Let (V, o) be an n-dimensional Euclidean Jordan algebra (EJA) with rank r equipped with the standard inner product (x, s) := tr([x.sub.0] s) and assume that K is the symmetric cone related to EJA (V, o).

A Jordan algebra has an identity element, if there exists a unique element e [member of] V such that x o e = e o x = x for all x [member of] V.

Therefore (A, (,,)) is a Lts since any

Jordan algebra is a Lts with respect to the operation (x, y, z) = [as.sup.J] (y, z, x) (see [1]).

By defining a new binary operation a-b = (1/2)(ab + ba) on an associative algebra over a field whose characteristic is not equal to 2, we obtain another important nonassociative algebra known as

Jordan algebra. It is worth mentioning here that the theory of nonassociative algebras is a fruitful branch of algebra.

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.

A complex

Jordan algebra C with product x [o] y and involution x [??] x* is called a JB*-algebra if C carries a Banach space norm ||dot|| satisfying ||x[o]y|| [less than or equal to] ||x||dot||y|| and ||{xx*x}|| = ||x||[.sup.3].

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.

He has also worked extensively with questioning -- and dismissing -- the assumption of the finite nature of "

Jordan algebras." Richard Brualdi of the University of Wisconsin-Madison, calls Zelmanov "one of the most brilliant mathematicians of this century."

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.