Algebres

Algebres, in French, refers to algebras in the mathematical sense. In English texts, the term algebras denotes similar structures, and the plural algebras is used. An algebra over a field F is a vector space over F equipped with a bilinear multiplication that maps a pair of elements to another element of the same space. If there is a multiplicative identity, the algebra is called unital. If the multiplication is associative, it is an associative algebra; if it is commutative, it is a commutative algebra.

Common examples include the field F itself viewed as a one-dimensional algebra, the algebra of n by n matrices M_n(F) with ordinary addition and multiplication, and the polynomial ring F[x] under usual addition and multiplication. Function algebras such as C(X) with pointwise operations, and group algebras F[G] formed from a group G, are also algebras over F. Many algebras are finite dimensional, and their structure is studied via modules, representations, and decomposition into simpler components.

Over a base field, one often distinguishes between associative algebras and non‑associative ones. Lie algebras, for example, are non‑associative algebras with a bilinear product called the Lie bracket that is antisymmetric and satisfies the Jacobi identity. Any associative algebra yields a Lie algebra via the commutator bracket [a,b] = ab − ba, linking the two theories.

Applications of algebras span mathematics and physics, including representation theory, algebraic geometry, functional analysis, and quantum mechanics. Specialized classes of algebras, such as C*-algebras and other operator algebras, play central roles in analysis and mathematical physics.