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© 2014 Elsevier B.V. We study categorial properties of the operadic twisting functor Tw. In particular, we show that Tw is a comonad. Coalgebras of this comonad are operads for which a natural notion of twisting by Maurer-Cartan elements exists. We give a large class of examples, including the classical cases of the Lie, associative and Gerstenhaber operads, and their infinity-counterparts Lie∞, As∞, Ger∞. We also show that Tw is well behaved with respect to the homotopy theory of operads. As an application we show that every solution of Deligne's conjecture is homotopic to a solution that is compatible with twisting.

We study prime ideals in skew power series rings T:= R[[y; τ, δ]], for suitably conditioned complete right Noetherian rings R, automorphisms τ of R, and τ-derivations δ of R. Such rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, assuming that τ extends to a compatible automorphsim of T, we prove: If I is an ideal of R, then there exists a τ-prime ideal P of T contracting to I if and only if I is a τ-δ-prime ideal of R. Consequently, under the more specialized assumption that δ = τ - id (a basic feature of the Iwasawa-theoretic context), we can conclude: If I is an ideal of R, then there exists a prime ideal P of T contracting to I if and only if I is a τ-prime ideal of R. Our approach depends essentially on two key ingredients: First, the algebras considered are Zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings. © 2012 Hebrew University Magnes Press.

Let n be a positive integer, and let k be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented k-algebra R has infinitely many equivalence classes of semisimple representations R → Mn (k′), where k′ is the algebraic closure of k. The test reduces the problem to computational commutative algebra over k, via famous results of Artin, Procesi, and Shirshov. The test is illustrated by explicit examples, with n = 3. © 2008 Elsevier Inc. All rights reserved.

Given an abelian category with arbitrary products, arbitrary co-products, and a generator, we show that the closed subspaces (in the sense of A. L. Rosenberg) are parameterized by a suitably defined poset of ideals in the generator. In particular, the collection of closed subspaces is itself a small poset. © 2006 American Mathematical Society.

Let n be a positive integer, and let R be a finitely presented (but not necessarily finite dimensional) associative algebra over a computable field. We examine algorithmic tests for deciding (1) if every at most-n-dimensional representation of R is semisimple, and (2) if there exist nonsplit extensions of non-isomorphic irreducible R-modules whose dimensions sum to no greater than n. © Elsevier B.V. All rights reserved.

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1, 1), provides a noetherian prime counterexample.

We describe an algorithmic test, using the "standard polynomial identity" (and elementary computational commutative algebra), for determining whether or not a finitely presented associative algebra has an irreducible n-dimensional representation. When n-dimensional irreducible representations do exist, our proposed procedure can (in principle) produce explicit constructions. © 2001 Academic Press.

The prime and primitive spectra of Oq(fn), the multiparameter quantized coordinate ring of affine n-space over an algebraically closed field k, are shown to be topological quotients of the corresponding classical spectra, specO(fn) and maxo(fn) ≈kn, provided the multiplicative group generated by the entries of q avoids -1. ©2000 American Mathematical Society.