Related papers: Diamond distances in Nottingham algebras
We prove several structural results on Nottingham algebras, a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series.…
Borrowing some terminology from pro-p groups, thin Lie algebras are N-graded Lie algebras of width two and obliquity zero, generated in degree one. In particular, their homogeneous components have degree one or two, and they are termed…
We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of Nottingham Lie…
The graded Lie algebra associated with the Nottingham group over a field of prime characteristic serves as a fundamental example of Nottingham algebras, a class of infinite-dimensional, positively graded thin algebras. This paper completes…
Thin Lie algebras are Lie algebras over a field, graded over the positive integers and satisfying a certain narrowness condition. In particular, all homogeneous components have dimension one or two, and are called diamonds in the latter…
Thin Lie algebras are Lie algebras L, graded over the positive integers, with all homogeneous components of dimension at most two, and satisfying a more stringent but natural narrowness condition modeled on an analogous one for pro-p…
A thin Lie algebras is a Lie algebra $L$, graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$, and such that each nonzero ideal of $L$ lies between consecutive terms of its lower…
Thin Lie algebras are infinite-dimensional graded Lie algebras $L=\bigoplus_{i=1}^{\infty}$, with $\dim(L_1)=2$ and satisfying a covering property: for each $i$, each nonzero $z\in L_i$ satisfies $[zL_1]=L_{i+1}$. It follows that each…
We answer an open question in the theory of transducer degrees initially posed in [1] on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which…
We introduce the notion of a Lie-like algebra$^{\diamond}$ (superalgebra$^{\diamond}$) for $\diamond\in\{^{1-st}, ^{2-nd}, ^{3-rd} \}$.
We answer an open question in the theory of transducer degrees on the existence of a diamond structure in the transducer hierarchy. Transducer degrees are the equivalence classes formed by word transformations which can be realized by a…
Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic…
A Boolean algebra $\mathcal A$ has the Nikodym property if every pointwise bounded sequence of bounded finitely additive measures on $\mathcal A$ is uniformly bounded. Assuming the Diamond Principle $\Diamond$, we will construct an example…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…
We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard…
In this paper we study a special subclass of real solvable Lie algebras having small dimensional or small codimensional derived ideal. It is well-known that the derived ideal of any Heisenberg Lie algebra is 1-dimensional and the derived…
The Nottingham group at 2 is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \cdots$ in the variable $t$ with coefficients $a_i$ from the field with two elements, where the group operation is given by composition of power series.…
We give an explicit description of the Lie algebra of derivations for a class of infinite dimensional algebras which are given by \'etale descent. The algebras under consideration are twisted forms of central algebras over rings, and…
Let 2<n\leq l<m< \omega. Let L_n denote first order logic restricted to the first n variables. We show that the omitting types theorem fails dramatically for the n--variable fragments of first order logic with respect to clique guarded…
We find all finite Ockham algebras that admit only finitely many compatible relations (modulo a natural equivalence). Up to isomorphism and symmetry, these Ockham algebras form two countably infinite families: one family consists of the…