Malcevtype

Malcevtype is a term used in algebra to describe a class of nonassociative structures that generalize Lie algebras in the spirit of Anatoly Malcev’s work. In this context, Malcevtype refers to algebras and related systems that exhibit Malcev-type identities, which generalize the Jacobi identity that characterizes Lie algebras. The label is often used descriptively to indicate Malcev-like behavior rather than to denote a single universal axiom system.

Most commonly, Malcevtype structures are anticommutative algebras whose multiplication satisfies the Malcev identity, a condition that extends the Lie algebra framework. This places Malcev algebras at the core of Malcevtype theory, while also including tangent algebras of Moufang loops and other nonassociative constructions that share analogous structural features. In many texts, Lie algebras are treated as a special case of Malcevtype, since the Jacobi identity implies the Malcev identity.

Key examples include the 7-dimensional simple Malcev algebra obtained from the imaginary octonions with the commutator multiplication, whose automorphism group is the exceptional Lie group G2. Tangent algebras of Moufang loops provide a geometric source for Malcev-type algebras and illustrate how nonassociative operations can give rise to rich algebraic structures in both algebra and geometry.

In summary, Malcevtype serves as a broad descriptive category for nonassociative algebras that extend Lie theory through Malcev identities, encompassing Malcev algebras and related tangent-algebra constructions. See also Malcev algebra, Moufang loop, and octonions.