subalgebras

A subalgebra is a subset of an algebraic structure that itself satisfies the defining rules of that structure. In the common setting of an associative algebra A over a field F, a subalgebra B is a subspace of A that is closed under the multiplication of A; equivalently, for all b1, b2 in B, b1 b2 lies in B. Since B is an F-subspace, closure under scalar multiplication is automatic. If A has a multiplicative identity 1_A, some authors require that 1_A belongs to B, calling B a unital subalgebra; other authors allow nonunital subalgebras.

In a Lie algebra L over F, a subalgebra is a linear subspace S ⊆ L that is closed under the Lie bracket: [x, y] ∈ S for all x, y in S. More generally, subalgebras can be defined for other algebraic structures with a bilinear product or bracket, always as subsets that inherit the same operation(s) as the ambient structure.

Examples include the diagonal or upper triangular matrices as subalgebras of the matrix algebra M_n(F); subalgebras of function algebras defined by imposing structural or symmetry conditions; and polynomial subalgebras in a commutative setting.

Key properties include: the intersection of subalgebras is a subalgebra; the subalgebra generated by a subset S is the smallest subalgebra containing S; in associative algebras this generated subalgebra consists of all finite F-linear combinations of finite products of elements of S. Subalgebras form a lattice under inclusion, with joins given by generated subalgebras and meets by intersections.