Donald S. Passman: Abstracts

Burnside's theorem for Hopf algebras

(with Declan Quinn) - A classical theorem of Burnside asserts that if X is a faithful complex character for the finite group G, then every irreducible complex character of G is a constituent of some power X^n of X. Fifty years after this appeared, Steinberg generalized it to a result on semigroup algebras K[G] with K an arbitary field and with G a semigroup, finite or infinite. Five years later, Rieffel showed that the theorem really concerns bialgebras and Hopf algebras. In this note, we simplify and amplify the latter work. PDF

Involutory Hopf algebras

(with Declan Quinn) - In 1975, Kaplansky conjectured that a finite-dimensional semisimple Hopf algebra is necessarily involutory. Twelve years later, Larson and Radford proved the conjecture in characteristic 0 and obtained significant partial results in positive characteristics. The goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites, no "harpoons," and gratifyingly few "hits." PDF

Derivations of skew polynomial rings

(with J. Marshall Osborn) - Let F be a field of characteristic 0 , let R be a skew polynomial ring over F in n variables, and let S be its 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 a sum of an inner derivation and a derivation that sends each variable to a scalar multiple of itself. In this paper, we extend these results to a more general situation. Specifically, we study twisted group algebras F^t[G], where G is a commutative group and F is a field of any characteristic. Furthermore, we consider certain subalgebras F^t[H], where H is a subsemigroup of G that generates G as a group. Finally, if e: G x G -> F is a skew-symmetric bilinear form, then we study the Lie algebra F_e[G] associated with e, and we consider its relationship to the Lie structure defined on various twisted group algebras F^t[G]. PDF

Delta ideals of Lie color algebras

(with Jeffrey Bergen) - Let L be a Lie color algebra (possibly restriced) over the field K. This paper considers several different Lie ideals of L associated with the Delta = D process. Specifically, D_\infty is the set of all x in L with dim_K [x,L] countable, D(L) is the set of all x in L with dim_K [x,L] finite, and D_L is the join of all finite dimensional Lie ideals of L. Note that the latter two ideals need not be equal. The two main results here are: 1) If D_\infty = 0, then U(L) is symmetrically closed. 2) [D(L),D(L)] is contained in D_L. We remark that these results are new and of interest even when L is an ordinary or super Lie algebra. In fact, we consider Lie color algebras here only because we can obtain the more general facts with little additional work. PDF

Semiprimitivity of group algebras: a survey

Let K be a field and let G be a multiplicative group. 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 40 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. PDF

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 that are either p', 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. PDF

The semiprimitivity problem for 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. 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. Here, we come tantalizingly close to a complete converse by showing that if G has no nonidentity locally subnormal subgroups, then K[G] is semiprimitive. The proof of this theorem uses the much earlier semiprimitivity results on locally finite, locally p-solvable groups, and the more recent results on locally finite, infinite simple groups. In addition, it uses the beautiful properties of finitary permutation groups. This paper is dedicated to the memory of Professor S. A. Amitsur. PDF

Representations of the Gupta-Sidki group

(with W. V. Temple) - If p is an odd prime, then the Gupta-Sidki group G is an infinite 2-generated p-group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree p. In this note, we make two observations concerning the irreducible representations of the group algebra K[G] with K an algebraically closed field. First, when the characteristic of K is different from p, we obtain a lower bound for the number of irreducible representations of any finite degree n. Second, when the characteristic is p, we show that if K[G] has one nonprincipal irreducible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact that the commutator subgroup H of G has a normal subgroup of finite index isomorphic to the direct product of p copies of H. PDF

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 middle 1970's. 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 nonidentity 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 K^t[G] is semiprimitive when G has no nontrivial 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. This paper is dedicated to the memory of Professor Brian Hartley. PDF

Ultraproducts and the group ring semiprimitivity problem

This expository paper is a slightly expanded version of the final talk I gave at the group rings conference celebrating the 25th anniversary of the Institute of Mathematics and Statistics at the University of Sao Paulo. The talk concerned an application of ultraproducts to the solution of the semiprimitivity problem for group algebras K[G] of locally finite groups G. It explained why the special cases of G locally solvable and of G infinite simple turn out to be the critical factors which must be studied. PDF

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

Multiplicative Jordan decomposition in group rings of 3-groups

(with Chia-Hsin Liu) - In this paper, we essentially classify those finite 3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. Thus, we are only concerned with the nonabelian case. Here we show that Z[G] has the MJD property for the two nonabelian groups of order 27. Furthermore, we show that there are at most three other specific nonabelian groups, all of order 81, with Z[G] having the MJD property. Unfortunately, we are unable to decide which, if any, of these three satisfies the appropriate condition. PDF

Multiplicative Jordan decomposition in group rings of 2,3-groups

(with Chia-Hsin Liu) - In this paper, we essentially finish the classification of those finite 2,3-groups G having integral group rings with the multiplicative Jordan decomposition property. If G is abelian or a Hamiltonian 2-group, then it is clear that Z[G] satisfies MJD. Thus, we need only consider the nonabelian case. Recall that the 2-groups with MJD were completely determined by Hales, Passi and Wilson, while the corresponding 3-groups were almost completely determined by the present authors. Thus, we are concerned here, for the most part, with groups whose order is divisible by 6. As it turns out, there are precisely three nonabelian 2,3-groups, of order divisible by 6, with Z[G] satisfying MJD. These have orders 6, 12, and 24. In view of another result of Hales, Passi and Wilson, this essentially completes the classification of all finite groups with MJD. PDF An unpublished Appendix includes a reasonably self-contained argument showing that the "generalized quaternion group" of order 12 satisfies MJD.

Invariant ideals of abelian group algebras under the torus action of a field, I

This paper is dedicated to Prof. Susan Montgomery. Let V = V_1 + V_2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F* by defining (v_1 + v_2)^g = v_1 g\inv + v_2 g. If K is a field of characteristic different from that of F, then G acts on the group algebra K[V] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we consider the special case when V_1 and V_2 are both 1-dimensional and we show that there are just four G-stable proper ideals of K[V]. PDF

Character theory and group rings

This paper is dedicated to Prof. I. Martin Isaacs. While we were graduate students, Marty Isaacs and I worked together on the character theory of finite groups, studying in particular the character degrees of finite p-groups. Somewhat later, my interests turned to ring theory and infinite group theory. On the other hand, Marty continued with character theory and soon became a leader in the field. Indeed, he has had a superb career as a researcher, teacher and expositor. In celebration of this, it is my pleasure to discuss three open problems that connect character theory to the ring-theoretic structure of group rings. The problems are fairly old and may now be solvable given the present state of the subject. A general reference for character theory is of course Marty's book, while my book affords a general reference for group rings. PDF

Invariant ideals of abelian group algebras under the torus action of a field, II

Let V = V_1 + V_2 be a finite-dimensional vector space over an infinite locally-finite field F. Then V admits the torus action of G = F* by defining (v_1 + v_2)^g = v_1 g\inv + v_2 g. If K is a field of characteristic different from that of F, then G acts on the group algebra K[V] and it is an interesting problem to determine all G-stable ideals of this algebra. In this paper, we show that, for almost all fields F, the G-stable ideals are uniquely writable as finite irredundant intersections of augmentation ideals of subspaces W_1 + W_2, with W_1 in V_1 and W_2 in V_2. As a consequence, the set of all G-stable ideals is Noetherian. PDF

Involutions and free pairs of Bass cyclic units in integral group rings

(with J. Z. Goncalves) - Let Z[G] be the integral group ring of the finite nonabelian group G over the ring of integers Z, and let * be an involution of Z[G] that extends one of G. If x and y are elements of G, we investigate when pairs of the form (u_{k,m}(x),u_{k,m}(x^{*})), or (u_{k,m}(x),u_{k,m}(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of Z[G]. PDF

Rewritable groups

(with M. I. Elashiry) - A group G is said to have the n-rewritable property Q_n if for all elements g_1,g_2,...,g_n in G, there exist two distinct permutations sigma,tau in Sym_n such that the two products g_{sigma(1)} g_{sigma(2)}...g_{sigma(n)} and g_{tau(1)} g_{tau(2)}...g_{tau(n)} are equal. We show here that if G satisfies Q_n, then G has a characteristic subgroup N such that |G:N| and |N'| are both finite and have sizes bounded by functions of n. This extends the result of Blyth which asserts that if G satisfies Q_n and if Delta is the finite conjugate center of the group, then |G:Delta | and |Delta'| are both finite with |G:Delta | bounded by a function of n. PDF

Elementary bialgebra properties of group rings and enveloping rings: An introduction to Hopf algebras

This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers. Let K be a field and let A be an algebra over K. Then the tensor product A o A is also a K-algebra, and it is quite possible that there exists an algebra homomorphism Delta: A -> A o A. Such a map Delta is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using Delta, one can define a tensor product on the collection of A-modules, and when A and Delta satisfy some rather mild axioms, then A is called a bialgebra. Classical examples of bialgebras include group rings K[G] and Lie algebra enveloping rings U(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Delta determines a convolution product on Hom_K(A,A) and this leads quite naturally to the definition of a Hopf algebra. PDF

Multiplicative Jordan decomposition in group rings of 3-groups, II

(with Chia-Hsin Liu) - In this paper we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that Z[G] satisfies MJD. In the nonabelian case, we show that Z[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 27. PDF

Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

(with Chia-Hsin Liu) - If G is a finite group whose integral group ring Z[G] has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring Q[G] have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central units in Z[G] cannot exist. With this, we are able to greatly simplify the argument that characterizes those 3-groups with integral group ring having MJD. Furthermore, we show that if G is a nonabelian semidirect product of a cyclic group of prime order p >7 with a cyclic 3-group, then Z[G] does not have MJD. PDF

Explicit free groups in division rings

(with J. Z. Goncalves) - Let D be a division ring of characteristic different from 2 and suppose that the multiplicative group D\0 has a subgroup G isomorphic to the Heisenberg group. Then we use the generators of G to construct an explicit noncyclic free subgroup of D\0. The main difficulty occurs here when D has characteristic 0 and the commutators in G are algebraic over the rational numbers. PDF

Involutions and free pairs of bicyclic units in integral group rings of non-nilpotent groups

(with J. Z. Goncalves) - If * 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. The main result here is that if G is a non-nilpotent group, then for any involution, Z[G] contains a free bicyclic pair. PDF

Groups with certain normality conditions

(with Chia-Hsin Liu) - We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings. PDF

Free groups in normal subgroups of the multiplicative group of a division ring

(with J. Z. Goncalves) - Let D be a division ring with center Z and multiplicative group D*, and let N be a normal subgroup of D*. We investigate various conditions under which N must contain a free noncyclic subgroup. In one instance, assuming that the transcendence degree of Z over its prime field is infinite, and that N contains a nonabelian solvable subgroup, we use a construction method due to Chiba to exhibit free generators of the free subgroup. PDF

Permutational and rewritable groups

Permutational groups and rewritable groups were introduced in 1985 and 1988, respectively. In this note, we briefly survey key properties of these groups. In addition, we consider certain related parameters and we compute a number of examples of interest. PDF

Free pairs of symmetric elements in normal subgroups of division rings

(with J. Z. Goncalves) - Let D be a division ring with central subfield k of characteristic not 2, let * be a k-involution of D, and let N be a normal subgroup of the multiplicative group of D. We show that if G is a *-stable nonabelian subgroup of N that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u, v) of *-symmetric elements that generate a noncyclic free group. One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper. PDF

Finite presentation

(with L. W. Small) - This paper surveys basic properties of finite presentation in groups, Lie algebras and rings. It includes some new results and also new, more elementary proofs, of some results that are already in the literature. In particular, we discuss examples of Stallings and of Roos on coherence and a recent theorem of Alahmadi and Alsulami on Morita invariance. PDF

On the counterexamples to the unit conjecture for group rings

We offer two comments on the beautiful papers of Giles Gardam and Alan Murray that yield counterexamples to the Kaplansky unit conjecture. First we discuss the determinants of these units in a certain 4 x 4 matrix representation of the group ring. Then we explain why there is a doubly infinite family of units in the Murray paper. PDF

Affine Noetherian algebras and extensions of the base field II

(with L. W. Small) - Paper I, written many years ago by Richard Resco and Lance Small, constructed an example in prime characteristic of a finitely generated Noetherian $K$-algebra that does not necessarily remain Noetherian under field extensions. After that work was completed, Yuri Medvedev visited Resco, read the paper, and suggested an approach using Jordan algebras that would yield a similar example in characteristic 0. This was announced in paper I, but to our knowledge the details were never published nor even entirely verified. We take the opportunity to do this now, but stress that the key $a_n$-sequence and its relevant properties are due to Medvedev. Also Resco suggested avoiding Jordan algebras and working ring-theoretically directly with the Grassman algebra. This is the approach we take. PDF

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