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We develop the bialgebra theory for two classes of non-associative algebras: nearly associative algebras and $LR$-algebras. In particular, building on recent studies that reveal connections between these algebraic structures, we establish that nearly associative bialgebras and $LR$-bialgebras are, in fact, equivalent concepts. We also provide a characterization of these bialgebra classes based on the coproduct. Moreover, since the development of nearly associative bialgebras - and by extension, $LR$-bialgebras - requires the framework of nearly associative $L$-algebras, we introduce this class of non-associative algebras and explore their fundamental properties. Furthermore, we identify and characterize a special class of nearly associative bialgebras, the coboundary nearly associative bialgebras, which provides a natural framework for studying the Yang-Baxter equation (YBE) within this context., Comment: 17 pages

We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$.

Let $A$ be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra $L$ by derivations, and let $c_n^L(A),$ $n\geq 1,$ be its differential codimension sequence. Such sequence is exponentially bounded and $\exp^L(A) = \lim_{n\to \infty}\sqrt[n]{c_n^L(A)}$ is an integer that can be computed, called differential PI-exponent of $A$. In this paper we prove that for any Lie algebra $L$, $\exp^L(A)$ coincides with $\exp(A)$, the ordinary PI-exponent of $A$. Furthermore, in case $L$ is a solvable Lie algebra, we apply such result to classify varieties of $L$-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth., Comment: 11 pages

In this paper we study the growth of the differential identities of some algebras with derivations, i.e., associative algebras where a Lie algebra $L$ (and its universal enveloping algebra $U(L)$) acts on them by derivations. In particular, we study in detail the differential identities and the cocharacter sequences of some algebras whose sequence of differential codimensions has polynomial growth. Moreover, we shall give a complete description of the differential identities of the algebra $UT_2$ of $2\times 2$ upper triangular matrices endowed with all possible action of a Lie algebra by derivations. Finally, we present the structure of the differential identities of the infinite dimensional Grassmann $G$ with respect to the action of a finite dimensional Lie algebra $L$ of inner derivations., Comment: Proceedings of workshop "Polynopmial identities in algebras", Springer INdAM Series

Let $A$ be an associative algebra over a field $F$ of characteristic zero and let $L$ be a Lie algebra over $F$. If $L$ acts on $A$ by derivations, then such an action determines an action of its universal enveloping algebra $U(L)$ and in this case we refer to $A$ as algebra with derivations or $L$-algebra. Here we give a characterization of the ideal of differential identities of finite dimensional $L$-algebras $A$ in case the corresponding sequence of differential codimensions $c_n^L (A)$, $n\geq 1$, is polynomially bounded. As a consequence, we also characterize $L$-algebras with multiplicities of the differential cocharacter bounded by a constant.