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.     

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.     

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.     

Quantum groups and quantum shuffles

Let U _q ⁺ be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix. We show that U _q ⁺ is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z ⁿ . This method gives supersymetric as well as multiparametric versions of U _q ⁺ in a uniform way (for a suitable choice of the Hopf bimodule). We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition. We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U _q sl(2) and a suitable irreducible finite dimensional representation. Oblatum 21-III-1997 & 12-IX-1997     

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.     

Orthosymplectic Quantum Function Superalgebras OSPq（2l＋ 1｜2n）

By the R-matrix of orthosymplectic quantum superalgebra U _q (osp(2l+1|2n)) in the vector representation, we establish the corresponding quantum Hopf superalgebra OSP _q (2l + 1|2n). Furthermore, it is shown that OSP _q (2l + 1|2n) is coquasitriangular.     

Module categories over pointed Hopf algebras

We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups u_q(\mathfraksl₂){u_q(\mathfrak{sl}_2)}.     

Approximate Subgroups of Linear Groups

We establish various results on the structure of approximate subgroups in linear groups such as SL _n (k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(\mathbb F_q){{\rm SL}_{n}({\mathbb {F}}_{q})} which generates the group must be either very small or else nearly all of SL_n(\mathbb F_q){{\rm SL}_{n}({\mathbb {F}}_{q})}. The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over an arbitrary field k and yields a classification of approximate subgroups of G(k). In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.     

Hodge and Laplace–Beltrami Operators for Bicovariant Differential Calculi on Quantum Groups

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace–Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite-dimensional. Using Jucys–Murphy elements of the Hecke algebra, the eigenvalues of the Laplace–Beltrami operator for the Hopf algebra (SL _q (N)) are computed.     

On the dimension of the module-varieties of tame and weld algebras

Inspired by a result in [Ga], we locate three integer forms of F_q[SL(n+ 1)] over k[q,q ^-1] wih a presentation by generators and relations, which for q=1 specialize to U(𝔥)), where 𝔥 is the Lie bialgebra of the Poisson Lie group dual to SL(n+1). In sight of this we prove two PBW-like theorems for F_q [SL(n+ 1)], both related to the classical PBW theorem for U(𝔥).     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.     

Braided bosonization and inhomogeneous quantum groups

We consider quasitriangular Hopf algebras in braided tensor categories introduced by Majid. It is known that a quasitriangular Hopf algebra H in a braided monoidal category C induces a braiding in a full monoidal subcategory of the category of H-modules in C. Within this subcategory, a braided version of the bosonization theorem with respect to the category C will be proved. An example of braided monoidal categories with quasitriangular structure deviating from the ordinary case of symmetric tensor categories of vector spaces is provided by certain braided supersymmetric tensor categories. Braided inhomogeneous quantum groups like the dilaton free q-Poincaré group are explicit applications.Supported in part by the Deutsche Forschungsgemeinschaft (DFG) through a research fellowship.     

Graded Quantum Groups and Quasitriangular Hopf Group-Coalgebras

ABSTRACT

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL _n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] ^l and generalize the Drinfeld-Jimbo quantum enveloping algebras.     

The formal theory of Hopf algebras Part II: The case of Hopf algebras

Abstract

The category Hopf_R of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category Hopf_R is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then Hopf_R is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of Hopf_R. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist.     

Generic representations of the finite general linear groups and the Steenrod algebra: II

The category of generic representations over the finite fieldF _q , used in PartI to study modules over the Steenrod algebra, is here related to the modular representation theory of the groups GL _n (F _q ). This leads to a simple and elegant approach to the classic objects of study: irreducible representations, extensions of modules, homology stability, etc. Connections to current research in algebraicK-theory involving stableK-theory and Topological Hochschild Homology are also explained.Partially funded by the NSF.     

Relative Completions of Linear Groups Over Coordinate Rings of Curves

We compute the completion of the special linear group over the coordinate ring A of a curve over a number field k relative to its representation in SL _n (A/𝔪), for 𝔪 a maximal ideal.     

Expansion in SLd(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL _d (Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of π _q (G) with respect to the generating set π _q (S) form a family of expanders, where π _q is the projection map Z→Z/q Z.