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In this paper we study algebras acted on by a finite group $G$ and the corresponding $G$-identities. Let $M_2( \mathbb{C})$ be the $2\times 2$ matrix algebra over the field of complex numbers $ \mathbb{C}$ and let $sl_2( \mathbb{C})$ be the Lie algebra of traceless matrices in $M_2( \mathbb{C})$. Assume that $G$ is a finite group acting as a group of automorphisms on $M_2( \mathbb{C})$. These groups were described in the Nineteenth century, they consist of the finite subgroups of $PGL_2( \mathbb{C})$, which are, up to conjugacy, the cyclic groups $ \mathbb{Z}_n$, the dihedral groups $D_n$ (of order $2n$), the alternating groups $ A_4$ and $A_5$, and the symmetric group $S_4$. The $G$-identities for $M_2( \mathbb{C})$ were described by Berele. The finite groups acting on $sl_2( \mathbb{C})$ are the same as those acting on $M_2( \mathbb{C})$. The $G$-identities for the Lie algebra of the traceless $sl_2( \mathbb{C})$ were obtained by Mortari and by the second author. We study the weak $G$-identities of the pair $(M_2( \mathbb{C}), sl_2( \mathbb{C}))$, when $G$ is a finite group. Since every automorphism of the pair is an automorphism for $M_2( \mathbb{C})$, it follows from this that $G$ is one of the groups above. In this paper we obtain bases of the weak $G$-identities for the pair $(M_2( \mathbb{C}), sl_2( \mathbb{C}))$ when $G$ is a finite group acting as a group of automorphisms.

Let $D$ be a Noetherian infinite integral domain, denote by $M_2(D)$ and by $sl_2(D)$ the $2\times 2$ matrix algebra and the Lie algebra of the traceless matrices in $M_2(D)$, respectively. In this paper we study the natural grading by the cyclic group $\mathbb{Z}_2$ of order 2 on $M_2(D)$ and on $sl_2(D)$. We describe a finite basis of the graded polynomial identities for the pair $(M_2(D), sl_2(D))$. Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for $(M_2(D), sl_2(D))$, is finitely generated. The polynomial identities for $M_2(D)$ are known if $D$ is any field of characteristic different from 2. The identities for the Lie algebra $sl_2(D)$ are known when $D$ is an infinite field. The identities for the pair we consider were first described by Razmyslov when $D$ is a field of characteristic 0, and afterwards by the second author when $D$ is an infinite field. The graded identities for the pair $(M_2(D), gl_2(D))$ were also described, by Krasilnikov and the second author. In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets.

Let $A=B+C$ be an associative algebra graded by a group $G$, which is a sum of two homogeneous subalgebras $B$ and $C$. We prove that if $B$ is an ideal of $A$, and both $B$ and $C$ satisfy graded polynomial identities, then the same happens for the algebra $A$. We also introduce the notion of graded semi-identity for the algebra $A$ graded by a finite group and we give sufficient conditions on such semi-identities in order to obtain the existence of graded identities on $A$. We also provide an example where both subalgebras $B$ and $C$ satisfy graded identities while $A=B+C$ does not. Thus the theorem proved by K\c{e}pczyk in 2016 does not transfer to the case of group graded associative algebras. A variation of our example shows that a similar statement holds in the case of graded group Lie algebras. We note that there is no known analogue of K\c{e}pczyk's theorem for Lie algebras.

Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with respect to the usual bracket (commutator), over an infinite field $F$. In this paper, we give a positive answer to the Specht property for the ideal of the $\mathbb{Z}_n$-graded identities of $UT_n(F)^{(-)}$ with the canonical grading when the characteristic $p$ of $F$ is 0 or is larger than $n-1$. Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of $UT_n(F)^{(-)}$, is finitely based. Moreover we show that if $F$ is an infinite field of characteristic $p=2$ then the $\mathbb{Z}_3$-graded identities of $UT_3^{(-)}(F)$ do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of $UT_3^{(-)}(F)$, and which is not finitely generated as an ideal of graded identities.

In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UT_n. For positive integers q \leq n, we classify these images on UT_n endowed with a particular elementary Z_q-grading. As a consequence, we obtain the images of multilinear graded polynomials on UT_n with the natural Z_n-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras UT_2 and UT_3, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra UJ_2, and also for UJ_3 endowed with the natural elementary Z_3-grading.

Let $K$ be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring $K[t]$, respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$, and we prove that they do not admit any finite basis., Comment: arXiv admin note: text overlap with arXiv:2107.10903

Let $E$ be the Grassmann algebra of an infinite dimensional vector space $L$ over a field of characteristic zero. In this paper, we study the $\mathbb{Z}$-gradings on $E$ having the form $E=E_{(r_{1},r_{2}, r_{3})}^{(v_{1},v_{2}, v_{3})}$, in which each element of a basis of $L$ has $\mathbb{Z}$-degree $r_{1}$, $r_{2}$, or $r_{3}$. We provide a criterion for the support of this structure to coincide with a subgroup of the group $\mathbb{Z}$, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in [11]., Comment: 20 pages