Good Reduction of Good Filtrations at Places

We consider filtered or graded algebras $A$ over a field $K$. Assume that there is a discrete valuation $O_v$ of $K$ with $m_v$ its maximal ideal and $k_v:=O_v/m_v$ its residue field. Let $\Lambda$ be $O_v$-order such that $\Lambda K=A$ and $\bar{\Lambda}:=k_v\otimes_{O_v}\Lambda$ the $\Lambda$-reduction of $A$ at the place $K\leadsto k_v$. Using the filtration of $A$ induced by $\Lambda$ we shall prove that for certain algebras $A$ their properties are related to $\bar{\Lambda}$.Comment: 17 page

Some bialgebroids constructed by Kadison and Connes-Moscovici are isomorphic

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.Comment: 4 page

Twisted algebras and Rota-Baxter type operators

We define the concept of weak pseudotwistor for an algebra $(A, \mu)$ 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 $(A, \mu \circ T)$ 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.Comment: 15 pages; continues arXiv:math/0605086 and arXiv:0801.2055, some concepts from these papers are recalled; we added a Note and some references. In this final version, accepted for publication in J. Algebra Appl., the title has been slighty modified and few little things have been adde

Valuation Extensions of Filtered and Graded Algebras

In this note we relate the valuations of the algebras appearing in the non-commutative geometry of quantized algebras to properties of sub-lattices in some vector spaces. We consider the case of algebras with $PBW$-bases and prove that under some mild assumptions the valuations of the ground field extend to a non-commutative valuation. Later we introduce the notion of $F$-reductor and graded reductor and reduce the problem of finding an extending non-commutative valuation to finding a reductor in an associated graded ring having a domain for its reduction.Comment: 12 page