Quantum group actions, twisting elements, and deformations of algebras

We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for catagories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.     

Construction of a Universal Twist Element from an R-Matrix

We propose a method for construction of a universal twist element based on a constant quasi-classical unifary matrix solution of the Yang–Baxter equation. The method is applied to few known R-matrices corresponding to Lie (super) algebras of rank one. Bibliography: 13 titles.     

Classical r-Matrices and Novikov Algebras

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.     

Integrable Deformations of Tops Related to the Algebra so(p,q)

We propose an integrable deformation of the known model of two interacting tops on the algebra so(p,q). We consider particular cases including the generalized Lagrange and Kovalevskaya tops. We construct the Lax matrices and the corresponding classical R-matrices.     

On flexible quadratic algebras

Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.      

R-matrix Poisson algebras and their deformations

We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory.     

On tensor representations of knot algebras

Summary The fact that a Yang-Baxter operator defines tensor representations of the Artin braid group has been used to construct knot invariants. The main purpose of this note is to extend the tensor representations of the Artin braid group to representations of the braid groupZ B _k associated to the Coxeter graphB _k. This extension is based on some fundamental identities for the standardR-matrices of quantum Lie theory, here called four braid relations. As an application, tensor representations of knot algebras of typeB (Hecke, Temperley-Lieb, Birman-Wenzl-Murakami) are derived.     

Quasi-hopf algebras associated withsl 2 and complex curves

We construct quasi-Hopf algebras quantizing double extensions of the Manin pairs of Drinfeld, associated to a curve with a meromorphic differential, and the Lie algebrasl ₂. This construction makes use of an analysis of the vertex relations for the quantum groups obtained in our earlier work, PBW-type results and computation ofR-matrices for them; its key step is a factorization of the twist operator relating “conjugated” versions of these quantum groups.     

The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras

Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras of finite dimensional k-Lie algebras in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of for semisimple a conjecture of Humphreys [28] on the block structure of is confirmed. In the case of for semisimple and an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of are determined, and a necessary condition (which may also be sufficient) for a baby Verma -module to be simple is obtained. Received: 24 June 1999; in final form: 30 March 2000 / Published online: 17 May 2001     

Classes of structurable algebras of skew-rank

Let R be a ring such that 2, 3 ∈ R ^×. We construct classes of structurable algebras over R whose residue class algebras have skew-dimension 1. These are matrix algebras or forms of matrix algebras which do not necessarily arise out of separable Jordan algebras of degree 3. As an application, we give canonical examples of structurable algebras of large dimension.     

Integral modular data and congruences

We compute all fusion algebras with symmetric rational S-matrix up to dimension 12. Only two of them may be used as S-matrices in a modular datum: the S-matrices of the quantum doubles of ℤ/2ℤ and S ₃. Almost all of them satisfy a certain congruence which has some interesting implications, for example for their degrees. We also give explicitly an infinite sequence of modular data with rational S- and T-matrices which are neither tensor products of smaller modular data nor S-matrices of quantum doubles of finite groups. For some sequences of finite groups (certain subdirect products of S ₃,D ₄,Q ₈,S ₄), we prove the rationality of the S-matrices of their quantum doubles.     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Twistors for modules over algebras

We study the concept of module twistor for a module over an algebra. This concept provides a unifying framework for various deformed constructions of modules over algebras, such as module R-matrices, (n-factor iterated) twisted tensor products and L-R-twisted tensor products of algebras. Among the main results, we find the relations among these constructions. Furthermore, we study some properties of module twistors.     

Loewy coincident algebra and <Emphasis Type="Italic">QF</Emphasis>-3 associated graded algebra

We prove that an associated graded algebra R _G of a finite dimensional algebra R is QF (= selfinjective) if and only if R is QF and Loewy coincident. Here R is said to be Loewy coincident if, for every primitive idempotent e, the upper Loewy series and the lower Loewy series of Re and eR coincide. QF-3 algebras are an important generalization of QF algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra R, the associated graded algebra R _G is QF-3 if and only if R is QF-3.     

R-Matrices for SU(2)-Invariant Two-Leg Spin Ladders

A list of R-matrices for the set of eight integrable spin ladders is presented. Four of these matrices are new. Bibliography: 13 titles.     

Perfect residuated lattice ordered monoids

Bounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).     

Tree Oriented Pullback

Given two epimorphisms of algebras A ? B and C ? B, we consider the pullback R. We introduce a particular class of algebras, the tree oriented pullback, where there is a close relationship between the category of indecomposable modules of these algebras. This leads us to prove that if A and C are hereditary algebras, then R is a tilted algebra.     

A proof that commutative artinian rings are noetherian

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.     

A Class of Auslander-Regular Macaulay Rings

Abstract

For a (left and right) noetherian semilocal ring R we analyse a regularity concept (called weak regularity) based on the equation gld R = dim R. Examples are regular Cohen-Macaulay orders over a regular local ring, localized enveloping algebras of finite dimensional Lie algebras, and the regular rings classified in Rump (2001b). We prove that weakly regular rings are Auslander-regular and Macaulay.     

Some Homological Properties of Skew PBW Extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well-known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.