Donald S. Passman: Abstracts

Group algebras whose units satisfy a group identity II

Let K[G] be the group algebra of a torsion group G over an infinite field K, and let U=U(G) denote its group of units. A recent paper of A. Giambruno, S. K. Sehgal, and A. Valenti proved that if U satisfies a group identity, then K[G] satisfies a polynomial identity, thereby confirming a conjecture of Brian Hartley. Here we add a footnote to their result by showing that the commutator subgroup G' of G must have bounded period. Indeed, this additional fact enables us to obtain necessary and sufficient conditions for U(G) to satisfy an identity. PDF

Construction of free subgroups in the group of units of modular group algebras

(with J. Z. Goncalves) - Let K[G] denote the group algebra of a p'-group G over a field K of characteristic p > 0 and let U = U(K[G]) be its group of units. If K[G] contains a bicyclic unit and if K is not algebraic over its prime subfield, then we prove that the free product P*P*P can be embedded in U, where P is the cyclic group of order p. In particular, U contains a noncyclic free group. PDF

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] of locally finite groups. Here we achieve our goal of describing the Jacobson radical JK[G] in terms of the radicals JK[A] of the group algebras of the locally subnormal subgroups A of G. More precisely, we show that if char K= p > 0 and if G has no normal p-subgroups, then the controller of JK[G] is the characteristic subgroup S(G) generated by the locally subnormal subgroups A of G having trivial p'-quotients. 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 manuscript 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. PDF

Semiprimitivity of group algebras: past results and recent progress

The group ring K[G] is an easily defined, rather attractive algebraic object. As the name implies, its study is a meeting place for two essentially different algebraic disciplines. Indeed, group ring results frequently require a blend of group theoretic and ring theoretic techniques. A natural, but surprisingly elusive, group ring problem concerns the semiprimitivity of K[G]. Specificially, we wish to find necessary and sufficient conditions on the group G for its group algebra to have Jacobson radical equal to zero. More generally, we wish to determine the structure of the ideal JK[G]. In the case of infinite groups, this problem has been studied with reasonable success during the past 45 years, and our goal here is to survey what is known. In particular, we describe some of the techniques used, discuss a number of the results which have been obtained, and mention several tantalizing conjectures. (Prepared for the 1996 Ring Theory Conference in Miskolc, Hungary.) PDF

Enveloping algebras of Lie color algebras: primeness versus graded-primeness

(with J. Bergen) - Let G be a finite abelian group and let L be a, possibly restricted, G-graded Lie color algebra. Then the enveloping algebra U(L) is also G-graded, and we consider the question of whether U(L) being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field K of characteristic not 2. Specifically, we show that if K contains the square root of -1 and if U(L) has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of U(L) whose square is the automorphism of the enveloping algebra associated with its Z_2-grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if U(L) is graded-prime and if most homogeneous components L_g of L are infinite dimensional over K, then U(L) is prime. Here we use Delta-methods to study the grading on the extended centroid C of U(L). In particular, if G is generated by the infinite support of L, then we prove that C is homogeneous. PDF

Group algebras with units satisfying a group identity II

(with Chia-Hsin Liu) - We classify group algebras of torsion groups over fields of characteristic p>0 with units satisfying a group identity. This is the last in a series of papers, and the new results here concern the case of finite fields. PDF

The semiprimitivity of group algebras

In this paper we briefly discuss some recent results on the semiprimitivity problem for group algebras. For the most part, we stress those topics which do not appear in the fairly complete Miskolc survey. In particular, we consider the controller of an ideal, examples associated with the controller subgroup of the Jacobson radical, and a new, but rather elementary, observation on Kaplansky's problem. PDF

Semi-Invariants and weights of group algebras of finite groups

(with P. Wauters) - We study the semi-invariants and weights of a group algebra K[G] over a field K of characteristic zero. Specifically, we show that certain basic results which hold when G is a polycyclic-by-finite group with no finite normal subgroups 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. PDF

Noetherian down-up algebras

(with Ellen Kirkman and Ian M. Musson) - Down-up algebras A= A(a,b,c) were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that b not 0 is equivalent to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3. PDF

Simple Lie algebras of Witt-type

Let K be a field, let A be an associative, commutative K-algebra and let D be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, the tensor product A o D = AD becomes a Lie algebra, and we obtain necessary and sufficient conditions here for this Lie algebra to be simple. With one minor exception in characteristic 2, simplicity occurs if and only if A is D-simple and B o D = BD acts faithfully as derivations on A, where B is the ring of D-constants in A. PDF

Simple Lie color algebras of Witt-type

Let K be a field and let e:G x G -> K be a bicharacter defined on the multiplicative group G. We suppose that A is a G-graded, associative K-algebra which is color commutative with respect to e. Furthermore, let D be a nonzero G-graded, K-vector space of color derivations of A and suppose that D is also color commutative with respect to the bicharacter e. Then, with a rather natural definition, the tensor product A o D = AD becomes a Lie color algebra, and we obtain necessary and sufficient conditions here for this Lie color algebra to be simple. With two minor exceptions when dim D = 1, simplicity occurs if and only if A is graded D-simple and B o D = BD acts faithfully as color derivations on A, where B is the ring of D-constants in A. PDF

Groups with all irreducible modules of finite degree

(with W. V. Temple) - Let K[G] denote the group algebra of the multiplicative group G over a field K of characteristic 0. As is well known, G has an abelian subgroup of finite index if and only if all irreducible modules of K[G] have finite bounded degree. An open question of interest is whether G must have an abelian subgroup of finite index if K[G] has all irreducible modules of finite degree, but without assuming a bound on these degrees. In this note, we discuss the unpublished thesis work of the second author which comes close to yielding an affirmative solution to this problem. PDF

Simple Lie algebras of special type

(with J. Bergen) - Let K be a field, let A be an associative, commutative K-algebra and let D be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, the tensor product W(A,D) = A o D = AD becomes a Lie algebra, a Witt type algebra. In addition, there is a map div: W(A,D) -> A called the divergence and its kernel S = S(A,D) is a Lie subalgebra, a special type algebra. In this paper, we offer sufficient conditions for the Lie simplicity of [S,S]. While the main result here is somewhat cumbersome to state, it does handle a number of examples in a fairly efficient manner. Furthermore, some of the preliminary lemmas are of interest in their own right and may, in time, lead to a more satisfactory answer. PDF

Trace methods in twisted group rings

In this brief note, we discuss trace methods in twisted group algebras. Specifically, we obtain information on the trace of idempotent and nilpotent elements. As is to be expected, if the ground field has positive characteristic, then the arguments used for ordinary group rings carry over to this context with little difficulty. On the other hand, lifting these results to characteristic zero algebras is not straightforward and requires a reduction to finitely presented groups. PDF

Unitary units in group algebras

(with J. Z. Goncalves) - Let K[G] denote the group algebra of the finite group G over the nonabsolute field K of characteristic not 2, and let * be the K-involution of K[G] determined by inverting group elements. In this paper, we study the group Un(K[G]) of unitary units of K[G] and we classify those groups G for which Un contains no nonabelian free group. If K is algebraically closed, then this problem can be effectively studied via the representation theory of K[G]. However, for general fields, it is preferable to take an approach which avoids having to consider the division rings involved. Thus, we use a result of Tits to construct fairly concrete free generators in numerous crucial special cases. PDF

Invariant ideals of abelian group algebras and representations of groups of Lie type

(with A. E. Zalesskii) - This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for G an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group A is the additive group of a finite-dimensional vector space V over a locally finite field F of prime characteristic p, and the automorphism group G is a simple infinite absolutely irreducible subgroup of GL(V). Thus, G is isomorphic to an infinite simple periodic group of Lie type, and G is realized in GL(V) via a twisted tensor product (phi) of infinitesimally irreducible representations. If S is a Sylow p-subgroup of G and if Fv is the unique line in V stabilized by S, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus T on Fv. We are able to handle the case where phi is a rational representation with character field F. PDF

Invariant ideals of abelian group algebras under the multiplicative action of a field, I and II

(with J. M. Osterburg and A. E. Zalesskii) - Let D be a division ring and let V be a finite-dimensional D-vector space, viewed multiplicatively. If G=D* is the multiplicative group of D, then G acts on V and hence on any group algebra K[V]. In these two papers we completely describe the semiprime G-stable ideals of K[V], and conclude that these ideals satisfy the ascending chain condition. As it turns out, this result follows from corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I (with Zalesskii) is concerned with the latter situation and part II (with Osterburg and Zalesskii) handles arbitrary division rings. PDF-I, PDF-II

Free unit groups in group algebras

(with J. Z. Goncalves) - Let K[G] denote the group algebra of the finite group G over the field K. If either K has characteristic 0 and G is nonabelian, or K is nonabsolute of prime characteristic p and G/O_p(G) is nonabelian, then it is well-known that the group of units U(K[G]) contains a nonabelian free group. For the most part, this follows from the fact that GL_2(K) contains such a free subgroup. In this paper, we refine the above result by showing that G has two cyclic subgroups X and Y of prime power order, and two units u in U(K[X]) and v in U(K[Y]), such that the group generated by u and v contains a nonabelian free group. Indeed, we use a theorem of Tits to obtain a rather concrete description of the two units u and v. PDF

Twisted group algebras satisfying a generalized polynomial identity

If K^t[G] is a twisted group algebra satisfying a nondegenerate multilinear generalized polynomial identity f=0, then we show that G has certain normal subgroups of finite index which can be viewed as being almost central. For example, there exists H normal in G with both |G:H| and |H'| bounded by a fixed function of the support sizes of the nonzero K^t[G]-terms involved in f. Indeed, we obtain a more precise version of this result, with the structure of H depending upon the specific twisting in the group algebra. We then go on to determine necessary and sufficient conditions for K^t[G] to satisfy such an identity. PDF

Polycyclic restricted Lie algebras

(with V. M. Petrogradsky) - We define a polycyclic restricted Lie algebra to be the Lie analog of a polycyclic group, and we describe the structure of poly(cyclic or finite-dimensional) restricted Lie algebras. In particular, we prove that these are precisely the restricted Lie algebras whose restricted enveloping algebras have polynomial growth. PDF

Trace methods in twisted group rings, II

(with J. M. Osterburg) - In this note, we continue our discussion of trace methods in twisted group algebras. Specifically, we obtain the twisted analog of Bass' theorem on the traces of idempotents in ordinary group algebras. Indeed, we show that with suitable normalization, the characteristic 0 trace values of an idempotent are all contained in a cyclotomic field. The proof is a variant of the original argument combined with a reduction to finitely presented groups. PDF

Invariant ideals and polynomial forms

Let K[H] denote the group algebra of an infinite locally finite group H. In recent years, the lattice of ideals of K[H] has been extensively studied under the assumption that H is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra K[V] stable under the action of an appropriate automorphism group of V. Specifically, in this paper, we let G be a quasi-simple group of Lie type defined over an infinite locally finite field F, and we let V be a finite-dimensional vector space over a field E of the same characteristic p. If G acts nontrivially on V by way of the homomorphism G -> GL(V), and if V has no proper G-stable subgroups, then we show that the augmentation ideal wK[V] is the unique proper G-stable ideal of K[V] when char K is different from p. The proof of this result requires, among other things, that we study characteristic p division rings D, certain multiplicative subgroups M of D, and the action of M on the group algebra K[A], where A is the additive group of D. In particular, properties of the quasi-simple group G come into play only in the last section of this paper. PDF

We remark that the final value of a polynomial form f: A -> S need not be a subgroup of S. See "Polynomial and inverse forms" below for a counterexample.

Free unit groups in algebras

(with J. Z. Goncalves) - Let R be an algebra over a field K, and let G be a finite group of units in R. Suppose that either K has characteristic 0 and G is nonabelian, or K is a nonabsolute field of characteristic p > 0 and G/O_p(G) is nonabelian. Then we show that there are two cyclic subgroups X and Y of G of prime power order, and two special units u in KX and v in KY, where KX is the K-linear span of X in R, such that the group generated by u and v contains a nonabelian free group. Indeed, we obtain a rather precise description of these units, generalizing an earlier result where R=K[G] was the group algebra of G over K. PDF

Invariant ideals of abelian group algebras under the action of simple linear groups

This is a survey of recent work. (Prepared for the "Around Group Rings Conference" in honor of S. K. Sehgal.) PDF

Finitely generated simple algebras: a question of B. I. Plotkin

(with A. I. Lichtman) In his recent series of lectures, Prof. B. I. Plotkin discussed geometrical properties of the variety of associative K-algebras. In particular, he studied geometrically noetherian and logically noetherian algebras and, in this connection, he asked whether there exist uncountably many simple K-algebras with a fixed finite number of generators. We answer this question in the affirmative using both crossed product constructions and HNN extensions of division rings. Specifically, we show that there exist uncountably many nonisomorphic 4-generator simple Ore domains, and also uncountably many nonisomorphic division algebras having 2 generators as a division algebra. PDF

Free products in linear groups

Let R be a commutative integral domain of characteristic 0, and let G be a finite subgroup of PGL(n,R), the projective general linear group of degree n over R. In this note, we show that if n>1, then PGL(n,R) also contains the free product G*T, where T is the infinite cyclic group generated by the image of a suitable transvection. PDF

Embedding free products in the unit group of an integral group ring

(with J. Z. Goncalves) - Let G be a finite group and let p be a prime. We show that the unit group of the integral group ring Z[G] contains the free product Z_p*Z if and only if G has a noncentral element of order p. Moreover, when this occurs, then the Z_p-part of the free product can be taken to be a suitable noncentral subgroup of G of order p. PDF

Filtrations in semisimple rings

In this paper, we describe the maximal bounded Z-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite Z-gradings. We also consider simple Artinian rings with involution, in characteristic not 2, and we determine those bounded Z-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the analogous questions for filtrations with respect to other Archimedean ordered groups. PDF

Filtrations in semisimple Lie algebras, I

(with Y. Barnea) - We study the maximal bounded Z-filtrations of a complex semisimple Lie algebra L. Specifically, we show that if L is simple of type A_n, B_n, C_n or D_n, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our results complete this classification in most cases. Finally, we briefly discuss the analogous question for filtrations with respect to other Archimedean ordered groups. PDF Computation files: OverView G2.txt F4.txt E6O3.txt E7O4.txt E8O5.txt

Filtrations in semisimple Lie algebras, II

(with Y. Barnea) - In this paper, we continue our study of the maximal bounded Z-filtrations of a complex semisimple Lie algebra L. Specifically, we discuss the functionals which give rise to such filtrations, and we show that they are related to certain semisimple subalgebras of L of full rank. In this way, we determine the "order" of these functionals and count them without the aid of computer computations. The main results here involve the Lie algebras of type E_6, E_7 and E_8, since we already know a good deal about the functionals for the remaining types. Nevertheless, we reinterpret our previous results into the new context considered here. Finally, we describe the associated graded Lie algebras of all of the maximal filtrations obtained in this manner. PDF

Linear groups and group rings

(with J. Z. Goncalves) - This paper consists of two parts. The first is concerned with free products in linear groups and uses the usual "ping pong" lemma and attractors to prove the results. What is new here is that we allow certain subspaces of V associated with the semisimple and generalized transvection operators to have dimensions larger than 1. The second part is concerned with applications of this machinery to integral groups rings Z[G] of finite groups. We show, for example, that if G is nonabelian of order prime to 6, then Z[G] contains two Bass cyclic units that generate a nonabelian free group. Corrected version PDF Errata for published version PDF

Polynomial and inverse forms

In a recent paper, this author studied invariant ideals in abelian group algebras under the action of certain infinite, locally finite, quasi-simple groups. While the main result was reasonably definitive, there are nevertheless certain natural extensions that should be considered. One approach to a proof of such extensions is to improve the basic tools that were used in the original work. However, we show here that the two obvious improvements to the polynomial form results fail in general. Specifically, we prove that the final value of a polynomial form f: A -> S is not necessarily a subgroup of S. We also show that polynomial forms cannot be extended to "rational function forms" without losing the key properties we require. PDF

Prime Lie rings of derivations of commutative rings in characteristic 2

(with Chia-Hsin Liu) - Let R be a commutative associative ring with 1 and let Der(R) be the Lie ring of all derivations of R. Suppose that D is a Lie subring and R-submodule of Der(R). When R is D-prime, we give necessary and sufficient conditions for D to be Lie prime. Since results of this nature are already known for rings R of characteristic different from 2, what is really new here is the characteristic 2 case. PDF

"Multiplicative Invariant Theory" by Martin Lorenz (a book review)

``Multiplicative Invariant Theory'' by Martin Lorenz is a beautiful book on an exciting new subject, written by an expert and major contributor to the field. Most of the proofs have been completely reworked, and many of the results appear to be new. The author is especially careful to explain where each chapter is going, why it matters, and what background material is required. The last chapter on open problems, with a good deal of annotation, is certainly welcome, since there is much yet to be done. Be aware, this is definitely a research monograph. The subject matter is broad and deep, and the prerequisites on the reader can sometimes be daunting. Still, it is wonderful stuff and well worth the effort. PDF

A description of incidence rings of group automata

(with A. V. Kelarev) - Group automata occur in the Krohn-Rhodes Decomposition Theorem and have been extensively investigated in the literature. The incidence rings of group automata were introduced by the first author in analogy with group rings and incidence rings of graphs. The main theorem of the present paper gives a complete description of the structure of incidence rings of group automata in terms of matrix rings over group rings and their natural modules. As a consequence, when the ground ring is a field, we can use known group algebra results to determine when the incidence algebra is prime, semiprime, Artinian or semisimple. We also offer sufficient conditions for the algebra to be semiprimitive. PDF

Free subgroups in linear groups and group rings

This paper is an extension of the talk I gave at the International Conference on Non-Commutative Rings, Group Rings, Diagram Algebras and Applications at the University of Madras in December 2006. It discusses certain techniques used to prove the existence of free subgroups in linear groups and in the unit group of integral group rings of finite groups. Its main focus is recent joint work with Jairo Goncalves on non-abelian free groups generated by Bass cyclic units. Additional concrete examples are also offered. PDF

Finitary actions and invariant ideals

If the group G acts on an abelian group V, then it acts naturally on any group algebra K[V], and we are concerned with classifying the G-stable ideals of K[V]. In this paper, we consider a rather concrete situation. We take G to be an infinite locally finite simple group acting in a finitary manner on V. When G is a finitary version of a classical linear group, then we show that the augmentation ideal is the unique proper G-stable ideal of K[V]. On the other hand, if G is a finitary alternating group acting on a suitable permutation module V, then there is a rich family of G-stable ideals of K[V], and we show that these behave like certain graded ideals in a polynomial ring. PDF

Involutions and free pairs of bicyclic units in integral group rings

(with J. Z. Goncalves) - If * : G -> G is an involution on the finite group G, then * extends to an involution on the integral group ring Z[G]. In this paper, we consider whether bicyclic units u in Z[G] exist with the property that the group, generated by u and u* is free on the two generators. If this occurs, we say that (u,u*) is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution *, the ring Z[G] contains a free bicyclic pair. PDF

Filtrations in semisimple Lie algebras, III

This paper is the third in a series. The first two, by Yiftach Barnea and this author, study the maximal bounded Z-filtrations of the finite-dimensional simple Lie algebras over the complex numbers. Those papers obtain a complete characterization for all but the five exceptional Lie algebras, namely the ones of type G2, F4, E6, E7 and E8. Here, we fill in the missing step for the algebra G2. The proof is computational and uses MAGMA, a computer algebra package, to handle the 7 x 7 matrices that occur. PDF The original version of this manuscript PDF contains annotated MAGMA code. Computation files: MAGMA in and MAGMA out

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