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.
We study the functional identity G(x)f(x)=H(x) on a division ring D, where f:D→D is an additive map and G(X)≠0,H(X) are generalized polynomials in the variable X with coefficients in D. Precisely, it ...is proved that either D is finite-dimensional over its center or f is an elementary operator. Applying the result and its consequences, we prove that if D is a noncommutative division ring of characteristic not 2, then the only solution of additive maps f,g on D satisfying the identity f(x)=xng(x−1) with n≠2 a positive integer is the trivial case, that is, f=0 and g=0. This extends Catalano and Merchán's result in 2023 to get a complete solution.
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 cnL(A), n≥1, be its differential codimension sequence. Such sequence is exponentially bounded and expL(A)=limn→∞cnL(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, expL(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.
We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3.
Let A be an algebra and let f be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between A,A, the linear span of commutators in A, and spanf(A), ...the linear span of the image of f in A. In particular, we show that A,A=A implies spanf(A)=A. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if C is a commutative unital algebra over a field F of characteristic 0, A is the matrix algebra Mn(C), and the polynomial f is neither an identity nor a central polynomial of Mn(F), then every commutator in A can be written as a difference of two elements, each of which is a sum of 7788 elements from f(A) (if C=F is an algebraically closed field, then 4 elements suffice). Similar results are obtained for some other algebras, in particular for the algebra B(H) of all bounded linear operators on a Hilbert space H.
Let F be an infinite field of characteristic
p
≠
2
, G be a group, and
*
be an involution of G extended linearly to an involution of the group algebra FG. In the literature, group identities on units
...U
(
FG
)
and on symmetric units
U
+
(
FG
)
=
{
α
∈
U
(
FG
)
|
α
*
=
α
}
have been considered. Here, we investigate normalized Laurent polynomial identities (as a generalization of group identities) on
U
+
(
FG
)
under the conditions that either p > 2 or F is algebraically closed. For instance, we show that if G is torsion and
U
+
(
FG
)
satisfies a normalized Laurent polynomial identity, then
U
+
(
FG
)
satisfies a group identity and FG satisfies a polynomial identity.
We study Jordan 3-graded Lie algebras satisfying 3-graded polynomial identities. Taking advantage of the Tits-Kantor-Koecher construction, we interpret the PI-condition in terms of their associated ...Jordan pairs, which allows us to formulate an analogue of Posner-Rowen Theorem for strongly prime PI Jordan 3-graded Lie algebras. Arbitrary PI Jordan 3-graded Lie algebras are also described by introducing the Kostrikin radical of the Lie algebras.
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.
In this article the smash product algebras as in the title due to V. V. Bavula and T. Lu have been studied at the root of unity. A full classification of simple modules over
along with a class of ...finite dimensional indecomposable modules are presented. Also simple X, Y-torsionfree modules over
are classified.
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