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     

An extension of Iwasawa's theorem on finitely generated modules over power series rings

This paper describes a structure theorem for finitely generated modules over power series rings O[[T]], where O is a maximal order in a semisimple Q_p-algebra of finite dimension over Q_p, extending Iwasawa's structure theorem (the case O=?p). A particular case of such power series ring is the ring Λ[Δ], where Λ is the power series ring ?_p?T? and Δ is a finite group of order prime to p. Several applications are given, including a new proof of a result of Iwasawa important for the relationship between Hecke characters and certain Galois representations for CM fields.     

Finite generation of Hochschild homology algebras

We prove converses of the Hochschild-Kostant-Rosenberg Theorem, in particular: If a commutative algebra S is flat and essentially of finite type over a noetherian ring , and the Hochschild homology HH _* (S|) is a finitely generated S-algebra for shuffle products, then S is smooth over . Oblatum 27-V-1999 & 27-IX-1999 / Published online: 24 January 2000     

Finite Generation of Units in Alternative Loop Rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z L of L over the ring of rational integers is finitely generated.     

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     

The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras

In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P ^<∞(Λ) if and only if P ^<∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]).     

Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra K[H _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H _p ]/J is a domain.     

The natural operators lifting 1-forms on manifolds to the bundles ofA-velocities

LetA be a Weil algebra withp variables. We prove that forn-manifolds (np+2) the set of all natural operatorsT ^*T ^* T ^A is a free finitely generated module over a ring canonically dependent onA. We construct explicitly the basis of this module.     

Algebras of singular integral operators generated by three orthogonal projections

The structure of theC ^*-algebra with identity generated by two orthogonal projections is well understood. All irreducible representations of this algebra are either two-dimensional or one-dimensional. The situation becomes unpredictable in the case of theC ^*-algebra generated by three orthogonal projections. Even in the more specific case when two of the projections commute, the algebra under consideration may have infinite dimensional irreducible representations.In this paper we produce three concrete realizations of the algebra generated by three orthogonal projections in that specific case. It turns out that these algebras have quite different structure.This work was partially supported by CONACYT Project 4069-E9404, México.     

Centrally large subgroups of finite p-groups

Let S be a finite p-group. We say that an abelian subgroup A of S is a large abelian subgroup of S if |A||A^*| for every abelian subgroup A^* of S. We say that a subgroup Q of S is a centrally large subgroup, or CL-subgroup, of S if |Q||Z(Q)||Q^*||Z(Q^*)| for every subgroup Q^* of S. The study of large abelian subgroups and variations on them began in 1964 with Thompson's second normal p-complement theorem [J.G. Thompson, Normal p-complements for finite groups, J. Algebra 1 (1964) 43–46]. Centrally large subgroups possess some similar properties. In 1989, A. Chermak and A. Delgado [A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989) 907–914] studied several families of subgroups that include centrally large subgroups as a special case. In this paper, we extend their work to prove some further properties of centrally large subgroups. The proof uses an analogue for finite p-groups of an application of Borel's Fixed Point Theorem for algebraic groups.     

True BRST symmetry algebra and the theory of its representations

A simple treatment of BRST symmetry is proposed. From the physical point of view, it expresses a symmetry between ghosts and spurions; from the mathematical point of view, the symmetry operations are linear transformations in the superspaceC _1,1. From this it follows that the true BRST symmetry algebra isl(1, 1), the Lie superalgebra of all linear endomorphisms ofC _1,1, which extends the usual BRST algebra of the generatorsQ andQ _c with two new generatorsK=Q ^* andR={Q,Q ^*}. The theory of the representations ofl(1, 1) is developed systematically. The sets of automorphisms and involutions ofl(1, 1) are described. Decompositions into irreducible and indecomposable components are constructed for large classes of representations, both finite-and infinite-dimensional. Particular attention is devoted to the analysis of the indecomposable representations (in particular, a connection between them and subspaces of the continuous spectrum of the generators is found) and also of the metric properties of the indefinite spaces of the representations. A class of physical representations is identified and described in detail.V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 3–16, April, 1992.     

On the Kronecker pairing for mixed cusp forms

LetG⊃PSL(2,R) be a Fuchsian group of the first kind with no elements of finite order, and letS ^2m V be the 2m-fold symmetric power of the standard representationV ofSL(2,R) on C². We determine the value of the Kronecker pairing between the canonical image of a mixed cusp formf of type (2,2m) inH ¹(G, S ^2m V) and a cycleg ⊗Q _g ^m inH ₁ (G, (S ^2m V)^*) for eachg inG, whereQ _g ^m is an element of (S ^2m V)^* associated tog, m and a monodromy representation ofG.     

A problem about the weak Hilbert property

In this paper, we prove the main result: Let both (K, S) and (K ^*,S ^*) be preordered fields, and let (K ^*,S ^*) be a finitely generated extension of (K, S). IfK ^* is transcendental overK, then (K ^*,S ^*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover, some results on the weak Hilbert property are established. Project supported by National Natural Science Foundation of China     

Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

Combinatorial aspects of free associative algebras and cohomologies of Lie <Emphasis Type="Italic">p</Emphasis>-algebras with one defining relation

Let R and Q be elements of a free, associative, finitely generated algebra A‹X› over a field k. Assume that a leading homogeneous part Q _v does not have two-sided divisors and RQ = QR and R _v = Q ^t _v . In this paper, the solutions of the equation Σ _i x _i Ry _i = z in A‹X› are found and with their help, the identity theorem and Freiheitssatz for a finitely generated associative algebra k‹X; R = 0›with one defining relation are proved. As a consequence, similar theorems for Lie p-algebras with one defining relation are proved; these results are applied to the proof of the periodicity of cohomologies of Lie p-algebras with one defining relation. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

A fully invariant ideal of alternative algebras

Let ϕ be an associative commutative ring with 1, containing 1/6, and A be an alternative ϕ-algebra. Let D be an associator ideal of A and H a fully invariant ideal of A, generated by all elements of the form h(y, z, t, x, x)=[{[y, z], t, x}-, x]+[{[y, x], z, x}-, t], where [x, y]=xy−yx, {x, y, z}-=[[x, y], z]−[[x, z], y]+2[x,[y, z]]. Here we consider an ideal Q=H∩D and prove that Q⁴=0 in the algebra A. If A is unmixed, then HD=0, DH=0, and Q²=0 in particular. If A is a finitely generated unmixed algebra, then the ideal H lies in its associative center and Q=0. It follows that any finitely generated purely alternative algebra satisfies the identity h(y,z,t,x,x)=0. We also show that a fully invariant ideal H₀ of the unmixed algebra A, generated by all elements of the form h(x, z, t, x, x), lies in its associative center and H₀∩D=0. Consequently, every purely alternative algebra satisfies the identity h(x,z,t,x,x)=0. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 323–340, May–June, 1997.     

Equimorphy in varieties of distributive double p-algebras

Any finitely generated regular variety V of distributive double p-algebras is finitely determined, meaning that for some finite cardinal n(V), any subclass S V of algebras with isomorphic endomorphism monoids has fewer than n(V) pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double p-algebras must be almost regular.