A finite commutative ring involution is the multiplicative inverse of the element attribute R is the element itself. This classical characteristic of a finite commutative ring makes Neutrosophic ...involutions possible, which are counted, listed and assessed in this work. Assume that the Neutrosophic ring R(I) is the finite commutative ring with unity 1 over the ring R under the indeterminate 𝑰 . We first establish some useful necessary and sufficient conditions for the Neutrosophic components of the type 𝒂 + 𝒃𝑰 is involutory in order to understand how to count Neutrosophic involutions of R(I). The behavior of the Neutrosophic composition table for identifying Neutrosophic involutions and counting the number of 1s that appear on the primary diagonal of the composition table of R(I) is also investigated in this work.
Let
K
be a field of characteristic different from 2 and let
G
be a group. If the algebra
U
T
n
of
n
×
n
upper triangular matrices over
K
is endowed with a
G
-grading
Γ
:
U
T
n
=
⊕
g
∈
G
A
g
, we give ...necessary and sufficient conditions on
Γ
that guarantees the existence of a homogeneous antiautomorphism on
A
, i.e., an antiautomorphism
φ
satisfying
φ
(
A
g
)
=
A
θ
(
g
)
for some permutation
θ
of the support of the grading. It turns out that
U
T
n
admits a homogeneous antiautomorphism if and only if the reflection involution of
U
T
n
is homogeneous. Moreover, we prove that if one homogeneous antiautomorphism of
U
T
n
is defined by the map
θ
, then any other homogeneous antiautomorphism is defined by the same map
θ
.
We define the notion of metabolicity index of involutions of the first kind in characteristic two. It is shown that this index is preserved over odd degree extensions of the base field. Also, its ...behavior over finite separable extensions is studied. As an application, it is shown that an orthogonal involution on a central simple algebra of degree a power of two which is either anisotropic or metabolic is totally decomposable if it is totally decomposable over some separable extension of the ground field. This result is then used to strengthen an earlier result of the author which proves a characteristic two counterpart of a conjecture concerning Pfister involutions formulated by Bayer-Fluckiger et al.
The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac–Moody algebras. The resulting quantum ...symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac–Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.
We study the construction of complex minimal smooth surfaces S of general type with pg(S)=0 and KS2=7. Inoue constructed the first examples of such surfaces, which can be described as Galois ...Z/2Z×Z/2Z-covers over the four-nodal cubic surface. Later the first named author constructed more examples as Galois Z/2Z×Z/2Z-covers over certain six-nodal del Pezzo surfaces of degree one.
In this paper we construct a two-dimensional family of minimal smooth surfaces of general type with pg=0 and K2=7, as Galois Z/2Z×Z/2Z-covers of certain rational surfaces with Picard number three, with eight nodes and with two elliptic fibrations. This family is different from the previous ones.
A matrix A is symplectic if
where
A symplectic matrix A is a commutator of symplectic involutions if
where X and Y are symplectic and
In this article, we prove that every complex symplectic matrix of ...size greater than 2 can be decomposed into a product of at most three commutators of involutions.
Communicated by Miriam Cohen
Let UT2n be the algebra of all 2n×2n upper triangular matrices over a field F whose characteristic is different from 2. In the present work, we will consider UT2n equipped with an involution of the ...first kind. In this setting, for certain subalgebras of UT2n (including UT2n itself), we will describe all the ⁎-polynomial identities which are of the form f±f⁎ where f is a product of commutators. In addition, we will exhibit sets of linear generators for the relatively free algebras induced from these types of identities.
Let
be the
identity matrix and
. A matrix A is called symplectic if
. A symplectic matrix A is a commutator of symplectic involutions if
, where X and Y are symplectic matrices satisfying
. In this ...article, we give necessary and sufficient condition for a symplectic matrix over the complex number field to be expressed as a product of two commutators of symplectic involutions.
In this paper, we introduce the left multiplication operators on the Riordan group, and carry out a systematic study on them. By using these operators, we obtain a family H of infinite Riordan ...subgroups. This family includes not only some well-known subgroups, but also the subgroup families introduced recently in literature. Fundamental properties of the left multiplication operators and the subgroup family H are presented. The Riordan involutions and pseudo-involutions, the stabilizer subgroups in family H, and the n-th commutator subgroup of any subgroup in family H are studied. Some further applications of the left multiplication operators are also provided. In particular, we study a class of left multiplication operators which are closely related to the diagonal translation operator and the m-complementary operator, and enrich the theory of complementary Riordan arrays. Moreover, we introduce and study the higher-level lifts of Riordan arrays as well as the vertical and horizontal 1/m Riordan arrays. As a result, it can be found that the left multiplication operators provide a unified approach to numerous subgroups, operators and transformations of the Riordan group.
Let F be an algebraically closed field of characteristic zero and G a cyclic group of odd prime order. We consider the class of finite dimensional (G,⁎)-algebras, namely G-graded algebras endowed ...with graded involution ⁎, and we characterize the varieties generated by algebras of this class which are minimal with respect to the (G,⁎)-exponent.
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