Lie Regular Generators of General Linear Groups

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ?_{2 ⁿ} ), GL(2, ? _{p ⁿ} ), and GL(2, ?_2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ?₄), GL(2, ?₆), GL(2, ?₈), and GL(2, ?₁₀) are also given.     

A set of maps from K to EndA(l2(A)) isomorphic to EndA(K)(l2(A(K))). Applikations

Let A be a C^*-algebra, K be a compact space, A(K) be the C^*-algebra of all continuous maps from K into A, 1₂(A) be the standard countably generated Hilbert A-module. We investigate a set of maps from K into End_A(1₂(A)), which is isomorphic to End_A(K)(1₂(A(K))). We describe the subsets which are isomorphic to End_fA(K) ^*(1₂(A(K))). GL_A(K)(1₂(A(K))) and GL_fA(K) ^*(1₂(A(K))), respectively. As an application we deduce a criterion for the self-duality of 1₂(A) in the commutative case.     

Homology stability for linear groups

Summary LetR be a commutative finite dimensional noetherian ring or, more generally, an associative ring which satisfies one of Bass' stable range conditions. We describe a modified version of H. Maazen's work [18], yielding stability for the homology of linear groups overR. Applying W.G. Dwyer's arguments (cf. [9]) we also get stability for homology with twisted coefficients. For example,H ₂(GL _n (R), R ⁿ ) takes on a stable value whenn becomes large.     

Distributivity conditions for lattices of dominions in quasivarieties of Abelian groups

Let ℳ be any quasivariety of Abelian groups, L_q(ℳ) be a subquasivariety lattice of ℳ, dom _G ^ℳ be the dominion of a subgroup H of a group G in ℳ, and G/dom _G ^ℳ (H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom _G ^N (H)| N ∈ L_q(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom _G ^ℳ (H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for its being distributive. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006.     

The Mathieu group M12

Let G be a finite simple non-Abelian group. t is an involution of G, and L=O² (C_G(t)/O · (C_G(t))). K the center Z(L) is cyclic and L/Z(L)-PGL (2, q), q odd, then either a Sylow 2-subgroup of G is semidihedral or C_G(t)Z₂×PGL (2.5) and G is isomorphic to the Mathieu group M₁₂ of degree 12.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

Comparison of germ expansion¶on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π^′ on GL _n (D) where D is a division algebra over k of index m and the germ expansion of the representation π of GL _mn (k) associated to π^′ by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GL _mn (k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GL _n (D). Received: 30 September 1999 / Revised version: 11 February 2000     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

Constructible Matrix Groups

We prove that the additive group of a ring K is constructible if the group GL ₂ (K) is constructible. It is stated that under one extra condition on K, the constructibility of GL ₂ (K) implies that K is constructible as a module over its subring L generated by all invertible elements of the ring L; this is true, in particular, if K coincides with L, for instance, if K is a field or a group ring of an Abelian group with the specified property. We construct an example of a commutative associative ring K with 1 such that its multiplicative group K* is constructible but its additive group is not. It is shown that for a constructible group G represented by matrices over a field, the factors w.r.t. members of the upper central series are also constructible. It is proved that a free product of constructible groups is again constructible, and conditions are specified under which relevant statements hold of free products with amalgamated subgroup; this is true, in particular, for the case where an amalgamated subgroup is finite. Also we give an example of a constructible group GL ₂ (K) with a non-constructible ring GL. Similar results are valid for the case where the group SL ₂ (K) is treated in place of GL ₂ (K) .     

NILPOTENT GROUPS AND ABELIANIZATION

We study the question of what properties of nilpotent groups are shared by their abelianizations. We identify two such properties—that of being a π-torsion group, where π is a family of primes, and that of having q^th roots, for some prime q. We use these properties to provide simplified proofs of the following theorems in the localization of nilpotent groups.

Let H, K be subgroups of the nilpotent group N and let P be a family of primes. Then [H, K] P = [H_P, K_p]

Let the group G act on the nilpotent group N. Then G acts compatibly on N_p and (Γ^G _i N)_P = Γ^G _i(N_p).

The second theorem above is then applied to the study of the localization of relative groups, in the sense of [4].     

Coinvariants for a coadjoint action of quantum matrices

Let K be a (algebraically closed ) field. A morphism A ⟼ g ⁻¹ Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL _q × GL _q -modules is a highest weight category.     

Untergruppen der GL2, welche durch homogene lineare Kongruenzen definiert sind

We generalize a result of Rankin [1]: Let R be a p — adic valuation ring or one of its factor rings and let G be GL₂(R) or SL₂(R). Then for M∈ M₂(R), G_M:={s∈G| trace MS=O} is a group iff trace M=0 and det M satisfies a simple condition (det M=O in most cases). We give similar conditions for several homogeneous linear equations defining subgroups of G.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

On the Selberg orthogonality for automorphic L-functions

For automorphic L-functions L(s, π) and L( s,p^￠){L( s,\pi^{\prime })} attached to automorphic irreducible cuspidal representations π and π′ of GL_m( \mathbbQ_A){GL_{m}( \mathbb{Q}_{A})} and GL_m^￠(\mathbbQ_A) {GL_{m^{\prime }}(\mathbb{Q}_{A}) }, we prove the Selberg orthogonality unconditionally for m ≤ 4 and m′ ≤ 4, and under hypothesis H of Rudnik and Sarnak if m > 4 or m′ > 4, without the additional requirement that at least one of these representations be self-contragradient.     

Equivalence classes of matrices inGL 2 (Q) andSL 2 (Q)

LetG denote either of the groupsGL ₂ (q) orSL ₂ (q). The mapping θ sending a matrix to its transpose-inverse is an automophism ofG and therefore we can form the groupG ⁺ =G. <θ>. In this paper conjugacy classes of elements inG ⁺ -G are found. These classes are closely related to the congruence classes of invertible matrices inG.     

Geometry of 2×2 Hermitian matrices over any division ring

Let D be a division ring with an involution-,H2(D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A-B) be the arithmetic distance between A,B ∈ H2(D) . In this paper,the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D(char(D) = 2) is proved:if  :H2(D) → H2(D) is the adjacency preserving bijective map,then  is of the form (X) = tP XσP +(0) ,where P ∈ GL2(D) ,σ is a quasi-automorphism of D. The quasi-automorphism of D is studied,and further results are obtained.     

Actions of the dipper-donkin quantization GL 2 on the clifford algebra C(l,3)

Following the method already developed for studying the actions of GL_q (2,C) on the Clifford algebra C(l,3) and its quantum invariants [1], we study the action on C(l, 3) of the quantum GL ₂ constructed by Dipper and Donkin [2]. We are able of proving that there exits only two non-equivalent cases of actions with nontrivial “perturbation” [1]. The spaces of invariants are trivial in both cases.

We also prove that each irreducible finite dimensional algebra representation of the quantum GL ₂ q^m ≠1, is one dimensional.

By studying the cases with zero “perturbation” we find that the cases with nonzero “perturbation” are the only ones with maximal possible dimension for the operator algebra ?.     

Actions of GLq (2,C) on C(1,3) and its four dimensional representations

A complete classification is given of all inner actions on the Clifford algebra C(l,3) defined by representations of the quantum group GL_q (2,C)q^m ≠1, which are not reduced to representations of two commuting “q-spinors”. As a consequence of this classification it is shown that the space of invariants of every GL_q (2,C)-action of this type, which is not an action of SL_q (2,C), is generatedby 1 and the value of the quantum determinant for the given representation.     

Uniqueness of twisted linear periods and twisted Shalika periods

Let k be a local field of characteristic zero.Letπbe an irreducible admissible smooth representation of GL2 n(k).We prove that for all but countably many charactersχ’s of GLn(k)×GLn(k),the space ofχ-equivariant(continuous in the archimedean case)linear functionals onπis at most one dimensional.Using this,we prove the uniqueness of twisted Shalika models.     

Finite arithmetic subgroups ofGL n,III

LetG be an algebraic group inGL _n (C) defined over Q, andK an algebraic number field with the maximal orderO _k . If the groupG(O _k ) of rational points ofG inM _n (O _k ) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(O _k ) is generated by roots of unity inK andG(Z). Dedicated to the memory of Professor K G Ramanathan