Our goal of the paper is to investigate the Waring problem for upper triangular matrix algebras, which gives a complete solution of a conjecture proposed by Panja and Prasad in 2023.

Let F be a field of characteristic zero, and let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft’s algebras $H_m$ on $UT_2$ and its $H_m$ -identities. In this paper, we give a complete description of the space of multilinear $H_m$ -identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of $H_m$ -module algebras generated by $UT_2$ has the Specht property, i.e., every $T^{H_m}$ -ideal containing the $H_m$ -identities of $UT_2$ is finitely based.

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.

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$-grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$, $b\in \mathbb {N}$, we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$, as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$-algebras which are not PI equivalent. Actually, we prove that the $T_{G}$-ideal of the former algebra is contained in the $T$-ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

We study an associative algebra $A$ over an arbitrary field that is a sum of two subalgebras $B$ and $C$ (i.e., $A\,=\,B+C$ ). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally, we generalize this result for semiprime algebras $A$ . Consider the class of all semisimple finite dimensional algebras $A\,=\,B+C$ for some subalgebras $B$ and $C$ that satisfy given polynomial identities $f\,=\,0$ and $g\,=\,0$ , respectively. We prove that all algebras in this class satisfy a common polynomial identity.

Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ that are minimal of polynomial growth (i.e., their sequence of codimensions grows like ${{n}^{k}}$, but any proper subvariety grows like ${{n}^{t}}$ with $t\,<\,k$). These varieties are the building blocks of general varieties of polynomial growth.

It turns out that for $k\,\le \,4$ there are only a finite number of varieties of polynomial growth ${{n}^{k}}$, but for each $k\,>\,4$, the number of minimal varieties is at least $\left| F \right|$, the cardinality of the base field, and we give a recipe for their construction.

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.

Let $\mathbb{K}$ be a field of characteristic zero, and $*\,=\,t$ the transpose involution for the matrix algebra ${{M}_{2}}\left( \mathbb{K} \right)$. Let $\mathfrak{U}$ be a proper subvariety of the variety of algebras with involution generated by $\left( {{M}_{2}}\left( \mathbb{K} \right),\,* \right)$. We define two sequences of algebras with involution ${{R}_{p}},\,{{S}_{q}}$, where $p,\,q\,\in \,\mathbb{N}$. Then we show that ${{T}_{*}}\left( \mathfrak{U} \right)$ and ${{T}_{*}}\left( {{R}_{p}}\oplus \,{{S}_{q}} \right)$ are $*$-asymptotically equivalent for suitable $p,\,q$.