Quantum Function Algebra Research Articles

Quantized Rank R Matrices

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n×r matrices as well as certain quantized factor algebras Mr+1q(n) of Mq(n) are analyzed. For r=1,…,n−1,Mr+1q(n) is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all (r+1)×(r+1) quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one. In almost all cases, the quantum parameter is a primitive mth root of unity. The degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m.

A global version of the quantum duality principle

The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, for any principal ideal domainR and for each primepeR we establish an “inner” Galois’ correspondence on the categoryHA of torsionless Hopf algebras overR, using two functors (fromHA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulop, respectively (i.e., they are“quantum function algebras” (=QFA) and“quantum universal enveloping algebras” (=QUEA), atp, respectively). In particular we provide a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebraH over a fieldk: apply the functors tok[ν] ⊗k H forp=ν. A relevant example occurring in quantum electro-dynamics is studied in some detail.

Cohomology of Quantized Function Algebras at Roots of Unity

Let $G$ be a simply-connected, semisimple algebraic group over $k$, an algebraically closed field of characteristic zero. Let ${\cal O}_{\epsilon}[G]$ be the quantized function algebra of $G$ at a primitive $\ell$th root of unity $\epsilon$, and let $\overline{{\cal O}_{\epsilon}[G]}$ be the `restricted' quantized function algebra at $\epsilon$, a finite-dimensional $k$-algebra obtained from ${\cal O}_{\epsilon}[G]$ by factoring out a centrally generated ideal. It is known that $\overline{{\cal O}_{\epsilon}[G]}$ is a Hopf algebra. We study the cohomology ring $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$, a graded commutative algebra, and, for any finite-dimensional $\overline{{\cal O}_{\epsilon}[G]}$-module $M$, the $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$. We prove that for sufficiently large $\ell$ there is an isomorphism of graded algebras\[ \mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k) \cong k[X_1,\ldots ,X_{2N}],\] where each $X_i$ is homogeneous of degree $2$, and $2N$ equals the number of roots associated to $G$. Moreover we show that in this case $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$is a finitely generated $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module. We also show under much less restrictive conditions on $\ell$ that $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$ continues to be a finitely generated graded commutative algebra over which $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$ is a finitely generated module.\vspace{6mm}\noindent 1991 Mathematics Subject Classification: 16W30, 17B37, 17B56.

The global quantum duality principle

Let R be an integral domain, let a non-zero h in R be such that k := R/hR is a field, and let HA be the category of torsionless (or flat) Hopf algebras over R. We call H in HA a "quantized function algebra" (=QFA), resp. "quantized restricted universal enveloping algebras" (=QrUEA), at h if - roughly speaking - H/hH is the function algebra of a connected Poisson group, resp. the (restricted, if R/hR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an "inner" Galois' correspondence on HA, via two endofunctors, ( )^\vee and ( )' , of HA such that H^\vee is a QrUEA and H' is a QFA (for all H in HA). In addition: (a) the image of ( )^\vee , resp. of ( )' , is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(R/hR) = 0 , the restrictions of ( )^\vee to the QFAs and of ( )' to the QrUEAs yield equivalences inverse to each other; (c) if p = 0 , starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra g, the functor ( )^\vee , resp. ( )' , gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [Ga2-4].

Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

We study finite centralizing extensions A ⊂ H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H [Rarr ]spec A to coincide with the X -orbits in spec H , where X denotes the finite group of characters of H that restrict to the counit of A . In particular, all of the fibres are X -orbits if and only if the fibre over the augmentation ideal of A is an X -orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.

Classification of differential calculi on Uq(b+), classical limits, and duality

We give a complete classification of bicovariant first order differential calculi on the quantum enveloping algebra Uq(b+) which we view as the quantum function algebra Cq(B+). Here, b+ is the Borel subalgebra of sl2. We do the same in the classical limit q→1 and obtain a one-to-one correspondence in the finite dimensional case. It turns out that the classification is essentially given by finite subsets of the positive integers. We proceed to investigate the classical limit from the dual point of view, i.e., with “function algebra” U(b+) and “enveloping algebra” C(B+). In this case there are many more differential calculi than coming from the q-deformed setting. As an application, we give the natural intrinsic four-dimensional calculus of κ-Minkowski space and the associated formal integral.

Links between cofinite prime ideals in quantum function algebras

We describe the skew primitive elements in a multiparameter enveloping algebraU=Uq,p−1 (g) and the links between cofinite maximal ideals in the corresponding quantum function algebra ℂq [G]. These results are applied to determine the coradical filtration forU, and to obtain a moduli space for multiparameter Drinfeld doubles.

Homological Aspects of Noetherian PI Hopf Algebras and Irreducible Modules of Maximal Dimension

We prove that a Noetherian Hopf algebra of finite global dimension possesses further attractive homological properties, at least when it satisfies a polynomial identity. This applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity, as well as to classical enveloping algebras in positive characteristic. In all three cases we show that these algebras are Auslander-regular and Macaulay. We derive representation theoretic consequences concerning the coincidence of the non-Azumaya and singular loci for each of the above three classes of algebras.

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is used to show that certain [Rscr ]-symmetric algebras S [Rscr ] ( V ) are Noetherian. This result applies in particular to the coordinate ring of quantum matrices A [Rscr ] ( V ) associated with an R -matrix [Rscr ] operating on the tensor square of a vector space V , to show that, under a natural set of hypotheses on [Rscr ], the algebra A [Rscr ] ( V ) is Noetherian and its augmentation ideal has a polynormal set of generators. As a corollary we deduce that these properties hold for the generic quantized function algebras R q [ G ] over any field of characteristic zero, for G an arbitrary connected, simply connected, semisimple group over [Copf ]. That R q [ G ] is Noetherian recovers a result due to Joseph [ 10 ], with a different proof.

Degrees of Quantum Function Algebras at Roots of 1

In this paper we will deal with quantum function algebrasFq[G] in the special case when the parameterqspecializes to a root of 1. Using a combinatorial technique, we will give general formulas for the degree of such algebras and of a particular family of quotients which are fundamental objects in representation theory.

On the Quantum Frobenius Map for General Linear Groups

For the quantum function algebras Oq(Mn) and Oq(GLn), atlth roots of unity whenlis odd, the image under theq-analog of the Frobenius morphism characterizes each prime (and primitive) ideal uniquely up to automorphisms obtained from row and column multiplication of the standard generatorsXijby powers ofq.

Quantum Function Algebra at Roots of 1

We introduce a form of the quantum function algebra on a Drinfeld-Jimbo quantum group over the ring Z[q, q−1]. Specializing q to a root of 1, we show that over the cyclotomic field this algebra is a projective module over its central sub-algebra, which is the usual coordinate algebra of the group. We study the induced Poisson-Lie structure of the group. A bundle of algebras on a complex simply connected Lie group with hamiltonian flows in the bundle is constructed. Some representations of the quantum function algebra in a root of 1 are constructed as an application. An estimate of the dimension of an arbitrary representation is given.

Algebras of functions on compact quantum groups, Schubert cells and quantum tori

The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R,\(u \in \Lambda ^2 \mathfrak{h}_R\), and\(\mathfrak{h}_R\) is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a,u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universalR-matrix, are given.