space; Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras
, and special linear groups; the Heisenbreg group and natural C*-algebras of a vector field in 3-space; the Schrodinger representation and the metaplectic representation; the Heisenberg group as a basic geometric background of signal analysis and geometric optics; quantization of quadratic polynomials; and field theoretic Weyl quantization of a vector field in 3-space.