Twisted Group Algebras Research Articles

The Jacobson radical of group rings of locally finite groups

This paper is the final installment in a series of articles, started in 1974, which study the semiprimitivity problem for group algebras K [ G ] K[G] of locally finite groups. Here we achieve our goal of describing the Jacobson radical J K [ G ] {\mathcal {J}}K[{G}] in terms of the radicals J K [ A ] {\mathcal {J}}K[{A}] of the group algebras of the locally subnormal subgroups A A of G G . More precisely, we show that if char ⁡ K = p > 0 \operatorname {char} K=p>0 and if O p ( G ) = 1 \mathbb {O}_{p}(G)=1 , then the controller of J K [ G ] {\mathcal {J}}K[{G}] is the characteristic subgroup S p ( G ) \mathbb {S}^{p}(G) generated by the locally subnormal subgroups A A of G G with A = O p ′ ( A ) A=\mathbb {O}^{p’}(A) . In particular, we verify a conjecture proposed some twenty years ago and, in so doing, we essentially solve one half of the group ring semiprimitivity problem for arbitrary groups. The remaining half is the more difficult case of finitely generated groups. This article is effectively divided into two parts. The first part, namely the material in Sections 2–6, covers the group theoretic aspects of the proof and may be of independent interest. The second part, namely the work in Sections 7–12, contains the group ring and ring theoretic arguments and proves the main result. As usual, it is necessary for us to work in the more general context of twisted group algebras and crossed products. Furthermore, the proof ultimately depends upon results which use the Classification of the Finite Simple Groups.

The Representation Ring of the Twisted Quantum Double of a Finite Group

AbstractWe provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.

The Semiprimitivity Problem for Twisted Group Algebras of Locally Finite Groups

Let K[G] be the group algebra of a locally finite group G over a field K of characteristic p > 0. In this paper, we show that K[G] is semiprimitive if and only if G has no locally subnormal subgroup of order divisible by p. Thus we settle the semiprimitivity problem for such group algebras by verifying a conjecture which dates back to the mid 1970s. Of course, if G has a locally subnormal subgroup of order divisible by p, then it is easy to see that the Jacobson radical jK[G] is not zero. Thus, the real content of this problem is the converse statement. Our approach here builds upon a recent paper where we came tantalizingly close to a complete solution by showing that if G has no non-identity locally subnormal subgroup, then K[G] is semiprimitive. In addition, we use a two step process, suggested by certain earlier work on semiprimitivity, to complete the proof. The first step is to assume that all locally subnormal subgroups are central. Since this is easily seen to reduce to a twisted group algebra problem, our goal for this part is to show that Kt[G] is semiprimitive when G has no non-trivial locally subnormal subgroup. In other words, we duplicate the work of the previous paper, but in the context of twisted group algebras. As it turns out, almost all of the techniques of that paper carry over directly to this new situation. Indeed, there are only two serious technical problems to overcome. The second step in the process requires that we deal with certain extensions by finitary groups, and here we use recent results on primitive, finitary linear groups to show that the factor groups in question have well-behaved subnormal series. With this, we can apply previous machinery to handle the extension problem and thereby complete the proof of the main theorem.

A Character Correspondence for Twisted Group Algebras

A Character Correspondence for Twisted Group Algebras

Perfect Isometries and Isotypies for Blocks with Abelian Defect Groups and the Inertial Quotients Isomorphic toZ3×Z3

Letbbe ap-block of a finite groupGwith abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P,e)/P·CG(P)) is isomorphic toZ3×Z3andp≠2,7, then there is a perfect isometry from the group of generalized characters of some twisted group algebra of the semidirect product ofEandPonto the group of generalized characters ofGinb, and, furthermore,band its Brauer correspondent inNG(P) are isotypic.

Perfect Isometries and Isotypies for Blocks with Abelian Defect Groups and the Inertial Quotients Isomorphic toZ4×Z2

Letbbe ap-block of a finite groupGwith abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P,e)/P·CG(P)) is isomorphic toZ4×Z2andp≧7, then there is a perfect isometry from the group of generalized characters of some twisted group algebra of the semidirect product ofEandPonto the group of generalized characters ofGinb, and, furthermore,band its Brauer correspondent inNG(P) are isotypic.

Semiprimitivity of group algebras of locally finite groups. II

Let K be a field of characteristic p > 0, let G be a locally finite group, and let K[ G] denote the group algebra of G over K. In this paper we study the Jacobson radical JK[ G] when G has a finite subnormal series with factors which are either p′-groups, infinite simple, or generated by locally subnormal subgroups. For example, we show that if such a group G has no finite locally subnormal subgroup of order divisible by p, then JK[ G] = 0. The argument here is a mixture of group ring and group theoretic techniques and requires that we deal more generally with twisted group algebras. Furthermore, the proof ultimately depends upon certain consequences of the classification of the finite simple groups. In particular, we use J.I. Hall's classification of the locally finite finitary simple groups.

On the Projective Schur Group of a Field

If k is a field, the projective Schur group PS(k) of k is the subgroup of the Brauer group Br(k) consisting of those classes which contain a projective Schur algebra, i.e., a homomorphic image of a twisted group algebra kαG with G finite, α ∈ H2(G, k*). It has been conjectured by Nelis and Van Oystaeyen (J. Algebra137 (1991), 501-518) that PS(k) = Br(k) for all fields k. We disprove this conjecture by showing that PS(k) ≠ Br(k) for rational function fields k0(x) where k0 is any infinite field which is finitely generated over its prime field.

Derivations of Skew Polynomial Rings

Let F be a field of characteristic 0 and let λi,j ∈ F. for 1 ≤ i,j ≤ n. Define R = F[x1,x2,..., xn] to be the skew polynomial ring with xixj = λi,jxjxi and let S = F[x1,x2,...,xn, x−11, x−12,...,x−1n] be the corresponding Laurent polynomial ring. In a recent paper, Kirkman, Procesi, and Small considered these two rings under the assumption that S is simple and showed, for example, that the Lie ring of inner derivations of S is simple. Furthermore, when n = 2, they determined the automorphisms of S, related its ring of inner derivations to a certain Block algebra, and proved that every derivation of R is the sum of an inner derivation and a derivation which sends each xi to a scalar multiple of itself. In this paper, we extended these results to a more general situation. Specifically, we study twisted group algebras Ft[G] where G is a commutative group and F is a field of any characteristic. Furthermore, we consider certain subalgebras Ft[H] where H is a subsemigroup of G which generates G as a group. Finally, if e: G × G → F is a skew-symmetric bilinear form, then we study the Lie algebra Fe[G] associated with e, and we consider its relationship to the Lie structure defined on various twisted group algebras Ft[G].

Twisted Dual-Group Algebras: Equivariant Deformations of C0(G)

We consider a family of twisted Fourier algebras A(G, ω) of a locally compact group G, which in the case of a abelian group G are the Fourier transforms of the usual twisted group algebras of ($) over cap Gô. The corresponding C*-algebras C*(Gô, ω) are deformations of C0(G), which are equivariant in the sense that G still acts by left translation. The main examples come from cocycles σ on the dual of an abelian subgroup H of G; we prove that for such cocycles the twisted dual-group algebras C*(Gô, ω) are induced from the twisted group algebras C*(Hô, σ), and we give detailed formulas for the multiplication on A(G, ω) which extend to larger dense subalgebras of C0(G) and Cb(G). We anticipate that these larger subalgebras will be useful for constructing deformations of homogeneous spaces C0(G/Γ).

Twisted Group Rings Whose Units Form an FC-Group

AbstractLet U(KλG) be the group of units of the infinite twisted group algebra KλG over a field K. We describe the FC-centre ΔU of U(KλG) and give a characterization of the groups G and fields K for which U(KλG) = ΔU. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus theorem.

Projective Schur Division Algebras Are Abelian Crossed Products

Let k be a field. A projective Schur Algebra over k is a finite-dimensional k-central simple algebra which is a homomorphic image of a twisted group algebra k α G with G a finite group and α ∈ H 2( G, k*). The main result of this paper is that every projective Schur division algebra is an abelian crossed product ( K/ k, ƒ), where K is a radical extension of k.

Semigroups of isometries, Toeplitz algebras and twisted crossed products

We study theC *-algebras generated by projective isometric representations of semigroups, using a dilation theorem and the stucture theory of twisted crossed products. These algebras include the Toeplitz algebras of noncommutative tori recently studied by Ji, and similar algebras associated to the twisted group algebras of other groups such as the integer Heisenberg group.

Perfect Isometries for Blocks with Abelian Defect Groups and Klein Four Inertial Quotients

Let b be a p-block of a finite group G with an abelian defect group P and e a root of b in C G ( P). If the inertial quotient E (= N G ( P, e)/ C G ( P)) is a Klein four group, there is a so called perfect isometry from the group of generalized characters of a suitable twisted group algebra of the semidirect product of E and P onto the group of generalized characters of G in b.

On the Structure of Twisted Group C ∗ -Algebras

We first give general structural results for the twisted group algebras C*(G, σ) of a locally compact group G with large abelian subgroups. In particular, we use a theorem of Williams to realise C*(G, σ) as the sections of a C*-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when Γ is a discrete subgroup of a solvable Lie group G, the K-groups K * (C*(Γ, σ)) are isomorphic to certain twisted K-groups K*(G/Γ, δ(σ)) of the homogeneous space G/Γ, and we discuss how the twisting class δ(σ) ∈ H 3 (G/Γ, Z) depends on the cocycle σ

Descending chain conditions for graded rings

It is proved that if R R is a perfect (resp. Artinian) strongly graded ring whose ground subring is, modulo its Jacobson radical, a finite direct product of finite-dimensional simple algebras over (nondenumerable) algebraically closed fields, then the grading group cannot contain an infinite abelian subgroup (resp. must be finite). These results extend those of A. Reid and D. S. Passman on twisted group algebras.

On the structure of twisted group 𝐶*-algebras

We first give general structural results for the twisted group algebras C ∗ ( G , σ ) {C^{\ast } }(G,\sigma ) of a locally compact group G G with large abelian subgroups. In particular, we use a theorem of Williams to realise C ∗ ( G , σ ) {C^{\ast }}(G,\sigma ) as the sections of a C ∗ {C^{\ast }} -bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when Γ \Gamma is a discrete subgroup of a solvable Lie group G G , the K K -groups K ∗ ( C ∗ ( Γ , σ ) ) {K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma )) are isomorphic to certain twisted K K -groups K ∗ ( G / Γ , δ ( σ ) ) {K^{\ast } }(G/\Gamma ,\delta (\sigma )) of the homogeneous space G / Γ G/\Gamma , and we discuss how the twisting class δ ( σ ) ∈ H 3 ( G / Γ , Z ) \delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb {Z}) depends on the cocycle σ \sigma . For many particular groups, such as Z n {\mathbb {Z}^n} or the integer Heisenberg group, δ ( σ ) \delta (\sigma ) always vanishes, so that K ∗ ( C ∗ ( Γ , σ ) ) {K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma )) is independent of σ \sigma , but a detailed analysis of examples of the form Z n ⋊ Z {\mathbb {Z}^n} \rtimes \mathbb {Z} shows this is not in general the case.

Schur and Projective Schur Groups of Number Rings

The Schur or projective Schur group of a field consists of the classes of central simple algebras which occur in the decomposition of a group algebra or a twisted group algebra. For number fields, the projective Schur group has been determined in [8], whereas the Schur group is extensively studied in [25]. Recently, some authors have generalized these concepts to commutative rings. One then studies the classes of Azumaya algebras which are epimorphic images of a group ring or a twisted group ring. Though several properties of the Schur or projective Schur group defined in this way have been obtained, they remain rather obscure objects.

On the indices of simple components of twisted group algebras of finite groups over a P-adic number field

It is shown that if 2≢0 (mod P) the set of indices of simple components of the twisted group algebra of finite nilpotent groups of class 2 over the field of P-adic numbers consists of all natural numbers.

Une correspondance de modules pour les blocs à groupes de défaut abéliens

Let b be a block of a finite group G with an abelian defect group P and an inertial quotient E. Let us denote by L the semi-direct product of P and E. If E is cyclic and acts freely on P−{1}, we prove that the stable categories of \(\mathcal{O}\)Gb- and \(\mathcal{O}\)L-modules are equivalent, as a consequence of a more general result, without any hypothesis on E, about partial covering exomorphisms relating \(\mathcal{O}\)Gb with a suitable twisted group algebra \(\mathcal{O}\)*\(\mathcal{O}\)