Smashed extensions for hopf algebraSL q (2)

The structure of the smashed extensions forSL ₂(2) over a fixed algebraC is discussed, and the isomorphic classes of smashed exrersions forSL _q(2) overC are determined. Project supported by the National Natural Science Foundation of China (Grant No. 19671070).     

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.     

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.     

Representations of the compact quantum groupSU q(2) and geometrical quantization

The Souriau—Kostant method of geometrical quantization is used to construct infinite-dimensional irreducible unitary representations of the algebra of functions of the compact quantum groupSU _q(2). The generalization to the case of the quantum groupSU _q (n) is discussed.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 163–172, August, 1994.     

A New Family of 2-Designs over GF (q) Admitting SLm(q1)

Given a 2-(l,3,q³(q^l-5-1/q-1);q) design for an integer l 5 mod 6(q-1) which admits the action of a Singer cycle Z_l of GL_l(q), we construct a 2-(ml,3,q³(q^l-5-1/q-1);q) design for an arbitrary integer m 3 which admits the action of SL_m(q^l). The construction applied to Suzuki's designs actually provides a new family of 2-designs over GF(q) which admit the SL_m(q^l) action.     

The Structure of Weak Hopf Algebras Corresponding to U q (sl 2)

The structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to U _q (sl ₂) can be written as the direct sum of U _q (sl ₂) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their Ext quiver. There are four types of such structures.     

Quotients of the congruence kernels of SL2 over arithmetic dedekind domains

LetC=C(C, P, k) be the coordinate ring of the affine curve obtained by removing a closed pointP from a (suitable) projective curveC over afinite fieldk. Let SL₂ (C,q) be the principal congruence subgroup of SL₂(C) andU ₂(C,q) be the subgroup generated by the all unipotent matrices in SL₂(C,q), whereq is aC-ideal. In this paper we prove that, for all but finitely manyq, the quotient SL₂(C,q)/U ₂(C,q) is a free group of finite,unbounded rank. LetC(SL₂(A)) be the congruence kernel of SL₂(A), whereA is an arithmetic Dedekind domain with only finitely many units. (e.g.A=C or ℤ) and letG be any finitely generated group. From the above (and previous results) we deduce that the profinite completion ofG,Ĝ, is a homonorphic image ofC(SL₂(A)). This is related to previous results of Lubotzky and Mel'nikov.     

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

The non commuting matrix elements of matrices from quantum groupGL _q (2;C) withq≡ω being then-th root of unity are given a representation as operators in Hilbert space with help ofC ₄ ⁽ⁿ⁾ generalized Clifford algebra generators appropriately tensored with unit 2×2 matrix infinitely many times. Specific properties of such a representation are presented. Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum alà Lévy-Leblond [10] and to polar decomposition ofSU _q (2;C) quantum algebra alà Chaichian and Ellinas [6] is also commented. The case ofq∈C, |q|=1 may be treated parallely.     

The representations ofU q(SO4) andU q(SO3,1)

Summary We have considered here the (unitary) irreducible representations of theq-deformed algebraU _q(SO₄) and of theq-deformed Lorentz algebraU _q(SO_3,1). Both of them contain, as subalgebra, the algebraU _q(SO₃) which is shown to be isomorphic to the Fairlie-Odesskii algebra. As the list of pairwise nonequivalent irreps of theU _q(SO_3,1) demonstrates, the set of the parameters, which characterize such irreps is somewhat reduced (due to periodicity properties of the function w(z)=[z]_q) in comparison with that of theq=1 (classical) case. From another side, the list of unitary irreps of theU _q(SO_3,1) contains the strange series which has no classical counterpart (disappears at q=1).Published in Teoreticheskaya i Matematicheskaya Fizika. Vol. 95, No. 2, pp. 251–257, May, 1993.     

2-Fusion Systems with Components of Type SL2(q)

We study a problem in fusion systems designed to mimic, simplify, and generalize parts of the Classification of Finite Simple Groups. A 2-fusion system is a 2-group S with a family of injective homomorphisms on subgroups of S. Fusion systems arise in the study of modular representations and classifying spaces, and our results have potential ramifications beyond finite group theory. One problem is to determine S or, whenever possible, the entire 2-fusion system from the knowledge of certain subgroups and maps. We consider the case where S contains a subgroup and maps that arise in the Classification with standard component of type SL₂(q).     

The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

In this note, we establish the connection between certain quantum algebras and generalized Clifford algebras (GCA). Precisely, we embed the quantum tori Lie algebra andU _q(sl (2)) in GCA.     

WEIGHT PROPERTY FOR IDEALS OF U q (sl(2))

Let k be an arbitrary field of characteristic zero, and U be the quantized enveloping algebra U _q (sl(2)) over k. The aim of this present paper is to study the ideals of U at q not a root of unity. It turns out that every non-zero ideal of U can be generated by at most two highest weight vectors under the adjoint action, and by a sum of two highest weight vectors. This weight property make it possible to give a complete list of all prime (primitive, maximal) ideals of U according to their generators.     

Representations of the Quantum Group SLq(2) in a Space of Functions

We construct representations of the quantum group SL _q(2) in a functional space. We consider the cases of general and exceptional values of the deformation parameter q.     

Exact calculability,semigroup of representations,and the stability property for representations of the algebra of functions on the quantum groupSU q(2)

An operation of the coproduct of representations of a bialgebra is defined. The coproduct operation of representations for the Hopf algebra of functions on the quantum groupSU _q (2) is investigated. For this Hopf algebra a representation II called the stable representation is constructed. This representation is invariant with respect to the coproduct with an arbitrary representation. An expression for the trace in the representation II is derived. The invariant Voronovich integal inSU _q (2) takes the form fdµ=tr(fcc ^*).V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 163–178, November, 1994.     

A Class of Finite-Dimensional Representations of U_q（sl_2）

In order to study a class of finite-dimensional representations of Uq（sl2）, we deal with the quotient algebra Uq （m, n, b） of quantum group Uq（sl2） with relations Kr=1, Emr=b, Fnr=0 in this paper, where q is a root of unity. The algebra Uq（m, n, b） is decomposed into a direct sum of indecomposable （left） ideals. The structures of indecomposable projective representations and their blocks are determined.     

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”.     

Matching Realization of Uq（sln＋1） of Higher Rank in the Quantum Weyl Algebra Wq（2n）

In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.     

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.     

A correspondence for the supercuspidal representations of $\overline {SL_2(F)}$, F p-adic

Following D. Manderscheid, we describe the supercuspidal representations of the n-fold metaplectic cover [`(SL<sub>2</sub>(F))]\overline {SL_2(F)}, where F is a p-adic field with (p, 2n) = 1. We prove a "Frobenius formula" for the character of a supercuspidal representation of [`(SL₂(F))]\overline {SL_2(F)}. Using this formula, we obtain a character relation between corresponding supercuspidal representations of [`(SL₂(F))]\overline {SL_2(F)} and of SL₂(F)> in the case n = 2.     

Generating Random Elements in SL n (F q ) by Random Transvections

This paper studies a random walk based on random transvections in SL _n(F _q ) and shows that, given > 0, there is a constant c such that after n + c steps the walk is within a distance from uniform and that after n – c steps the walk is a distance at least 1 – from uniform. This paper uses results of Diaconis and Shahshahani to get the upper bound, uses results of Rudvalis to get the lower bound, and briefly considers some other random walks on SL _n(F _q ) to compare them with random transvections.