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.     

Examples of quantum cluster algebras associated to partial flag varieties

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiß, Leclerc and Schröer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.     

On the defining relations for generalized q-Schur algebras

We show that the defining relations needed to describe a generalized q-Schur algebra as a quotient of a quantized enveloping algebra are determined completely by the defining ideal of a certain finite affine variety, the points of which correspond bijectively to the set of weights. This explains, unifies, and extends previous results.     

Wild automorphisms of free metabelian algebras

We establish a necessary condition for the invertibility of an endomorphism of a free associative algebra. As an application, we offer examples of wild automorphisms of certain free metabelian algebras.     

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let A_n be the nth Weyl algebra and P_m be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {A_n⊗P_m} is proved: an algebraAadmits a finite setδ₁,…,δ_sof commuting locally nilpotent derivations with generic kernels andiffA?A_n⊗P_mfor somenandmwith2n+m=s, and vice versa. The inversion formula for automorphisms of the algebra A_n⊗P_m (and for ) has been found (giving a new inversion formula even for polynomials). Recall that (see [H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (New Series) 7 (1982) 287-330]) given, then (the proof is algebro-geometric). We extend this result (using [non-holonomic] D-modules): given, then. Any automorphism is determined by its face polynomials [J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J. Pure Appl. Algebra 52 (1988) 102-119], a similar result is proved for .One can amalgamate two old open problems (the Jacobian Conjecture and the Dixmier Problem, see [J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968) 209-242. [6]] problem 1) into a single question, (JD): is aK-algebra endomorphismσ:A_n⊗P_m→A_n⊗P_man algebra automorphism providedσ(P_m)⊆P_mand? (P_m=K[x₁,…,x_m]). It follows immediately from the inversion formula that this question has an affirmative answer iff both conjectures have (see below) [iff one of the conjectures has a positive answer (as follows from the recent papers [Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42(2) (2005) 435-452. [10]] and [A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier Conjecture. ArXiv:math.RA/0512171. [5]])].     

Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras

For a commutative algebra the shuffle product is a morphism of complexes. We generalize this result to the quantum shuffle product, associated to a class of non-commutative algebras (for example all the Hopf algebras). As a first application we show that the Hochschild-Serre identity is the dual statement of our result. In particular, we extend this identity to Hopf algebras. Secondly, we clarify the construction of a class of quasi-Hopf algebras.     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

Elliptic Lie algebras and tubular algebras

This article is to study relations between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito-Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel-Ringel, we prove that the elliptic Lie algebra of type , , or is isomorphic to the Ringel-Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel-Malkin-Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac-Moody Lie algebra by using indecomposable representations of the corresponding affine quiver.     

Presenting degenerate Ringel-Hall algebras of cyclic quivers

Using the generators labelled by simple and sincere semisimple modules for the Ringel-Hall algebra H_q(n) of a cyclic quiver Δ(n), we give a presentation for the degenerate algebra H₀(n). This is achieved by establishing a presentation for the generic extension monoid algebra of Δ(n). As an application, we show that both the degenerate Ringel-Hall algebra and the degenerate quantum affine sl_n admit multiplicative bases.     

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Let g be a finite-dimensional simple Lie algebra and let Sg be the locally finite part of the algebra of invariants (EndCV⊗Sg(g)) where V is the direct sum of all simple finite-dimensional modules for g and S(g) is the symmetric algebra of g. Given an integral weight ξ, let Ψ=Ψ(ξ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra of Sg and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.     

Generalized tilting functors and Ringel-Hall algebras

It is known from [M. Auslander, M.I. Platzeck, I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1-46] and [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] that the Bernstein-Gelfand-Ponomarev reflection functors are special cases of tilting functors and these reflection functors induce isomorphisms between certain subalgebras of Ringel-Hall algebras. In [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] the result from [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] is generalized to the tilting module case by giving an isomorphism between two Ringel-Hall subalgebras. In [J. Miyashita, Tilting Modules of Finite Projective Dimension, Math. Z. 193 (1986) 113-146] Miyashita generalized the tilting theory by introducing the tilting modules of finite projective dimension. In this paper the result in [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] is generalized to the tilting modules of finite projective dimension.     

Prime ideals in the quantum grassmannian

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a natural torus action of on the quantum grassmannian and the cell decomposition of the set of -primes leads to a parameterisation of the -spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the nonnegative cells in recent studies concerning the totally nonnegative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.     

Classification of arithmetic root systems

Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the construction of quantized enveloping algebras, in the noncommutative differential geometry of quantum groups, and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the present paper arithmetic root systems are classified in full generality. As a byproduct many new finite dimensional pointed Hopf algebras are obtained.     

Hopf algebras constructed by the FRT-construction

Given any bialgebra A and a braiding product 〈|〉 on A, a bialgebra U_〈|〉 was constructed in [R. Larson, J. Towber, Two dual classes of bialgebras related to the concepts of “quantum group” and “quantum Lie algebra”, Comm. Algebra 19 (1991) 3295-3345], contained in the finite dual of A. This construction generalizes a (not very well known) construction of Fadeev, Reshetikhin and Takhtajan [L.D. Faddeev, N.Yu. Reshetikhin, L.A. Takhtajan, Quantum Groups. Braid Group, Knot Theory and Statistical Mechanics, in: Adv. Ser. Math. Phys., vol. 9, World Sci. Publishing, Teaneck, NJ, 1989, pp. 97-110]. In the present paper it is proved that when U_〈|〉 is finite-dimensional (even if A is not), then it is a quasitriangular Hopf algebra.     

Associative conformal algebras with finite faithful representation

The classification of irreducible subalgebras of the associative conformal algebra Cend_N is presented in this paper. The structure theory of associative conformal algebras with finite faithful representation is developed.     

Automorphism groups of Weyl-type algebras

Let F be a field of characteristic 0, be n commuting variables over F, and be the field of all rational functions. Let . We have the simple Weyl type algebra . In this paper, the automorphism group of the associative algebra and the automorphism group of the Lie algebra are determined, and it is proved that . Received: 4 October 2001 / Revised version: 5 November 2001     

On Lie algebras associated with representation-directed algebras

Let B be a representation-finite C-algebra. The Z-Lie algebra L(B) associated with B has been defined by Riedtmann in [Ch. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994) 526-546]. If B is representation-directed, there is another Z-Lie algebra associated with B defined by Ringel in [C.M. Ringel, Hall Algebras, vol. 26, Banach Center Publications, Warsaw, 1990, pp. 433-447] and denoted by K(B).We prove that the Lie algebras L(B) and K(B) are isomorphic for any representation-directed C-algebra B.     

Skew differential operator algebras of twisted Hopf algebras

By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac-Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like.