Maximal(Graded) Subalgebras Of The Special Linear Lie(Super) Algebras Over(Super) Rings

The maximal sub-systems of algebras can deeply reflect the intrinsic properties of algebras,so it is very important to study maximal sub-systems of algebras.In this paper,we determine all maximal subalgebras of the special linear Lie algebras containing the canonical torus over a unital commutative ring with 2,3 be the unit and we also determine all maximal graded subalgebras of the special linear Lie superalgebras containing the canonical torus over a unital supercommutative superring with 2,3 be the unit.Firstly,the maximal subalgebras is constructed by using the maximal ideals of the rings.Secondly,we give the maximal subalgebras on the field and use it to give the maximal subalgebras in the rings.In the end,we calculate the number of maximal subalgebras and prove that each maximal subalgebra is conjugate under a permutation matrix to a standard one.