Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

Zero divisors and units with small supports in group algebras of torsion-free groups

Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field 𝔽, the group ring 𝔽[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in 𝔽[G] whose supports have size 3. For any field 𝔽 and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥10. If 𝔽 = 𝔽₂ is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields.     

On commutative group algebras of mixed groups

Suppose F is a perfect field of characteristic p 0 and G is an abelian group whose torsion subgroup Gt is p-primary. If Gt is totally projective of countable length, it is shown that G is a direct factor of the group of normalized units V(G) of the group algebra F(G) and that V(G)/G is a totally projective p-group. The proof of this result is based on a new characterization of the class of totally projective p-groups of countable length. Li addition, the same result regarding V(G) is obtained if G has countable torsion-free rank and Gt is totally projective of length less than ω₁ + ω₀ . Finally, these results are applied to the question of whether the existence of an F-i pomorphism F(G) ? F(H) for some group H implies that G?H.     

Invariants for group algebras of Abelian groups

LetF be a field of characteristicp>0 and letG be an arbitrary abelian group written multiplicatively withp-basis subgroup denoted byB. The first main result of the present paper is thatB is an isomorphism invariant of theF-group algebraFG. In particular, thep-local algebraically compact groupG can be retrieved fromFG. Moreover, for the lower basis subgroupB ₁ of thep-componentG _p it is shown thatG _p/B_l is determined byFG. Besides, ifH is (p-)high inG, thenG _p/H_p andH ^{p n}[p] for ℕ₀ are structure invariants forFG, andH[p] as a valued vector space is a structural invariant forN ₀ G, whereN _p is the simple field ofp-elements. Next, presume thatG isp-mixed with maximal divisible subgroupD. ThenD andF(G/D) are functional invariants forFG. The final major result is that the relative Ulm-Kaplansky-Mackeyp-invariants ofG with respect to the subgroupC are isomorphic invariants of the pair (FG, FC) ofF-algebras. These facts generalize and extend analogous in this aspect results due to May (1969), Berman-Mollov (1969) and Beers-Richman-Walker (1983). As a finish, some other invariants for commutative group algebras are obtained.     

Probability measures on [SIN] groups and some related ideals in group algebras

Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L ¹(G), of the form J _μ = [L¹(G) * (δ _e − μ)]⁻, where μ is a probability measure on G. Let J_d(G)={\cal J}_d(G)= {J_m;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , J_a(G)={J_m;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=J_d(G){\cal J}(G)={\cal J}_d(G) and that J_a(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J _μ in general locally compact second countable groups are also obtained.     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany     

On group algebras with metabelian unit groups

We characterize the group algebras of finite groups over a field of characteristic 2 with metabelian groups of units.Dedicated to Professor László Fuchs on his 70th birthdayResearch was supported by Hungarian National Fund for Scientific Research grant No. T014279     

Cartan invariants of group algebras of finite groups

We give a result on Cartan invariants of the group algebra of a finite group over an algebraically closed field , which implies that if the Loewy length (socle length) of the projective indecomposable -module corresponding to the trivial -module is four, then has characteristic 2. The proof is independent of the classification of finite simple groups.

     

Weighted group algebras on groups of polynomial growth

Let G be a compactly generated group of polynomial growth and a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L ¹ (G,). In particular, if the weight is sub-exponential, then the algebra L ¹ (G,) is symmetric. For these weights we develop a functional calculus on a total part of L ¹ (G,) and use it to prove the Wiener property. Mathematics Subject Classification (2000):43A20, 22D15, 22D12.Supported by the Austrian Science Foundation project FWF P-14485.Supported by the research grants MEN/CUL/98/007 and CUL/01/014.     

Action of some braid groups on hodge algebras

In this paper we show that the braid groups B _n and the symmetric automorphism groups H(n) of the free group F _n,n = 3,4 act in a non-linear way on an algebra with straightening law (ordinal Hodge algebra). We indicate various properties of these rings.     

Quasisimple classical groups and their complex group algebras

Let H be a finite quasisimple classical group, i.e., H is perfect and S:= H/Z(H) is a finite simple classical group. We prove that, excluding the open cases when S has a very exceptional Schur multiplier such as PSL₃(4) or PSU₄(3), H is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.     

Semi-invariants and weights of group algebras of finite groups

We study the semi-invariants and weights of a group algebra over a field of characteristic zero. Specifically, we show that certain basic results which hold when is a polycyclic-by-finite group with need not hold in the case of group algebras of finite groups. This turns out to be a purely group theoretic question about the existence of class preserving automorphisms.

     

On local automorphisms of group algebras of compact groups

We show that with few exceptions every local isometric automorphism of the group algebra of a compact metric group is an isometric automorphism.