Subvarieties of the Varieties Generated by the SuperalgebraM_{1, 1}(E) orM₂(𝒦)

The $G$-graded identities of the Grassmann Algebra

“…The Y -proper polynomials (see [7,Section 2]) are the elements of the unitary F -subalgebra B of F X generated by the elements of Z and by all non-trivial commutators. More…”

“…It is well known (see, for instance, Lemma 1 Section 2 in [7]) that all the graded polynomial identities of a superalgebra A follow from the Y -proper ones. This means that the set T Z2 (A) ∩ B generates the whole T Z2 (A) as a T Z2 -ideal.…”

Z_2-graded Gelfand-Kirillov dimension of the Grassmann algebra

“…One defines G-graded subspaces of A, G-graded A-modules, G-graded homomorphisms and so on, in a standard way (see for example [11] or [8] for more details).…”

Z2-graded cocharacters for superalgebras of triangular matrices

“…It is well known (see, for instance, Lemma 1, Section 2 in [10]) that all graded polynomial identities of a superalgebra A follow from the Y -proper ones. This means that the set…”

“…The following result relates the structure of V h; k (A) with the structure of h; k (A) (see [10] Proposition 1, Section 2): Proposition 2.3. Let A be a unitary superalgebra, and let i; k (A) = m ; ⊗ be the sequence of proper cocharacters of A, Then…”