Liealgebran

Liealgebran is a term used in abstract algebra to denote a class of binary-bracket algebraic structures that generalize Lie algebras. In its simplest form, a Liealgebran consists of a finite-dimensional vector space L over a field K together with a bilinear operation [ , ]: L × L → L, called the Lie bracket.

The ungraded variant requires antisymmetry [x,y] = -[y,x] and the Jacobi identity [x,[y,z]] = [[x,y],z] + [y,[x,z]] for all x, y, z in L. A graded or colored version introduces a decomposition of L into homogeneous components and replaces the ordinary antisymmetry and Jacobi identity with graded or sign-twisted analogues. In addition, there are n-ary generalizations where the binary bracket is replaced by a multilinear operation [x1, ..., xn] satisfying appropriate generalized Jacobi identities.

Relationship to Lie algebras: the ungraded Liealgebran recovers the familiar notion of a Lie algebra when the bracket is antisymmetric and satisfies the standard Jacobi identity. Many properties and constructions from Lie theory, such as representations and universal enveloping algebras, extend to Liealgebrans in their respective variants.

Examples: any classical Lie algebra (for instance gl(n, K) with the commutator [A,B] = AB − BA) is a Liealgebran in the ungraded sense. Graded or colored versions lead to Lie superalgebras, while certain n-ary versions yield additional generalizations used in specialized contexts.

Applications: Liealgebrans appear in differential geometry, theoretical physics (including symmetry and conservation laws), and representation theory, where generalized bracket structures provide frameworks for studying complex symmetries and their realizations.

History and usage: the term Liealgebran is not uniformly standardized and appears sporadically in expository or exploratory discussions of generalizations of Lie theory. Definitions vary by author and context. See also Lie algebra, Lie superalgebra, Leibniz algebra, Malcev algebra.