We investigate the ε-Lie structure of K and K,K; here K denotes the skew-symmetric elements of an (ε,G)-Lie color algebra (obtained from an associated algebra A) with an ε-involution. The ...relationship with the (associative) ideals of A is also explored.
Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division ...rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.
Let UT
m
be the algebra of all m × m upper triangular matrices over a field
whose characteristic is different from 2. Given an involution * of the first kind on UT
m
, we will obtain the minimal ...integer t such that the Lie polynomial z
1
, z
2
, ..., z
2t−1
, z
2t
is an identity for
. Afterwards, under a mild technical restriction on
, we will describe all multilinear Lie polynomials whose image evaluated on K(m, *) is the space formed by all strictly upper triangular matrices of K(m, *).
Let A be a non-commutative prime ring with involution ∗, of characteristic ≠2(and3), with Z as the center of A and Π a mapping Π:A→A such that Π(x),x∈Z for all (skew) symmetric elements x∈A. If Π is ...a non-zero CE-Jordan derivation of A, then A satisfies s4, the standard polynomial of degree 4. If Π is a non-zero CE-Jordan ∗-derivation of A, then A satisfies s4 or Π(y)=λ(y−y*) for all y∈A, and some λ∈C, the extended centroid of A. Furthermore, we give an example to demonstrate the importance of the restrictions put on the assumptions of our results.
In this paper we determine the Jordan algebras associated to ad-nilpotent elements of index at most 3 in Lie algebras R− and Skew(R,⁎) for semiprime rings R without or with involution ⁎. To do so we ...first characterize these ad-nilpotent elements.
In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings
R
free of 2-torsion. With the idea of keeping under control the torsion of
R
, we introduce a ...more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring
R
in order to describe its ad-nilpotent elements. If
R
is a semiprime ring and
a
∈
R
is a pure ad-nilpotent element of
R
of index
n
with
R
free of
t
and
n
t
-torsion for
t
=
n
+
1
2
, then
n
is odd and there exists
λ
∈
C
(
R
)
such that
a
-
λ
is nilpotent of index
t
. If
R
is a semiprime ring with involution
∗
and
a
is a pure ad-nilpotent element of
Skew
(
R
,
∗
)
free of
t
and
n
t
-torsion for
t
=
n
+
1
2
, then either
a
is an ad-nilpotent element of
R
of the same index
n
(this may occur if
n
≡
1
,
3
(
mod
4
)
) or
R
is a nilpotent element of
R
of index
t
+
1
, and
R
satisfies a nontrivial GPI (this may occur if
n
≡
0
,
3
(
mod
4
)
). The case
n
≡
2
(
mod
4
)
is not possible.
Let
RG
be the group ring of a finite group
G
over a commutative ring
R
with 1. An element
x
in
RG
is said to be skew-symmetric with respect to an involution
σ
of
RG
if
σ
(
x
)
=
-
x
.
A structure ...theorem for the Lie algebra of skew-symmetric elements of
FG
is given where
F
is an algebraic extension of
Q
which generalizes some previously known results in this direction.
We study the Lie nilpotency and Lie n-Engel properties of symmetric elements under oriented involutions. In some cases, the results obtained are similar to the ones in the case of the classical ...involution, but interesting new situations arise.
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