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Let U T 2 be the algebra of 2 × 2 upper triangular matrices over a field F of characteristic zero. Here we study the generalized polynomial identities of U T 2 , i.e., identical relations holding for U T 2 regarded as U T 2 -algebra. We determine the generator of the T U T 2 -ideal of generalized polynomial identities of U T 2 and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in n variables in the language of Young diagrams through the representation theory of the symmetric group S n. Finally, we prove that, unlike the ordinary case, the generalized variety of U T 2 -algebras generated by U T 2 has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth. [ABSTRACT FROM AUTHOR]

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 ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n 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. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra with involution * over a field of characteristic zero. A central *-polynomial of A is a polynomial in non-commutative variables that takes central values in A. Here we prove the existence of two limits called the central *-exponent and the proper central *-exponent that give a measure of the growth of the central *-polynomials and proper central *-polynomials, respectively. Moreover, we compare them with the PI-*-exponent of the algebra. [ABSTRACT FROM AUTHOR]

Let A be an algebra with involution ⁎ over a field of characteristic zero. In this paper we characterize in two different ways when the multiplicities of the ⁎-cocharacter of A are bounded by a constant. As a consequence, we characterize the algebras with involution of bounded colength. [ABSTRACT FROM AUTHOR]

We present a comprehensive study of nearly associative algebras. Our research shows that these algebras are power associative Lie-admissible for which the related Lie algebra is solvable, and are also Jordan-admissible. Furthermore, we describe the features of finite-dimensional nearly associative algebras that are nilpotent. In addition, we define the concepts of radical and semisimplicity for this type of algebras. We give a characterization of semisimple nearly associative algebras and prove the Wedderburn Principal Theorem for them. Lastly, we deal with quadratic nearly associative algebras. We provide a characterization and then an inductive description of them through the process of double extension, enriching our understanding of the structural intricacies of this class of algebras. [ABSTRACT FROM AUTHOR]

Abstract We study the differential identities of the algebra U T 2 of 2 × 2 upper triangular matrices over a field of characteristic zero. We let the Lie algebra L = Der (U T 2) of derivations of U T 2 (and its universal enveloping algebra) act on it. We study the space of multilinear differential identities in n variables as a module for the symmetric group S n and we determine the decomposition of the corresponding character into irreducibles. If V is the variety of differential algebras generated by U T 2 , we prove that unlike the other cases (ordinary identities, group graded identities) V does not have almost polynomial growth. Nevertheless we exhibit a subvariety U of V having almost polynomial growth. [ABSTRACT FROM AUTHOR]

Let G be a finite abelian group and let A be an associative G -graded algebra over a field of characteristic zero. A central G -polynomial is a polynomial of the free associative G -graded algebra that takes central values for all graded substitutions of homogeneous elements of A. We prove the existence and the integrability of two limits called the central G -exponent and the proper central G -exponent that give a quantitative measure of the growth of the central G -polynomials and the proper central G -polynomials, respectively. Moreover, we compare them with the G -exponent of the algebra. [ABSTRACT FROM AUTHOR]