Projective bases of division algebras and groups of central type

Letk be a field. For each finite groupG and two-cocylef inZ ² (G, k ^x ) (with trivial action), one can form the twisted group algebra wherex _σ x _τ =f(σ,τ)x _στ for all σ, τ∃G. Our main result is a short list ofp-groups containing all thep-groupsG for which there is a fieldk and a cocycle such that the resulting twisted group algebra is ak-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Haile) that everyk-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

On the structure theory of the Iwasawa algebra of a p-adic Lie group

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, Λ of a p-adic analytic group G. For G without any p-torsion element we prove that Λ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-nullΛ-module. This is classical when G=ℤ ^k _p for some integer k≥1, but was previously unknown in the non-commutative case. Then the category of Λ-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the ℤ _p -torsion part of a finitely generated Λ-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere. Received May 12, 2001 / final version received July 5, 2001?Published online September 3, 2001     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Group algebras

Given a group G and a commutative ring k with identity, one can define an k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X]. We will discuss some of the developments in the study of algebraic elements in group algebras.     

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

Finite <Emphasis Type="Italic">p</Emphasis>-central groups of height <Emphasis Type="Italic">k</Emphasis>

A finite group G is called p ⁱ -central of height k if every element of order p ⁱ of G is contained in the k ^th -term ζ _k (G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω₁(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P ^p is less than or equal to the order of the subgroup Ω₁(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N _G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl _p (G) is p-central of height p − 2, then N _G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

Piecewise Hereditary Nakayama Algebras

Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver \mathbb A_n\mathbb A_n for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category D ^b (Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category H\mathcal H as triangulated category.     

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C ^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ^*-algebra C ^*(S,α), and we describe a setting which permits us to conclude that this C ^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-singular dichotomy for orbital measures of classical simple Lie algebras

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ ^k is singular to Lebesgue measure or μ ^k ∈ L ². The minimum k for which μ ^k ∈ L ² is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure. S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. K. E. Hare was supported in part by NSERC.     

Highest weight representations of a Lie algebra of Block type

For a field F of characteristic zero and an additive subgroup G of F, a Lie algebra B(G) of the Block type is defined with the basis {Lα,i, c|α∈G, -1≤i∈Z} and the relations [Lα,i,Lβ,j] = ((i 1)β- (j 1)α)Lα β,i j αδα,-βδi j,-2c,[c, Lα,i] = 0. Given a total order (?) on G compatible with its group structure, and anyα∈B(G)0*, a Verma B(G)-module M(A, (?)) is defined, and the irreducibility of M(A,(?)) is completely determined. Furthermore, it is proved that an irreducible highest weight B(Z )-module is quasifinite if and only if it is a proper quotient of a Verma module.     

Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)^* has the weak uniform A(G × H)^** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC ₂(G × H)^* is equal to the Fourier–Stieltjes algebra B(G × H).     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.     

Gradings and Derived Categories

Let G{{\mathcal G}} be a group, Λ a G{{\mathcal G}}-graded Artin algebra and gr(Λ) denote the category of finitely generated G{{\mathcal G}}-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and to study connections between the tilting theories of D^b(L){{\mathcal D}}^b(\Lambda) and D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)). In particular, using that if T is a gradable Λ-module, then a grading of T induces a G{{\mathcal G}}-grading on End_Λ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories D^b(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and D^b(gr(End_L(T))){{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T))).     

Structures and Representations of Generalized Path Algebras

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {e _ii } _I with each ē _ii in the center of if Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over a perfect field k (e.g., a field with characteristic zero or a finite field) if Λ is isomorphic to a generalized path algebra k (D, Ω, ρ) of finite directed graph with weak relations and dim < ∞; Λ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents if Λ is isomorphic to a path algebra with relations. Presented by Idun Reiten.     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

On <Emphasis Type="Italic">α</Emphasis>-complete retracts of specker lattice ordered groups

Let α be a cardinal. The notion of α-complete retract of a Boolean algebra has been studied by Dwinger. Specker lattice ordered groups were investigated by Conrad and Darnel. Assume that G is a Specker lattice ordered group generated by a Boolean algebra B(G). The notion of α-complete retract of G can be defined analogously as in the case of Boolean algebras. In the present paper we deal with the relations between α-complete retracts of G and α-complete retracts of B(G).     

Variations on polynomial subgroup growth

A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ ^a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).     

On the sum of the reciprocals of cycle lengths in sparse graphs

For a graphG let ℒ(G)=Σ{1/k contains a cycle of lengthk}. Erdős and Hajnal [1] introduced the real functionf(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviouslyf(1)=0. We provef (k+1/k)≧(300k logk)⁻¹ for all sufficiently largek, showing that sparse graphs of large girth must contain many cycles of different lengths.