quadratic Jordan algebras are a generalization of Jordan algebras introduced by Kevin McCrimmon (1966). The fundamental identities of the quadratic representation...

In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: x y = y x {\displaystyle...

mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...

with quadratic map Q then (V,Q,e) is a quadratic Jordan algebra. In the opposite direction, if (V,Q,e) is a separable quadratic Jordan algebra with inversion...

called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and...

real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism...

→ A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped...

mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies...

*-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings. Quaternions, split-complex numbers...

words, it is a 8-dimensional unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm form) such that N ( x y ) =...

{g}}_{-1}} . Associator E7 (mathematics) Quadratic Jordan algebra Bertram, Wolfgang (2000), The geometry of Jordan and Lie structures, Lecture Notes in Mathematics...

enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...

direct sum of algebras C ⊕ C. The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the...

{Q(a)Q(a^{-1}+b^{-1})Q(b)=Q(a+b).}} In fact in this case the relations for a quadratic Jordan algebra imply R ( a , b ) = 2 Q ( a ) Q ( a − 1 , b ) = 2 Q ( a , b − 1...

In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are...

Octonion algebra Pre-Lie algebra Poisson algebra Process algebra Quadratic algebra Quaternion algebra Rees algebra Relation algebra Relational algebra Rota–Baxter...

nonassociative. Also like the octonions, they form a composition algebra since the quadratic form N is multiplicative. That is, N ( x y ) = N ( x ) N ( y...

part of linear algebra, which is used in many parts of mathematics, including geometric transformations, coordinate changes, quadratic forms, and many...

quadratic form ‖ z ‖ 2 = z z ∗ = z ∗ z = x 2 − y 2   . {\displaystyle \lVert z\rVert ^{2}=zz^{*}=z^{*}z=x^{2}-y^{2}~.} It has the composition algebra...

McCrimmon (2004) A Taste of Jordan Algebras, pp 64, Universitext, Springer ISBN 0-387-95447-3 MR2014924 Albert, A. A. (1942), "Quadratic forms permitting composition"...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

American mathematician, specializing in Jordan algebras. He is known for his introduction of quadratic Jordan algebras in 1966. McCrimmon attended secondary...

quadratic forms. The trace is a map of Lie algebras tr : g l n → K {\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K} from the Lie algebra...

law of inertia (quadratic forms) Witt's theorem (quadratic forms) Artin–Wedderburn theorem (abstract algebra) Artin–Zorn theorem (algebra) Brauer–Cartan–Hua...

bioctonion z is z z* = p p* + q q*, which is a complex quadratic form with eight terms. The bioctonion algebra is sometimes introduced as simply the complexification...

types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures...

or a matrix of dimension ⁠ 2 × 3 {\displaystyle 2\times 3} ⁠. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric...

introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations...

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A  is symmetric ⟺ A = A T . {\displaystyle A{\text{...

based on Jordan's Traité, made Galois theory accessible to a wider German and American audience as did Heinrich Martin Weber's 1895 algebra textbook....