Associative algebra in the context of Scalar multiplication

In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and ${\displaystyle x\cdot (2+x)}$ is the product of ${\displaystyle x}$ and ${\displaystyle (2+x)}$ (indicating that the two factors should be multiplied together).When one factor is an integer, the product is called a multiple.

The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well.

In mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial ${\displaystyle g(x)\in K[x]}$ with coefficients in K such that g(a) = 0. Elements of A that are not algebraic over K are transcendental over K. A special case of an associative algebra over ${\displaystyle K}$ is an extension field ${\displaystyle L}$ of ${\displaystyle K}$.

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with C being the field of complex numbers and Q being the field of rational numbers).

In 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 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".

The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). Given an integer n, the ring of real square matrices of order n is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.

In 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 of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879).

The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space ${\displaystyle {\mathfrak {g}}}$ together with an operation called the Lie bracket, an alternating bilinear map ${\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}}$, 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 bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors ${\displaystyle x}$ and ${\displaystyle y}$ is denoted ${\displaystyle [x,y]}$. 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, ${\displaystyle [x,y]=xy-yx}$.

Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the real or complex numbers, there is a corresponding connected Lie group, unique up to covering spaces (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra.

In mathematics, the exterior algebra or Grassmann algebra of a vector space ${\displaystyle V}$ is an associative algebra that contains ${\displaystyle V,}$ which has a product, called exterior product or wedge product and denoted with ${\displaystyle \wedge }$, such that ${\displaystyle v\wedge v=0}$ for every vector ${\displaystyle v}$ in ${\displaystyle V.}$ The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol ${\displaystyle \wedge }$ and the fact that the product of two elements of ${\displaystyle V}$ is "outside" ${\displaystyle V.}$

The wedge product of ${\displaystyle k}$ vectors ${\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}}$ is called a blade of degree ${\displaystyle k}$ or ${\displaystyle k}$-blade. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude of a 2-blade ${\displaystyle v\wedge w}$ is the area of the parallelogram defined by ${\displaystyle v}$ and ${\displaystyle w,}$ and, more generally, the magnitude of a ${\displaystyle k}$-blade is the (hyper)volume of the parallelotope defined by the constituent vectors. Its bilinearity, expected from such a generalization of volume, and its alternating property that ${\displaystyle v\wedge v=0}$ implies a skew-symmetric property that ${\displaystyle v\wedge w=-w\wedge v,}$ and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.

In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.

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.

Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups and their representations.