A Class of Nongraded Simple Lie Algebras

Abstract

Nongraded simple Lie algebras appear naturally in mathematical physics. In this paper, a new class of nongraded simple Lie algebras are presented based on the pairs (𝒜, 𝒟) consisting of a commutative associative unital algebra 𝒜 and a finite dimensional commutative derivation subalgebra 𝒟 such that 𝒜 is 𝒟-simple. The isomorphism classes of these nongraded Lie algebras are also determined and the structure space of these algebras is given explicitly.     

Commuting Graphs of Group Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.     

Unitary Units Satisfying a Group Identity

Let K be a nonabsolute field of characteristic p ≠ 2, G a locally finite group and KG its group algebra. Let ?: KG → KG denote the K-linear extension of an involution ? defined on G. In this article, we prove that if the subgroup 𝒰_?(KG), i.e., the ?-unitary units of KG, satisfies a group identity, then KG satisfies a polynomial identity. Moreover, in case the prime radical of KG is nilpotent, we characterize the groups G for which 𝒰_?(KG) satisfies a group identity.     

Minimal Idempotents of Twisted Group Algebras of Cyclic 2-Groups

For a field K of characteristic different from 2, we find the explicit form of the minimal idempotents of the twisted group algebra K_tg of a cyclic 2-group g over K.AMS Subject Classification (1991): primary 16S35, secondary 16U60.Partially supported by the Fund “NIMP’’ of Plovdiv University.     

On Constantive Simple and Order-Primal Algebras

A finite algebra is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.** Research supported by the Thailand Research Fund.     

Central Units in Metacyclic Integral Group Rings

In this article, we give a method to compute the rank of the subgroup of central units of ? G, for a finite metacyclic group, G, by means of ?-classes and ?-classes. Then we construct a multiplicatively independent set  ? (U(? C _{p, q} )) and by applying our results, we prove that  generates a subgroup of finite index.     

非交换群上一类代数的不可约表示

在一类无限维非交换Hopf代数上,借助其Hopf理想,构造出商Hopf代数,讨论了此商代数上的有限维不可约模,得出此非平凡不可约模的维数一定是2.     

Identities on Units of Algebraic Algebras

Let be an algebraic algebra over an infinite field K and let ( ) be its group of units. We prove a stronger version of Hartley's conjecture for , namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in ( ), then satisfies a polynomial identity (PI). We also show that if is non-commutative, then is a PI-ring, provided f = 0 is satisfied by the non-central units of . In particular, is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in ( ) if and only if the same identity is satisfied in . The last fact remains true for generalized Laurent polynomial identities, provided that is locally finite.     

弱可约极大三角代数的上同调

In this paper, we introduce the concept of weakly reducible maximal triangular algebras φwhich form a large class of maximal triangular algebras. Let B be a weakly closed algebra containing 5φ, we prove that the cohomology spaces H^n(φ, B) (n≥1) are trivial.     

极大三角代数的交换子和上同调

设M为闭极大三角代数S的σ-弱闭双边模且满足MS,证明了模M交换子C(A,M)= M．进而,如果MAlgLat S,得到H~n(S,AlgLat S)=H~n(S,M)(n≥1)；若dimH〧≤1,则H~n(S,M)=0．     

（n-3）-Filiform李代数的极大环面

求出了（n-3）-filiform李代数的极大环面,并证明了（n-3）-filiform李代数是可完备化的．     

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

On Central Elements in the Universal Enveloping Algebras of the Orthogonal Lie Algebras

We present an analogy of the famous formula that the square of the Pfaffian is equal to the determinant for an alternating matrix for the case where the entries are the generators of the orthogonal Lie algebras. This identity clarifies the relation between the two sets of central elements in the enveloping algebra of the orthogonal Lie algebras. We employ systematically the exterior calculus for the proof.     

On the Structure of the Units of Group Algebra of Dihedral Group

In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

On 2-Homogeneity of Monounary Algebras

Fraïssé introduced the notion of a k-set-homogeneous relational structure. In the present paper the following classes of monounary algebras are described: --the class of all algebras which are 2-set-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively, and --the class of all algebras which are 2-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively.     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.