Z2-graded cocharacters for superalgebras of triangular matrices

AbstractLet K be a field of characteristic zero, let A, B be K-algebras with polynomial identity and let M be a free (A,B)-bimodule. The algebra R=A0MB can be endowed with a natural Z2-grading. In this paper, we compute the graded cocharacter sequence, the graded codimension sequence and the superexponent of R. As a consequence of these results, we also study the above PI-invariants in the setting of upper triangular matrices. In particular, we completely classify the algebra of 3×3 upper triangular matrices endowed with all possible Z2-gradings

(δ,ε)(\delta,\varepsilon)-Differential Identities of UTm(F)UT_m(F)

Let δ and ε be the inner derivations of UT m(F) induced by the unit matrices e 1m and e mm respectively. We study the differential polynomial identities of the algebra UT m(F) under the coupled action of δ and ε. We produce a basis of the differential identities, then we determine the S n-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence

ON THE EXISTENCE OF THE GRADED EXPONENT FOR FINITE DIMENSIONAL MATHBBZPMATHBB{Z}_P-GRADED ALGEBRAS

Let F be an algebraically closed field of characteristic zero, and let A be an associative unitary F-algebra graded by a group of prime order. We prove that if A is finite dimensional then the graded exponent of A exists and is an integer

Graded Polynomial Identities of Triangular Algebras

Let F be any field, G a finite abelian group and let A, B be F-algebras graded by subgroups of G. If M is a G-graded free (A, B)-bimodule, we describe the G-graded polynomial identities of the triangular algebra of M and, in case the field F has characteristic zero, we provide the description of its G-graded cocharacters by means of the graded cocharacters of A and B

Comparing the Z_2-Graded Identities of Two Minimal Superalgebras with the Same Superexponent

Let F be a field of characteristic zero. We study two minimal superalgebras A and B having the same superexponent but such that T2(A) â«\u8b T2(B), thus providing the first example of a minimal superalgebra generating a non minimal supervariety. We compare the structures and codimension sequences of A and B

mathbbZ2mathbb{Z}_2-graded cocharacters for superalgebras of triangular matrices

Let K be a field of characteristic zero, let A, B be K-algebras with polynomial identities and let M be a free (A; B)-bimodule. The algebra R of 2x2 upper triangular matrices, having the elements of the algebras A and B on the main diagonal and the elements of the free module M on the (1,2) position can be endowed with a natural Z2-grading. In this paper, we compute the graded cocharacter sequence, the graded codimension sequence and the superexponent of R. As a consequence of these results, we also study the above PI-invariants in the setting of upper triangular matrices over the field K. In particular, we completely classify these invariants for the algebra of 3 × 3 upper triangular matrices endowed with all possible Z2-gradings

On a Regev-Seeman conjecture about Z_2-graded tensor products

In the Theory of Polynomial Identities of algebras, superalgebras play a key role, as emphasized by the celebrated Kemer’s results on the structure of T -ideals of the free associative algebra. Kemer succeeded in classifying the T –prime algebras over a field of characteristic zero, and all of them possess a natural superalgebra structure. In a celebrated work Regev proved that the tensor product of PI-algebras is again a PI-algebra, and the so-called Kemer’s Tensor Product Theorem shows that the tensor product of T -prime algebras is again PI-equivalent to a T -prime algebra, explicitly described. When dealing with superalgebras, however, it is possible to define an alternative tensor product, sometimes called super, or graded, or signed tensor product. In a recent paper Regev and Seeman studied graded tensor products, and they proved that the graded tensor product of PI-algebras is again PI, as for the ordinary case. Then natural questions arise: is the graded tensor product of T -prime algebras again T -prime? If so, do the graded and ordinary tensor products of T -prime algebras give the “same results” up to PI-equivalence? Among their results, Regev and Seeman noticed cases for which a graded version of Kemer’s Tensor Product Theorem does hold. More precisely, the re- sulting algebra is still a T -prime algebra, possibly “different” from the “natural” one. Then they conjectured this should be true in general. The present paper positively solves the conjecture. More precisely, we can prove that, in zero characteristic, the graded tensor product of T -prime algebras “is” again T -prime, and we describe the resulting algebra up to PI-equivalence