In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, .
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional.
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result. The identity is named after the German mathematician Carl Gustav Jacob Jacobi.
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions.
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang. Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra.
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
- .
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals. The theorem was first proven by Emanuel Lasker for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether.
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication operation A × A → A which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (ab)(cd), (a(bc))d and a(b(cd)) may all yield different answers.
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
In mathematics, and in particular the study of algebra, an Akivis algebra is a nonassociative algebra equipped with a binary operator, the commutator and a ternary operator, the associator that satisfy a particular relationship known as the Akivis identity. They are named in honour of Russian mathematician Maks A. Akivis.
