arXiv subjects
Publications and source records attributed to Freddy Van Oystaeyen.
The glider Brauer-Severy variety GBS(A) of a central simple algebra A over a field K is introduced as the set of all irreducible left glider ideals in A for some filtration FA. For fields we deduce that GBS(K) equals R(K) x Z, the product of the Riemann surface of K and the ring of integers Z. For a csa A over K it turns out that GBS(A) = BS(A) x GBS(K), where BS(A) denotes the classical Brauer-Severi variety of A.
We define the concept of weak pseudotwistor for an algebra $(A, μ)$ in a monoidal category $\mathcal{C}$, as a morphism $T:A\otimes A\rightarrow A\otimes A$ in $\mathcal{C}$, satisfying some axioms ensuring that $(A, μ\circ T)$ is also an algebra in $\mathcal{C}$. This concept generalizes the previous proposal called pseudotwistor and covers a number of exemples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota-Baxter operators and some of their relatives (such as Leroux's TD-operators and Reynolds operators). By using weak pseudotwistors, we introduce an equivalence relation (called "twist equivalence") for algebras in a given monoidal category.
We look at value functions of primes in simple Artinian rings and associate arithmetical pseudo-valuations to Dubrovin valuation rings which, in the Noetherian case, are $\mathbb{Z}$-valued. This allows a divisor theory for bounded Krull domains.
Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B^{co(H)}# H which is an H-bicomodule algebra, isomorphic to B.
Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and right-left Long module structures over H, with the property that, if (H, D) is an L-R-admissible pair, then D is a bialgebra in {\cal LR}(H).
Let H be a Hopf algebra with bijective antipode, let α, βbe two Hopf algebra automorphisms of H and M a finite dimensional (α, β)-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H.
We study some classes of lazy cocycles, called pure (respectively neat), together with their categorical counterparts, entwined (respectively strongly entwined) monoidal categories.
A laycle is the categorical analogue of a lazy cocycle. Twines (as introduced by Bruguieres) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If $c$ is a braiding, the double braiding $c^2$ is always a twine; we prove that it is a strong twine if and only if $c$ satisfies a sort of modified braid relation (we call such $c$ pseudosymmetric, as any symmetric braiding satisfies this relation). It is known that symmetric Yetter-Drinfeld categories are trivial; we prove that the Yetter-Drinfeld category $_H{\cal YD}^H$ over a Hopf algebra $H$ is pseudosymmetric if and only if $H$ is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the $2^{n+1}$-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by properties of laycles and twines.
We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra $(A, μ, u)$ in a monoidal category, as a morphism $T:A\otimes A\to A\otimes A$ satisfying a list of axioms ensuring that $(A, μ\circ T, u)$ is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich's braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.
It is a fairly known fact that most of the algebras appearing in the theory of rings of differential operators, quantized algebras of different kinds (including many quantum groups), regular algebras in projective non-commutative geometry, etc... come equipped with a natural gradation or filtration controlled by some finite dimensional vector space(s), e.g. the degree one part of filtration or gradation. In this note we relate the valuations of the algebras considered to unramified sub-lattices in some vector space(s).
The purely mathematical root of the dequantization constructions is the quest for a sheafification needed for presheaves on a noncommutative space. The moment space is constructed as a commutative space, approximating the noncommutative space appearing as a dynamical space, via a stringwise construction. The main result phrased is purely mathematical, i.e. the noncommutative stalks of some sheaf on the noncommutative space can be identified to stalks of some sheaf associated to it on the commutative geometry (topology) of the moment space. This may be seen as a (partial) inverse to the deformation--quantization idea, but in fact with a much more precise behaviour of stalks of sheaves. The method, based on minimal axiomatics necessary to rephrase continuity principles in terms of partial order (noncommutative topology) exclusively, leads to the appearance of objects like strings and (M--)branes. Also, spectral families and observables may be defined and studied as separated filtrations on the noncommutative topology! We highlight the relation with pseudo--places and generalized valuation theory. Finally, we hint at a new notion of ``space'' as a dynamical system of noncommutative topologies with the same commutative shadow (4 dimensional space--time if you wish) and variable (but isomorphic) moment spaces, which for special choices may be thought of as a higher dimensional brane--space.
We introduce a new class of graded rings extending the class of generalized Weyl algebras. These rings are orders in crossed products of the most general type, and we introduce their basic structure theory. We provide an extensive list of examples, some completely new but also some considered earlier, and highlight some specific structure results. Many new and interesting problems about the crystalline graded rings may be identified.
Let $G$ be an abelien group, $ε$ an anti-bicharacter of $G$ and $L$ a $G$-graded $ε$ Lie algebra (color Lie algebra) over $\K$ a field of characteristic zero. We prove that all $G$-graded, positive filtered $A$ such that the associated graded algebra is isomorphic to the $G$-graded $ε$-symmetric algebra $S(L)$, there is a $G$- graded $ε$-Lie algebra $L$ and a $G$-graded scalar two cocycle $ω\in\mathrm{Z}_{gr}^2(L,\K)$, such that $A$ is isomorphic to $ U_ω(L)$ the generalized enveloping algebra of $L$ associated with $ω$. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra $U(L)$ and the generalized cohomology of color Lie algebra $L$.
Let $A$ be a $(G, χ)$-Hopf algebra with bijection antipode and let $M$ be a $G$-graded $A$-bimodule. We prove that there exists an isomorphism \mathrm{HH}^*_{\rm gr}(A, M)\cong{\rm Ext}^*_{A{-}{\rm gr}} (\K, {^{ad}(M)}), where $\K$ is viewed as the trivial graded $A$-module via the counit of $A$, $^{ad} M$ is the adjoint $A$-module associated to the graded $A$-bimodule $M$ and $\mathrm{HH}_{\rm gr}$ denotes the $G$-graded Hochschild cohomology. As an application, we deduce that the cohomology of color Lie algebra $L$ is isomorphic to the graded Hochschild cohomology of the universal enveloping algebra $U(L)$, solving a question of M. Scheunert.
We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.
We introduce a more general version of the so-called L-R-smash product and study its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product). We also give an interpretation of the L-R-smash product in terms of an L-R-twisting datum.
In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may be applied to H^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the "generating matrix" formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.
If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v:H\to B a morphism of right H-comodule algebras, we prove that there exists a left H-module algebra A such that B\simeq A# H. The main difference comparing to the Hopf case is that, from the multiplication of B, which is associative, we have to obtain the multiplication of A, which in general is not; for this we use a canonical projection E arising from the fact that B becomes a quasi-Hopf H-bimodule.