mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure...

, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie...

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero...

A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative...

associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus...

is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory. Fields of sets play an essential...

related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which...

lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more...

mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix...

unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra g {\displaystyle {\mathfrak...

abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings R and A, where...

operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas...

In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle...

quantum field theory to string theory and LHC phenomenology. Formally, a Hopf algebra is an (associative and coassociative) bialgebra H over a field K together...

mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically...

Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this...

3-dimensional Euclidean space Algebra over a field – a vector space equipped with a bilinear product Any of the original vector algebras of the nineteenth century...

both are subgroups of G {\displaystyle G} . If R is a ring or an algebra over a field, and S is a subset of R, then the centralizer of S is exactly as...

that a simple ring is an associative algebra over this field. It is then called a simple algebra over this field. Several references (e.g., Lang (2002)...

analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A {\displaystyle A} over the real or complex numbers (or over a non-Archimedean...

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

overview, see Algebra. The bare word "algebra" may refer to: Elementary algebra Abstract algebra Algebra over a field In universal algebra, algebra has an axiomatic...

function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description...

In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from...

In 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...

In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"...

mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is,...

to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this...

mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices...

Linear algebra is the branch of mathematics concerning linear equations such as: a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...