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.     

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 ?.     

Quantum anchor:U q (sl(2)) case

We introduce the tangent spaceT(H _q ) on the quantum hyperboloid (A _0,q ^c ) and equip it with an action onA _0,q ^c being a deformation of the action of vectors fields on functions. An embeddingsl(2) _q →T(H _q ) ofq-deformed Lie algebrasl(2) being an analogue of the anchorsl(2)→Vect (H) is called “quantum anchor”.     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

Skew derivations andU q (sl(2))

This note first describes the basic properties of the skew derivations on the polynomial ringk[X]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U _q(sl(2)), has a unique action on C[X], where “action” means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU _q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it. Both authors were supported by the NSF, S. Montgomery by grant DMS 87-00641, and S. P. Smith by DMS 87-02447.     

On quantum mechanics and generalized Clifford algebras

In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular quantum mechanical models. First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ _n is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and Berry [2], as shown in [3]. Second the coherent state picture of the finite dimensional “Z _n — Quantum Mechanics” is presented. In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings. Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176     

The old subvariety ofJ 0(pq) and the Eisenstein kernel in Jacobians

Whenp, q are distinct odd primes, and γ:J ₀(p)²×J ₀(q)²→J ₀(pq) is the natural map defined by the degeneracy maps, Ribet [10] determined the odd part of the kernel of γ. We study the 2-primary part of this kernel through its intersection with the Eisenstein kernelJ ₀(p)[I _p )²×J ₀(q)[I _q ]². We determine this intersection forp≢1 mod 16,q≢1 mod 16, and also produce new elements of ker γ wheneverp≡9 mod 16 orq≡9 mod 16. These sharpen Ribet's results in [10].     

Minimum deformations of commutative algebra and linear groupGL(n)

In the algebra of formal seriesM _q (x ⁱ ), the relations of generalized commutativity that preserve the tensorI _q grading and depend on parametersq(i, k) are considered. A norm of the differential calculus onM _q consistent with theI _q grading is chosen. A new construction of a symmetrized tensor product of algebras of the typeM _q (x ⁱ ) and a corresponding definition of the minimally deformed linear groupQGL(n) and Lie algebraqgl(n) are proposed. A study is made of the connection ofQGL(n) andqgl(n) with the special matrix algebra Mat(n, Q), which consists of matrices with noncommuting elements. The deformed determinant in the algebra Mat(n, Q) is defined. The exponential mapping in the algebra Mat(n, Q) is considered on the basis of the Campbell-Hausdorff formula.Institute of Applied Physics, Tashkent State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 403–417, June, 1993.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

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.     

On aschbacher’s localC(G; T) theorem

Aschbacher’s localC(G; T) theorem asserts that ifG is a finite group withF*(G)=O ₂(G), andTεSyl₂(G), thenG=C(G; T)K(G), whereC(G; T)=〈N _G (T ₀)|1≠T ₀ charT〉 andK(G) is the product of all near components ofG of typeL ₂(2 ⁿ ) orA ₂ ⁿ ₊₁. Near components are also known asχ-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher’s theorem in the case thatG is aK-group, i.e., in the case that every simple section ofG is isomorphic to one of the known simple groups. Our proof relies on a result of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained. For John Thompson Supported in part by NSF grant #DMS 89-03124, by DIMACS, an NSF Science and Technology Center, funded under contract STC-88-09648, and by NSA grant #MDA-904-91-H-0043. Prof. Gorenstein died on August 26, 1992.     

The Compact Quantum Group Uq（2） （Ⅱ）

In this paper, we first prove that the θ-deformation Uθ（2） of U（2） constructed by Connes and Violette is our special case of the quantum group Uq（2） constructed in our previous paper. Then we will show that the set of truces on the C^＊ -algebra Uθ, θ irrational, is determined by the set of the truces on a subalgebra of Uθ.     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

LatticeW algebras and quantum groups

We present Feigin's construction [Lectures given in Landau Institute] of latticeW algebras and give some simple results: lattice Virasoro andW ₃ algebras. For the simplest caseg=sl(2), we introduce the wholeU _q(2)) quantum group on this lattice. We find the simplest two-dimensional module as well as the exchange relations and define the lattice Virasoro algebra as the algebra of invariants ofU _q(sl(2)). Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine groupU _q(ñ₊). We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 132–147, July, 1994.     

Curves with infinitely many points of fixed degree

Thed-th symmetric productC ^(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang [Hi] proved by Faltings [Fa2] implies ifC ^(d) (K) is an infinite set, then there is aK-rational covering ofC → ℙ _|K ¹ of degree ≤2d. As an application one gets that for fixed fieldK and fixedd there are only finitely many primes ι such that the set of all elliptic curves defined over some extensionsL ofK with [L∶K]≤d and withL-rational isogeny of degree ι is infinite.     

Basic sets of Brauer characters of finite groups of Lie type,III

LetG(F ^q ) be a finite classical group whereq is odd and the centre ofG is connected. We show that there exists a set of irreducible characters ofG(F ^q ) such that the corresponding matrix of scalar products with the characters of Kawanaka’s generalized Gelfand-Graev representations is square unitriangular. This uses in an essential way Lusztig’s theory of character sheaves. As an application we prove that there exists an ordinary basic set of 2-modular Brauer characters and that the decomposition matrix of the principal 2-block ofG(F ^q ) has a lower unitriangular shape.