Irreducible Subgroups of GL 1(D) Satisfying Group Identities

ABSTRACT

Let D be a finite dimensional F -central division algebra and G an irreducible subgroup of D*: = GL ₁(D). Here we investigate the structure of D under various group identities on G. In particular, it is shown that when [D:F] = p ², p a prime, then D is cyclic if and only if D* contains a nonabelian subgroup satisfying a group identity.     

Central Units in Metacyclic Integral Group Rings

In this article, we give a method to compute the rank of the subgroup of central units of ? G, for a finite metacyclic group, G, by means of ?-classes and ?-classes. Then we construct a multiplicatively independent set  ? (U(? C _{p, q} )) and by applying our results, we prove that  generates a subgroup of finite index.     

Some Skew Linear Groups Satisfying Generalized Group Identities

In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GL_n(D) satisfying a generalized group identity over GL_n(D) is central.     

On Prime Rings Satisfying a Certain Generalized Polynomial Identity

Abstract

Recently, Beidar, Fong and Bokut proved that a prime ring satisfies a nontrivial semigroup generalized identity if and only if its central closure is a primitive ring with nonzero socle and the associated skew field is a field. We shall extend their result to the case when generalized polynomial contains three summands.     

On Units of Group Algebras of 2-Groups of Maximal Class

Abstract

We determine the number of elements of order two in the group of normalized units V(𝔽₂ G) of the group algebra 𝔽₂ G of a 2-group of maximal class over the field 𝔽₂ of two elements. As a consequence for the 2-groups G and H of maximal class we have that V(𝔽₂ G) and V(𝔽₂ H) are isomorphic if and only if G and H are isomorphic.     

A Family of Complex Irreducible Characters Possessed by Unitary and Special Unitary Groups Defined over Local Rings

Abstract

Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 ^l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U _n (R) be the subgroup of GL(V) that preserves ( , ). Let SU _n (R) be the subgroup of all g ∈ U _n (R) whose determinant is equal to one. Let Ψ be the Weil character of U _n (R).

All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU _n (R).     

Unitary Modules for EALAs Co-ordinatized by a Quantum Torus

Abstract

A class of unitary modules is constructed for an extended affine Lie algebra (EALA) co-ordinatized by a quantum torus for |q| = 1 by extending the method developed by Kac and Raina (Kac, V. G., Raina, A. K. (1987 Kac, V. G. and Raina, A. K. 1987. “Highest weight representations of infinite dimensional Lie algebras”. Singapore: World Scientific.  [Google Scholar]). Highest Weight Representations of Infinite Dimensional Lie Algebras. Singapore: World Scientific.) which uses infinite wedge products. By twisting the module by an automorphism (defined by the author Eswara Rao (Eswara Rao, S. (2001 Eswara Rao, S. 2001. “A class of integrable modules for the core of EALA co-ordinated by Quantum Tori”. TIFR preprint [Google Scholar]). A class of integrable modules for the core of EALA co-ordinated by Quantum Tori. TIFR preprint.) which commutes with anti-linear anti-automorphism, we construct a large class of unitary modules for the EALA.     

Bireflectionality of the Orthogonal Group over a Principal Ideal Domain

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Traces of Torsion Units

A conjecture due to Zassenhaus asserts that if G is a finite group then any torsion unit in ?G is conjugate in ?G to an element of G. Here, a weaker form of this conjecture is proved for some infinite groups.     

Identities on Units of Algebraic Algebras

Let be an algebraic algebra over an infinite field K and let ( ) be its group of units. We prove a stronger version of Hartley's conjecture for , namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in ( ), then satisfies a polynomial identity (PI). We also show that if is non-commutative, then is a PI-ring, provided f = 0 is satisfied by the non-central units of . In particular, is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in ( ) if and only if the same identity is satisfied in . The last fact remains true for generalized Laurent polynomial identities, provided that is locally finite.     

Group algebras with units satisfying a group identity II

We classify group algebras of torsion groups over a field of characteristic with units satisfying a group identity.

     

Unitary designs and codes

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code—a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values—and give an upper bound for the size of a code with s inner product values in U(d), for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.      

Unitary equivalence to a complex symmetric matrix: An algorithm

We present a necessary and sufficient condition for a 3×3 matrix to be unitarily equivalent to a symmetric matrix with complex entries, and an algorithm whereby an arbitrary 3×3 matrix can be tested. This test generalizes to a necessary and sufficient condition that applies to almost every n×n matrix. The test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix.     

Homology Stability for Unitary Groups II

In this paper, the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied. We obtain a much better range of homology stability compared to the results existing in the literature. An application of our results is given.     

EXAMPLES OF FCR-ALGEBRAS

Abstract

Let R be a prime ring and 𝒰(R) its group of units. We prove that if 𝒰(R) satisfies a group identity and 𝒰(R) generates R,then either R is a domain or R is isomorphic to the algebra of n × n matrices over a finite field of order d. Moreover the integers n and d depend only on the group identity satisfed by 𝒰(R). This result has been recently proved by C. H. Liu and T. K. Lee (Liu,C. H.; Lee,T. K. Group identities and prime rings generated by units. Comm. Algebra (to appear)) and here we present a new different proof.     

Some Results on Hypercentral Units in Integral Group Rings

Abstract

In this note we investigate the hypercentral units in integral group rings ?G,where G is not necessarily torsion. One of the main results obtained is the following (Theorem 3.5): if the set of torsion elements of G is a subgroup T of G and if Z ₂(𝒰) is not contained in C _𝒰(T),then T is either an Abelian group of exponent 4 or a Q* group. This extends our earlier result on torsion group rings.