Journal of Algebra and Its Applications

Cartan–Brauer–Hua Theorem is a well-known theorem which states that if $R$ is a subdivision ring of a division ring $D$ which is invariant under all elements of $D$ or $dR{d}^{-1}\subseteq R$ for all $d\in D\setminus \{0\}$, then either $R=D$ or $R$ is contained in the center of $D$. The invariance idea of this basic theorem is the main notion of this paper. We prove that if $D$ is a division ring with involution ${}^{\ast }$ and $M$ is a subspace of $D$ which is invariant under all symmetric elements of $D$, then either $M$ is contained in the center of $D$ or is a Lie ideal of $D$. Also, we show that if $T$ is a self-invariant subfield of a non-commutative division ring $D$ with a nontrivial automorphism, then $T$ contains at least one non-central proper subfield of $D$.

Let ($A$, $\alpha$) be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel’d double $D(A)=(A^{\mathrm{\text{op}}}\bowtie A^{\ast },\alpha \otimes {({\alpha }^{-1})}^{\ast })$ in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid’s bicrossproduct for Hopf algebras (see [S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995)]) and another one is to introduce the notion of dual pairs of Hom-Hopf algebras. Then we study the relation between the Drinfel’d double $D(A)$ and Heisenberg double $H(A)=A\#A^{\ast }$, generalizing the main result in [J. H. Lu, On the Drinfel’d double and the Heisenberg double of a Hopf algebra, Duke Math. J. 74 (1994) 763–776]. The examples given in the paper are especially, not obtained from the usual Hopf algebras.

We provide a class of Artin–Schelter regular algebras of global dimension 5 with four generators, which is obtained by parametrizing a finite-dimensional graded Lie algebra. Moreover, the algebras are piecewise-Koszul under a slight constraint on the parameters.

Let $(S,\alpha )$ be a partial Galois extension where $\alpha$ is a partial action of a finite group on a ring $S$ such that the associated ideals are generated by central idempotents. We determine the set of all Galois extensions in $(S,\alpha )$, and give an orthogonality criterion for nonzero elements in the Boolean semigroup generated by those central idempotents. These results lead to a structure theorem for $(S,\alpha )$.

Let $G$ be a group and $h,g\in G$. The 2-tuple $(h,g)$ is said to be an $n$-Engel pair, $n\ge 2$, if $h=[h{,}_{n}g]$, $g=[g{,}_{n}h]$ and $h\ne 1$. Let SL$(2,F)$ be the special linear group of degree $2$ over the field $F$. In this paper, we show that given any field $L$, there is a field extension $F$ of $L$ with $[F:L]\le 6$ such that SL$(2,F)$ has an $n$-Engel pair for some integer $n\ge 4$. We will also show that SL$(2,F)$ has a $5$-Engel pair if $F$ is a field of characteristic $p\equiv \pm 1mod5$.

A proper subgroup $H$ of a group $G$ is said to be: $\mathbb{P}$-subnormal in $G$ if there exists a chain of subgroups $H=H_0<H_1<\cdots <H_n=G$ such that $|H_i:H_{i-1}|$ is a prime for $i=1,\dots ,n$; $\mathbb{P}$-abnormal in $G$ if for every two subgroups $K\le L$ of $G$, where $H\le K$, $|L:K|$ is not a prime. In this paper we describe finite groups in which every non-identity subgroup is either $\mathbb{P}$-subnormal or $\mathbb{P}$-abnormal.

We consider properties and extensions of PBW deformations of Artin–Schelter regular algebras. PBW deformations in global dimension two are classified and the geometry associated to the homogenizations of these algebras is exploited to prove that all simple modules are one-dimensional in the non-PI case. It is shown that this property is maintained under tensor products and certain skew polynomial extensions.

In this note, we show that for the product of some smooth projective varieties (e.g. projective spaces) the Rouquier dimension and the Krull dimension are equal. This verifies a conjecture of Orlov in these examples.

In this paper, we introduce the smallest equivalence relation ${\tau }^{\ast }$ on a hypergroup $H$ such that the quotient $H/{\tau }^{\ast }$, the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relations is investigated and several results on the topic are presented.

Let $G$ be a finite group. There is a natural Galois correspondence between the permutation groups containing $G$ as a regular subgroup, and the Schur rings (S-rings) over $G$. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group $G$ is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

Let $(H,\alpha )$ be a monoidal Hom-Hopf algebra, $(A,\beta )$ a right $(H,\alpha )$-Hom-comodule algebra and $(C,\gamma )$ a right $(H,\alpha )$-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of $(A,\beta )$ in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral $\mathrm{{\rm Y}}:C\to \mathrm{\text{Hom}}(C,A)$ if and only if any representation of the pair $(H,A,C)$ is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of $(H,A,C)$-Doi Hom-Hopf modules a result of Doi on projectivity of every relative $(A,H)$-Hopf module as an $(A,\beta )$-module.

Let $(R,𝔪)$ be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and $I$ an ideal of $R$. Let $M$ be a nonzero finitely generated $R$-module and $i\ge 0$ be an integer. In this paper we show that, the $R$-module $H_{𝔪}^i(M)$ is nonzero and $I$-cofinite if and only if $Rad(I+0{:}_R{H}_{𝔪}^i(M))=𝔪$. Also, several applications of this result will be included.

We determine the skew fields of fractions of the enveloping algebra of the Lie superalgebra $𝔬𝔰𝔭(1,2)$ and of some significant subsuperalgebras of the Lie superalgebra $𝔬𝔰𝔭(1,4)$. We compare the kinds of skew fields arising from this “super” context with the Weyl skew fields in the classical Gelfand–Kirillov property.

The second author and his collaborators, Lauve and Rodriguez-Romo (see [A class of left quantum groups modeled after SL$q$($n$), J. Pure Appl. Algebra 208(3) (2007) 797–803; A left quantum group, J. Algebra 286 (2005) 154–160), have constructed left Hopf algebras which are not Hopf algebras modeled after ${\mathrm{\text{SL}}}_q(n)$. In particular, they constructed the left quantum group $\stackrel{̃}{{\mathrm{\text{SL}}}_q(2)}$, along with an epimorphism to the quantum group ${\mathrm{\text{SL}}}_q(2)$, the latter being a Hopf algebra. The current work began as a search for a left Hopf algebra (which is not a Hopf algebra) containing the quantum group $U_q({\mathrm{\text{sl}}}_2)$, the latter being a Hopf algebra. The natural choice was to look for the dual of $\stackrel{̃}{{\mathrm{\text{SL}}}_q(2)}$. We show that the Hopf dual of $\stackrel{̃}{{\mathrm{\text{SL}}}_q(2)}$ is equal to the Hopf dual of ${\mathrm{\text{SL}}}_q(2)$, which is of independent interest. Thus the Hopf dual of $\stackrel{̃}{{\mathrm{\text{SL}}}_q(2)}$ is a Hopf algebra. The original search of a left Hopf algebra which is not a Hopf algebra, larger than $U_q({\mathrm{\text{sl}}}_2)$, is still open.

Let $G$ be a finite solvable group, let Irr${}_m(G)$ be the set of all irreducible monomial characters of $G$ and let $p$ be a prime. We prove that if $p|\chi (1)$ for every nonlinear $\chi \in$$\mathrm{\text{Irr}}$${}_m(G)$, then $G$ has a normal $p$-complement, and if $p$ is relatively prime to $\chi (1)$ for every $\chi \in {\mathrm{\text{Irr}}}_m(G)$, then $G$ has a normal Sylow $p$-subgroup.

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of $L^2$ in terms of the Fourier transform.

Simple restricted modules are considered for the restricted contact Lie superalgebras of odd type over an algebraically closed field with characteristic $p>2$. In particular, a sufficient and necessary condition in terms of typical or atypical weights is given for the restricted Kac modules to be simple. Furthermore, the number of the simple restricted Kac modules is obtained.

In this paper, we introduce and study a new class of Cayley graphs.