Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An add...

We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalising the one introduced by Squier for word rewriting systems. We characterise this property by using the notion of critical branching. In particular, we define sufficient conditions f...

Most Lie group integrators can be expanded in series indexed by the set of ordered rooted trees. To each tree one can associate two distinct higher order derivation operators, which we call frozen and unfrozen operators. Composition of frozen operators induces a concatenation product on the trees, whereas composition of unfrozen operators induces a somewhat more complicated product known as the...

We prove that the separating space of an epimorphism from a Lie–Banach algebra onto the (continuous) derivation algebra Der(A) of a Banach algebra A consists of derivations which map into the radical of A.

Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations higher Frobenius kernels reduces to first kernel. In preceding paper in this series, deforming distribution algebra a kernel yielded family deformations called reduced enveloping algebras. we prove Steinberg decomposition can be similarly deformed, allowing us reduce representa...

In the late 1980s, Friedlander and Parshall studied representations of a family algebras which were obtained as deformations distribution algebra first Frobenius kernel an algebraic group. The representation theory these tells us much about Lie in positive characteristic. We develop analogue this for higher kernels, answering 30 year old question posed by Parshall. also examine their case speci...