Frattini subalgebra

A notion imported from group theory (cf. also Group), where the Frattini subgroup of a group plays an important role. Due to the many close connections which Lie algebras have with groups, these concepts have been mainly studied for Lie algebras.

The Frattini subalgebra of a non-associative (that is, not necessarily associative) algebra (cf. also Associative rings and algebras; Non-associative rings and algebras) is the intersection of its maximal subalgebras. Unlike the group case, where the Frattini subgroup is always a normal subgroup, the Frattini subalgebra is not an ideal in general; the Frattini ideal is then the largest ideal contained in the Frattini subalgebra.

Key features.

See also [a3], [a1].

i) An element $x$ belongs to the Frattini subalgebra of a non-associative algebra $A$ if and only if whenever $S\cup\{x\}$ generates $A$, $S$ alone suffices to generate $A$.

ii) For finite-dimensional algebras from the usual varieties (associative, Lie, Jordan, alternative or Mal'tsev algebras), the Frattini ideal is nilpotent and an algebra is nilpotent if so is its quotient modulo the Frattini ideal (cf. also Nilpotent ideal; Nilpotent algebra).

iii) Finite-dimensional algebras with trivial Frattini ideal have some nice decomposition properties based on the fact that any ideal whose square is zero is complemented by a subalgebra.

Generalizations.

Related concepts are being studied in very general algebraic systems [a2].

References

[a1]	A. Elduque, "A Frattini theory for Malcev algebras" Algebras, Groups, Geom. , 1 (1984) pp. 247–266
[a2]	L.A. Shemeltkov, A.N. Skiba, "Formations of algebraic systems" , Nauka (1989) (In Russian)
[a3]	D.A. Towers, "A Frattini theory for algebras" Proc. London Math. Soc. , 27 : 3 (1973) pp. 440–462