Finite subgroups occuring in finite-dimensional division algebras

Irreducible strongly indecomposable torsion-free abelian groups of finite rank are called Reid groups, and a full subring R of a division algebra D is called Reid-realizable if it is isomorphic to the endomorphism ring of a Reid group. The main results is that a full subring R of a finite dimensional division algebra D is Reid-realizable if for infinitely many prime numbers p the quotient of the p-adic localizations D p * / div R p * > either is the product of at least two p-adic number fields, or is not a product of division algebras. In particular, a full subring of a finite dimensional central division algebra over an algebraic number field is Reid-realizable if it is locally free for infinitely many prime numbers.

Division algebras D with valuation v are studied, where D is lSnite-dimensional and totally ramified over its center F (i.e., the ramification index of v over wlF equals [D: F]). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When vlF is Henselian a full classification of the F-subalgebras of D is given. When F has a Henselian valuation v with separably closed residue field and A is any tame central simple F-algebra, an algorithm is given for computing the underlying division algebra of A from a suitable subgroup of A*/F*. Some examples are constructed using this valuation theory, including the first example of finitbdimensional F-central division algebras D1 and D2 with D1 XF D2 not a division ring, but D1 and D2 having no common subfield K D F. Valuation theory, long a basic tool in commutative algebra, has been relatively neglected in the study of division algebras, until quite recently. Nontheless, valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties, particularly algebras over iterated Laurent power series fields. For example, such division algebras have been key ingredients in Amitsur's noncrossed product construction [Am] and in Platonov's construction [P] of division algebras D with SK1(D) 7& 1. Valuations are not so prevalent on division algebras as on fields. But if a division algebra D does have a valuation, this structure contains a substantial amount of information about D which would scarcely be accessible otherwise. We consider here valued divisioll algebras D for which D is totQlly ramified and tame over its center F, i.e., for which the ramification index IrD rFI equals the dimension [D: F] of D over F and the characteristic char(D) does not divide [D: F]. (Here rD iS the value group of the valuation on D, and D is the residue division algebra. We assume throughout that [D: F] < oo.) Valued division algebras of this type appear, e.g., in Amitsur's noncrossed product paper [Am, §2], in Saltman's work on indecomposable division algebras [Sa], in certain of the MalcevNeumann division algebras considered by the first author and Amitsur [TA2,§4], etc. However, the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before. This may be because most past work on valued division algebras has concentrated on discrete valuations, when rD -Z; for such a valuation D is never totally ramified over its center F unless D = F (cf. (3.2) below). Received by the editors June 26, 1986. 1980 Mathematics Subject Clmsification (1985 Reon). Primary 16A39. 2Supported in part by F.N.R.S. 2Supported in part by the National Science Foundation. (:)1987 American Mathematlcal Society 0002-9947/87 $}.00 + $ 25 per page

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Let D be a finite dimensional division algebra over a local field of characteristic p and let SL 1(D) denote the group of elements of reduced norm 1 in D. In this paper we prove that SL 1(D) is finitely presented as a profinite group.

Even-order subgroups of finite-dimensional division rings

The structure of finite quotients of “large” subgroups of the multiplicative group of a finite dimensional division algebra is interesting and is related to the Margulis-Platonov conjecture. We develop machinery to handle such quotients and we conjecture that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. The proofs rely on Wedderburn's Factorisation Theorem.

The authors present a new way to construct finite dimensional estimation algebra of nonmaximal rank. For a given finite dimensional estimation algebra E, one can construct a finite dimensional estimation algebra E' of nonmaximal rank which is isomorphic to E. The interesting fact is that even if E comes from a linear filtering system, there are arbitrary nonlinear filtering systems associated to E. Since every finite dimensional estimation algebra gives rise to a finite dimensional filter, we have constructed quite a large class of finite dimensional nonlinear filters.

Finite Dimensional Division Algebras

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

Let D be a finite dimensional division algebra with center K and let G be a finite odd order subgroup of the multiplicative group £>* of D. This note is concerned with the following:CONJECTURE.If K contains no nonidentity odd order roots of unity, then G is cyclic.We announce here some results and raise several questions about this conjecture.In [3] and [4] we proved this conjecture if K is either an algebraic number field or the completion of an algebraic number field.(The converse is also proved in [4] ; if K is an algebraic number field which does contain an odd order nonidentity root of unity, then there is a finite dimensional division algebra central over K containing a noncyclic odd order subgroup.)In this note we will consider the more general case where K is an arbitrary field of characteristic zero.By a X-division ring we mean a finite dimensional division algebra with center K. Let G be a finite subgroup of the multiplicative group of a Â-division ring D and, for L a subfield of D, denote by i^L(G) the division subring of D generated by L and G. Let 2£ L denote the center of i^L(G) and e L the exponent of Î^L^)-The following result is basic to our approach to the above conjecture : THEOREM 1.With notation as above, let £ be a primitive e L th root of unity and let <f> be an L-automorphism ofi^jjfi).Then <j>{t)=t ) .

R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with maximal rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all maximal rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >

The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a non-constructive way, the basic building blocks of the respective algebras (the radical and the simple components of the factor by the radical). In view of the extensive computations done in such algebras, it seems important to design efficient algorithms to find these building blocks.We find polynomial time solutions to a substantial part of these problems. We restrict our attention to algebras over finite fields and over algebraic number fields. We succeed in determining the radical (the “bad part” of the algebra) in polynomial time, using (in the case of prime characteristic) some new algebraic results developed in this paper. For associative algebras we are able to determine the simple components as well. This latter result generalizes factorization of polynomials over the given field. Correspondingly, our algorithm over finite fields is Las Vegas.Some of the results generalize to fields given by oracles.Some fundamental problems remain open. An example: decide whether or not a given rational algebra is a noncommutative field.

On the images of multilinear maps of matrices over finite-dimensional division algebras

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

A Decomposition Theory for Differentiable Systems