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     

DOI-KOPPINEN HOPF BIMODULES ARE MODULES

I construct a generalized twisted smash product A ^H B, which gives an abstract structure of Cibils-Rosso's algebra X associated to a finite-dimensional Hopf algebra H, for the H-bimodule algebra A and H-bicomodule algebra B. I show that the Doi-Koppinen Hopf (H, B, D)-bimodules are modules over a certain algebra which is of this type. Moreover, if D is finitely generated projective as a k-module, there exists a k-module-preserving equivalence of categories between the category of Doi-Koppinen (H, B, D)-Hopf bimodules and the category of left (D ^*op ? D ^*) ^{H?H op} (B ? B ^op )-modules.     

The Ring of Modules with Endo-permutation Source

In this paper we consider certain subalgebras of the Green algebra (representation algebra) of a finite group G. One such algebra is spanned by the isomorphism classes of all indecomposable modules whose source is an endo-permutation module. This algebra can be embedded into a finite direct product of Laurent polynomial rings in finitely many variables over a field. Another such algebra is spanned by the isomorphism classes of all irreducibly generated modules. When G is p-solvable then this algebra is finite-dimensional and split semisimple.R. Boltje was supported by the NSF, DMS-0200592 and 0128969. B. Külshammer was supported by the DAAD.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

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     

Action on flag varieties: 2-Dimensional case

LetA be a finite dimensional commutative semisimple algebra over a fieldk and letV be a finitely generatedA-module. We examine the action of the general linear group GL _A (V) on the set of flags ofk-subspaces ofV. Also, let (V, B) be a finitely generated symplectic module overA. We also investigate the action of the symplectic group Sp _A (V, B) on the set of flags ofB-isotropick-subspaces ofV, whereB=°B is thek-symplectic form induced by a nonzerok-linear map :A k. In both cases, the orbits are completely classified in terms of certain integer invariants provided that dim _{k A}=2.This work is partially supported by a KOSEF research grant.     

Schur Subalgebras

Let k be a field of prime characteristic and let r be a positive integer. In this paper, we study the Schur algebra S(2, r) over k and consider certain natural subalgebras.     

Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions

Let k denote an algebraically closed field of arbitrary characteristic. Let C denote the set of all commutative, finite dimensional, local k-algebras of the form (B, m, k) with i(m) ?2. Here i(m) denotes the index of nilpotency of the maximal ideal m. A Akalgebra (R, J,k)∈L is called a (c₁-construction if there exists (B, m, k)∈ £ ? {(k, (0), k)} and a finitely generated, faithful B-module N such that R,?B?(the idealization of N). (R.J.k) is called a (c_2::-construction if there exist a (B,m k)∈ L, a positive integer p $ge;2 and a nonzero z £ S_B(the socle of B) such that R?B[x]/(mX, X^p- z). Let M_n×n(K) denote the set of all n x n matrices, over k with n≥2. Let .M_n(k) denote the set of all maximal, commutative A;-subalgebras of M_n×n(k). In this paper, we show any (R J, k) ∈£?M_n;(k) with n>5 is a C₁ or C₂ -construction except for one isomorphism class. The one exception occurs when n = 5.     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].     

Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors ? ? _k V and V ? _k  ? are known to be left adjoint to some kind of Hom-functors as endofunctors of _H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case.

In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors ? ?V and ? ?V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors ? ?V* and Hom _k (V, ?) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.     

Serre Theorem for involutory Hopf algebras

We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel’d modules over H the dimension of which are invertible in k are Serre categories.     

On some types of maximal abelian subalgebras

Let H be a Hilbert space and B(H) the algebra of all bounded linear operators on H. It is known that there are two kinds of maximal abelian sub-algebras in B(H), to one of which there exists a unique faithful normal projection of norm one from B(H) and to the other any projection of norm one is singular. Any maximal abelian subalgebra A contains a projection e such that Ae is a maximal abelian subalgebra of B(eH) of the first kind and A(1 − e) is the one of the second kind in B((1 − e)H). This will be generalized to an arbitrary von Neumann algebra together with the existence problem of those kinds of maximal abelian subalgebras.     

A module structure on cyclic cohomology of group graded algebras

Letk be a field of characteristic 0, and letB be an algebra overk which is graded by a discrete groupG. Let HC*(A) denote the cyclic cohomology of an algebraA overk. We prove that there is an HC*(kG)-module structure on HC*(B) which generalizes Connes' periodicity operator on HC*(B). This module structure also decomposes with respect to conjugacy classes and results in a natural generalization of the results of Burghelea and Nistor in the cases of group algebras and algebraic crossed product algebras, respectively. Moreover, the proofs given in this paper are purely analytic with explicit constructions which can be used in the calculation of the cyclic cohomology of topological twisted crossed product algebras.Research sponsored in part by NSF Grant DMS-9204005.     

Semilocal Group Algebras

Let k[G] be a semilocal group algebra. It is shown that if k is an algebraically closed field, then every finitely generated flat k[G]-module is projective.     

Structure theorems over polynomial rings

Given a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint, http://www.ma.umist.ac.uk/pas/preprints].     

On hyperfinite factors of type III0 and Krieger's factors

A factor M is the cross product of an abelian von Neumann algebra by a single automorphism iff there exists an increasing sequence of normal conditional expectations of M onto finite-dimensional subalgebras N_k with (U N_k)^? = M. Assuming the uniqueness of the hyperfinite factor of type II_∞, we prove then that any hyperfinite factor of type III₀ is the cross product of an abelian von Neumann algebra by a single automorphism.     

On the growth of algebras with bialgebra action

Ifk is a field,A ak-algebra, andB ak-bialgebra which acts onA, we study the rate of growth ofA under its algebra structure together with the action ofB. We then briefly place our results in the more general context of a vector spaceA on which anoperad acts, and sketch an application to a (still open) question on finitely generated subalgebras of free associative algebras. Dedicated to the memory of Shimshon Amitsur 1991Mathematics Subject Classifications: Primary: 11N45, 16P90, 16W30; secondary: 08B20, 08C15, 11P99, 13N99, 17A01. Work done while the author held NSF contract DMS 93-03379. An erratum to this article is available at .     

Free Akivis Algebras, Primitive Elements, and Hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved.The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed.Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.     

The Category of Firm Modules for Nonunital Monomial Algebras

Let k be a field and X a set and P be a set of words over X. Consider the free nonunital k-algebra over X generated by the nonempty words over X and let R be the quotient of this algebra modulo the ideal generated by the words in P. R is called a “nonunital monomial algebra”. A right R-module M is said to be “firm” if M? _R R → M given by m ? r? mr is an isomorphism. In this article we prove that if R is a nonunital monomial algebra, the category of firm modules is Grothendieck.