Hall algebras,hereditary algebras and quantum groups

4-VIII-1994 & 4-X-1994     

Basic Hopf algebras and quantum groups

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra structure are found. In particular, it is shown that only the quivers given in terms of a finite group G and sequence of elements of G in the following way can occur. The quiver has vertices and arrows , where the set is closed under conjugation with elements in G and for each g in G, the sequences W and are the same up to a permutation. We show how is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not characteristic free. All the quivers , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction. For closely related results of Cibils and Rosso see [Ci-R]. Received August 15, 1994; in final form May 16, 1997     

Weyl algebras over quantum groups

The term “Weyl algebras” is proposed for differential algebras associated with dual pairs of Hopf algebras. The principle of complete reducibility for the category of “admissible” modules over Weyl algebras is proved. Comodule structures that connect Weyl algebras with the Drinfeld quantum double are investigated. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 118, No. 2, pp. 190–204, February, 1999.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

On locally compact quantum groups whose algebras are factors

In this paper we are interested in examples of locally compact quantum groups (M,Δ) such that both von Neumann algebras, M and the dual , are factors. There is a lot of known examples such that are respectively of type (I_∞,I_∞) but there is no example with factors of other types. We construct new examples of type (I_∞,II_∞), (II_∞,II_∞) and (IIIλ,IIIλ) for each λ∈[0,1]. We also show that there is no such example with M or a finite factor.     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

Multiplier completion of Banach algebras with application to quantum groups

Archiv der Mathematik - Let $${{mathcal {A}}}$$ be a Banach algebra and let $$varphi $$ be a non-zero character on $${{mathcal {A}}}$$ . Suppose that $${{mathcal {A}}}_M$$ is the closure of the...     

The q-difference Noether problem for complex reflection groups and quantum OGZ algebras

For any complex reflection group G = G(m,p,n), we prove that the G-invariants of the division ring of fractions of the n:th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl field. This shows that the q-difference Noether problem has a positive solution for such groups, generalizing previous work by Futorny and the author [10 Futorny, V., Hartwig, J. T. (2014). Solution to a q-difference Noether problem and the quantum Gelfand–Kirillov conjecture for 𝔤𝔩_N. Math. Z. 276(1–2):1–37. [Google Scholar]]. Moreover, the new result is simultaneously a q-deformation of the classical commutative case and of the Weyl algebra case recently obtained by Eshmatov et al. [8 Eshmatov, F., Futorny, V., Ovsienko, S., Fernando Schwarz, J. (2015). Noncommutative Noether’s Problem for Complex Reflection Groups. Available at: http://arxiv.org/abs/1505.05626 [Google Scholar]].

Second, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk [18 Mazorchuk, V. (1999). Orthogonal Gelfand-Zetlin algebras, I. Beiträge Algebra Geom. 40(2):399–415. [Google Scholar]] originating in the classical Gelfand–Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of 𝔤𝔩_n and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko [11 Futorny, V., Ovsienko, S. (2010). Galois orders in skew monoid rings. J. Algebra 324:598–630.[Crossref], [Web of Science ®] , [Google Scholar]], with symmetry group being a direct product of complex reflection groups G(m,p,r_k).

Finally, using these results, we prove that the quantum OGZ algebras satisfy the quantum Gelfand–Kirillov conjecture by explicitly computing their division ring of fractions.     

Many-valued quantum algebras

We deal with algebras of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety. Supported by the Research and Development Council of the Czech Government via the project MSM6198959214.     

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field $\mathbb{K}$, we investigate a class of noncommutative $\mathbb{K}$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots ,x_n\}$ such that each pair satisfies a relation of the form $x_i{x}_j=q_{ij}{x}_j{x}_i+r_{ij}$, where $q_{ij}\in {\mathbb{K}}^{?}$ and $r_{ij}\in \mathbb{K}$, with, in some sense, sufficiently many pairs for which $r_{ij}\ne 0$. For such an algebra it turns out that there is a single parameter q such that each $q_{ij}=q^{\pm 1}$. Assuming that $q\ne \pm 1$, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let $n\ge 3$ be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type $A_{n?1}$ as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver $P_{n+1}^{(1)}$ identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in $n+2$ variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of $A_{n?1}$ and $P_{n+1}^{(1)}$ are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Representations of quantum algebras

Inventiones mathematicae -     

Commutative quantum operator algebras

A key notion bridging the gap between quantum operator algebras [26] and vertex operator algebras [4, 9] is the definition of the commutativity of a pair of quantum operators (see Section 2). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. In [26] we give a definition of a commutative quantum operator algebra. We show in [26] that a vertex operator algebra gives rise to a special case of a CQOA. The main purpose of the current paper is to further develop the foundations for a complete mathematical theory of CQOAs. We give proofs of most of the relevant results announced in [26], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators.