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Topic: Jordan algebra

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	Algebra over a field (Site not responding. Last check: 2007-11-03)
		In mathematics, an algebra over a field K, or a K -algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A.
		Algebras can also more generally be defined over any commutative ring K : we need a module A over K and a bilinear multiplication operation which satisfies the same identities as above; then A is a K -algebra, and K is the base ring of A.
		For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A.
	www.sciencedaily.com /encyclopedia/algebra_over_a_field (1258 words)

	Map Zones : Jordan Map
		Jordan had four universities with a combined enrollment of nearly 29,000; more than one-third of the students were women (11,000).
		Jordan's location as a buffer zone between the settled region of the Mediterranean littoral west of the Jordan River and the major part of the desert to the east contributed significantly to the country's experience in ancient and more recent times.
		Jordan, Hashemite Kingdom of (in Arabic, al-Mamlakah al-Urdunniyah al-Hashemiyah), kingdom in the Middle East, bordered on the north by Syria, on the east by Iraq and Saudi Arabia, on the south by Saudi Arabia and the Gulf of Aqaba, and on the west by Israel and the West Bank.
	kids.mapzones.com /world/jordan (6287 words)

	PlanetMath: Jordan algebra
		A Jordan algebra isomorphism is just a bijective Jordan algebra homomorphism.
		However, unlike Lie algebras, not every Jordan algebra is embeddable in an associative algebra.
		This is version 7 of Jordan algebra, born on 2004-12-07, modified 2004-12-17.
	planetmath.org /encyclopedia/JordanAlgebra.html (200 words)

	Algebra over a field - Articles and Information (Site not responding. Last check: 2007-11-03)
		Oftentimes, the word algebra is used to refer to an associative algebra or even a unital associative algebra, but this is NOT the use of the word in this article.
		Algebras can also more generally be defined over any commutative ring K : we need a module A over K and a multiplication operation which satisfies the same identities as above; then A is a K -algebra, and K is the base ring of A.
		algebras of functions, such as the R -algebra of all real-valued continuous functions defined on the interval [0,1], or the C -algebra of all holomorphic functions defined on some fixed open set in the complex plane.
	www.breakpt.org /article/Algebra_over_a_field (1059 words)

	PlanetMath:
		Jordan algebra homomorphism (in Jordan algebra) owned by CWoo
		Jordan algebra isomorphism (in Jordan algebra) owned by CWoo
		Jordan's theorem (multiply transitive groups) owned by bwebste
	planetmath.org /encyclopedia/J (312 words)

	Algebra -- Resources
		Algebraic Areas of Mathematics - Topics include number theory, groups and sets, commutative rings, algebraic geometry, and linear algebra.
		Robbins Algebras Are Boolean - A web text by William McCune describing the solution of this problem by a theorem-proving program, with input files and the proofs.
		Universal Algebra is a technical branch of mathematics related to algebra and model theory.
	www.edinformatics.com /algebra.htm (669 words)

	Operations on Structure Constant Algebras and their Elements (Site not responding. Last check: 2007-11-03)
		The basis of the new algebra is the concatenation of the bases of A and B and the products a * b where a in A and b in B are defined to be 0.
		The integral elements in this algebra are those where either all coefficients are integral or exactly 4 coefficients lie in 1/2 + Z in positions i_1, i_2, i_3, i_4, such that i_1, i_2, i_3, i_4 are a basis of one of the 7 quaternion algebras or a complement of such a basis.
		In the case of the Cayley algebra over GF(3) we obtain a permutation representation of degree 351, which is in fact the smallest possible degree (corresponding to the representation on the cosets of the largest maximal subgroup U_3(3):2).
	mad.epfl.ch /magma/text893.htm (1179 words)

	EXPLORE: Jordan algebra - Dictionary of Mathematics (Site not responding. Last check: 2007-11-03)
		Jordan algebras were first introduced by Pascual Jordan in quantum mechanics.
		A Jordan algebra that is isomorphic to an algebra of the form
		Despite the similarity to the previous example, this is an exceptional Jordan algebra.
	www.explore-math.com /mathematics/J/Jordan_algebra.html (165 words)

	week193
		Now, whenever you have a simple Lie algebra it's a direct sum of "root spaces", one for each root, together with an n-dimensional subspace called the Cartan algebra, where n is the called the "rank" of the Lie algebra.
		My goal in life is now to define a set of algebraic varieties, one for each root in L(1) and L(2), so I can write a paper entitled "57 Varieties" and get sued for trademark infringement by Heinz.
		Jordan triple systems are a generalization of Jordan algebras - and I'm sort of confused about why this generalization also turns up here.
	math.ucr.edu /home/baez/week193.html (4715 words)

	week162
		We can then try to construct a projective space whose points are the 1-dimensional projections and whose lines are the 2-dimensional projections, with the relation of "lying on" given by the partial order in our Jordan algebra.
		However, the exceptional Jordan algebra is secretly related to string theory, so there's a sense in which it's still lurking in the collective subconscious.
		Though Jordan algebras were invented to study quantum mechanics, the spin factors are also deeply related to special relativity: we can think of J(V) = V + R as "Minkowski spacetime", with V as space and R as time.
	math.ucr.edu /home/baez/week162.html (3662 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Hebrew University The Amitsur Algebra Seminar On Tuesday, May 27th, at 12:15, in room 209 of the Mathematics building (Note the change of day!) Issai Kantor will speak about Representations of Jordan and Lie algebras Abstract Contrary to the Lie algebra case, the theory of Jordan algebras is poor in representations.
		Moreover, there is the so-called exceptional Jordan algebra E_3, which has no representations at all.
		The idea of this point of view is that the representations of a Jordan algebra A have to be in 1-1 correspondence with representations of a graded Lie algebra L(A), obtained from A by the Tits-Kantor-Koeher construction.
	www.math.technion.ac.il /~techm/20030527121520030527kan (134 words)

	Exceptional algebraic structures in physics
		A reason for going up to the Freudenthal algebra and not just using the Jordan algebra is so that related symmetric spaces have complex structure, and the math of Hermitian symmetric spaces and bounded complex domains can be used in calculations of force strength constants and particle masses.
		E6 is a 78-dimensional Lie algebra, whose lowest dimensional non-trivial representations are 27-dimensional, corresponding to the two 27-dim Jordan subspaces of the 56-dimensional Freudenthal algebra Fr3(O).
		Note that Q is 27-complex-dimensional, and that the 27-dim of Q are related to the 27-dim of the Jordan algebra J3(O) and that J3(O) has a 26-dim traceless subalgebra J3(O)o and that world-lines in Q (lines of states that could form a world-line succession of states) look sort of like bosonic strings in 26-dim J3(O)o.
	www.lns.cornell.edu /spr/2001-01/msg0030378.html (829 words)

	R-matrices and the magic square (Site not responding. Last check: 2007-11-03)
		For example, the Lie algebras SL(n) are regarded as a series in the sense that the irreducible representations can be parametrised so that tensor product decompositions and plethysms can be written uniformly and dimensions of representations are written as rational functions of n.
		In these cases the Yang-Baxter equation is a powerful tool for working with the centraliser algebras of the tensor powers of the distinguished representation V. In this paper we look at each row of the Freudenthal magic square from the point of view of the R-matrix with spectral parameter.
		Let V and W be representations of a simple Lie algebra which admit an action of the affine Lie algebra which extends the action of the Lie algebra.
	www.maths.warwick.ac.uk /~bww/Rmatrix_H.html (3195 words)

	[No title]
		He invented the definition of what is now called a "formally real Jordan algebra": a commutative (but not necessarily associative) unital algebra over the real numbers such that: x o (y o x^2) = (x o y) o x^2 and also: a^2 + b^2 + c^2 +...
		An "ideal" in the Jordan algebra A is a subspace B such that if b is in B, a o b lies in B for all a in A. A Jordan algebra A is "simple" if its only ideals are {0} and A itself.
		Now, the relation between Jordan algebras and quantum logic is already interesting in itself: 8) G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.
	math.ucr.edu /home/baez/twf_ascii/week162 (2913 words)

	Interior-Point Abstracts (Site not responding. Last check: 2007-11-03)
		In the present paper we explain that Euclidean Jordan algebras play essentially the same role for symmetric cones as the (Jordan) algebra of real symmetric matrices plays for the cone of positive definite symmetric matrices.
		We describe the optimal barrier function of arbitrary symmetric cone in terms of the attached Jordan algebra (this includes for the first time quaternionic and octonion co nes).
		Necessary results from the theory of Euclidean Jordan algebras are briefly described.
	www-unix.mcs.anl.gov /otc/InteriorPoint/abstracts/Faybusovich.html (178 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-11-03)
		The Jordan Canonical Form is an example of an upper bidiagonal matrix.
		Conversely, if a matrix is defective, then it must have at least one repeated eigenvalue, that is, an eigenvalue of algebraic multiplicity greater than 1.
		A matrix is defective if and only if its Jordan Canonical Form has at least one nonzero entry on the superdiagonal.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13934 words)

	Nat' Academies Press, Biographical Memoirs V.51 (1980) (Site not responding. Last check: 2007-11-03)
		An algebra is an abstract mathematical entity with ele- ments and operations fulfilling the familiar laws of algebra, with one important qualification the commutative law of multiplication is waived.
		Another thing that happened was that Albert was introduced to Jordan algebras.
		The physicist Pascual Jordan had suggested that a certain kind of algebra, inspired by using the operation xy +yx in an associative algebra, might be useful in quantum mechanics.
	www.nap.edu /books/0309028884/html/2.html (3569 words)

	REVIEWS
		"Jordan Triple Systems with Completely Reducible Derivation or Structure Algebras," Erhard Neher, (August 1985).
		"Radicals of Jordan Algebras of Degree 3," H.P. Petersson and M.L. Racine (August 1988).
		"An Expansion Formula for the Norm of a Jordan Algebra," Erhard Neher, (April 1998).
	faculty.goucher.edu /blewand/lewandr.htm (264 words)

	[No title]
		F4 is the group of >>automorphisms of the exceptional Jordan algebra, and it is >>52-dimensional.
		Is there any natural way to write >>the Lie algebra of F4 as the direct sum of two copies of the space >>of traceless elements in the exceptional Jordan algebra?
		Thus the answer to John's question "Is there any natural way to write the Lie algebra of G as the direct sum of two copies of V?" is: not quite, but in characteristic p the Lie algebra can be obtained as an extension of V by the almost-isomorphic representation V'!
	www.math.niu.edu /~rusin/known-math/00_incoming/f4 (1477 words)

	David Jordan's Research Publications
		(with C.R. Jordan) "A note on semiprimitivity of Ore extensions", Communications in Algebra 4 (7) (1976), 647-656.
		(with C.R. Jordan) "The Lie structure of a commutative ring with a derivation", J. London Math.
		International Algebraic Conference on the Occasion of the 90th Birthday of A. Kurosh, Moscow, Russia, May 25-30 1998, ed Y. Bahturin, Walter de Gruyter (2000), 145-153.
	www.shef.ac.uk /~daj/public.html (496 words)

	Bibliography
		Existence and conjugacy of Cartan subalgebras of Jordan algebras.
		(with A. Fernández and E. García) Annihilators of elements of the socle of a Jordan algebra.
		On the set of invertible elements in Banach Jordan algebras.
	informatik.uibk.ac.at /~loos/bibl.html (290 words)

	index
		My research interests lie in the area of Algebra: algebras with polynomial identities (2000 MSC 16Rxx), Lie and Jordan algebras (2000 MSC 17Bxx and 17Cxx).
		Algebraic extensions and Baer and Jacobson radicals, Proc.13th Spring Conf., UBM, Sofia, 1984, 152-156.
		Central polynomials in the matrix algebra of order two, with J. Colombo, Atas, IX ENAL, USP, 2001, 89-92, Linear and Multilinear Algebra, 2003, to appear.
	www.ime.unicamp.br /~plamen (422 words)

	On the optimal parameter of a self-concordant barrier over a symmetric cone
		Abstract : The properties of the barrier F(x)=-log(det(x)), defined over the cone of squares of an Euclidean Jordan algebra, are analyzed using pure algebraic techniques.
		Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained.
		Namely, it is proved that the Carathéodory number of the cone of squares of an Euclidean Jordan algebra is equal to the rank of the algebra.
	www.optimization-online.org /DB_HTML/2003/11/774.html (132 words)

	[No title]
		Jordan Algebras, Generalizations, and Algebraic Geometry: notes and links by Tony Smith.
		Includes preprints and course notes on Group Theory, Fields and Galois Theory, Algebraic Geometry, Algebraic Number Theory, Modular Functions and Modular Forms, Elliptic Curves, Abelian Varieties, Etale Cohomology, and Class Field Theory.
		Directory listings and categorizations are provided by the Open Directory Project with user enhancements and proprietary search functions developed by Juvio Corporation.
	kib6.juvioservice.com /search/directory.asp?c=128916 (195 words)

	The real Cayley algebra
		Some structure constants are defined more than once and we have to get rid of these when defining the algebra.
		are a basis of one of the 7 quaternion algebras or a complement of such a basis.
		In the case of the Cayley algebra over GF (3) we obtain a permutation representation of degree 351, which is in fact the smallest possible degree (corresponding to the representation on the cosets of the largest maximal subgroup
	magma.maths.usyd.edu.au /magma/Examples/node75.html (539 words)

	Operations on Structure Constant Algebras and their Elements (Site not responding. Last check: 2007-11-03)
		True if the algebra A is commutative; otherwise false.
		True if the algebra A is a Lie algebra; otherwise false.
		The degree (= dimension) of the module underlying the algebra A. Degree(a) : AlgGenElt -> RngIntElt
	www.msri.org /info/computing/docs/magma/text889.htm (1199 words)

	Near Infrared Reflectance Analysis by Gauss-Jordan Linear Algebra - Storming Media
		However, because near-infrared bands from solid samples are both abundant and broad, the reflectance at a given wavelength usually contains contributions from several sample components, requiring extensive calculations on overlapped bands.
		In the present study, these calculations have been performed using an approach similar to that employed in multi-component spectrophotometry, but with Gauss-Jordan linear algebra serving as the computational vehicle.
		Using this approach, correlations for percent protein in wheat flour and percent benzene in hydrocarbons have been obtained and are evaluated.
	www.stormingmedia.us /78/7874/A787421.html (241 words)

	Linear Algebra Syllabus
		A more specific list of topics is included later in this document.
		Objectives : The primary objective of this course is to understand thoroughly (with proofs) the basic material on vector spaces, linear transformations and their matrix representations, eigenvalues, and inner products.
		Textbook : The textbook for this class is Linear Algebra, 3rd edition, by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence, published by Prentice Hall (1997).
	www.puc.edu /Faculty/Steve_Waters/Syllabi/M465.html (690 words)

	Computing the derivation Lie algebra of the quadratic Jordan algebra H3(Os,-) at any characteristic. (Site not responding. Last check: 2007-11-03)
		Computing the derivation Lie algebra of the quadratic Jordan algebra H3(Os,-) at any characteristic.
		We have implemented the quadratic Jordan structure of H3(Os,-), where Os denotes the split octonions, into a computer system (Mathematica).
		We have used this implementation to compute a generic expression of any element of f4(Os) at any characteristic.
	mathematik.uibk.ac.at /jordan/archive/imH3split/index.html (66 words)

	Octonions and other Division Algebras
		My book ("Division Algebras:...") is now published by Springer (which recently bought Kluwer).
		Algebraic Spinor Reduction Yields the Standard Symmetry and Family Structure
		Let me know if you'd like to be included.
	www.7stones.com /Homepage/AlgebraSite/algebra0.html (114 words)

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