Differential identities and almost polynomial growth. Star algebras and cocharacters.

The purpose of this thesis is to present some recent results on the polynomial identities of algebras with derivations and of algebras with involution. First, we study in detail the differential polynomial identities of the algebra of $2\times 2$ upper triangular matrices over a field of characteristic zero when two distinct Lie algebras of derivations act on it. We explicitly determine a basis of the corresponding differential identities, the sequence of codimensions and the sequence of cocharacters in both cases. Furthermore, we study the growth of differential identities in both cases. In particular we prove that when the Lie algebra $L$ of all derivations acts on $UT_2$, then the variety of differential algebras with $L$ action generated by $UT_2$ has no almost polynomial growth. Afterwards, we study of the differential identities of the infinite dimensional Grassmann algebra over a field $F$ of characteristic different from two with respect to the action of a finite dimensional Lie algebra $L$ of inner derivations. We explicitly determine a set of generators of the ideal of differential identities of $G$. Also in case $F$ is of characteristic zero, we study the space of multilinear differential identities in $n$ variables as a module for the symmetric group $S_n$ and we compute the decomposition of the corresponding character into irreducibles. Furthermore, we prove that unlike the ordinary case the variety of differential algebras with $L$ action generated by $G$ has no almost polynomial growth. Finally, we study and characterize the algebras with involution over a field $F$ of characteristic zero satisfying a polynomial identity such that the multiplicities in the corresponding $*$-cocharacter are bounded by a constant.