We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan–Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan–Lusztig cell modules for Hecke algebras of type A from [V. Mazorchuk and C. Stroppel, Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (2008), 1363–1426].

This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1⊗A and ϵ(A), determine one another, functorially in A. The object S1⊗A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 :=Bℤ, while the object ϵ(A)is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a ϵ-dg-algebra (see the main text). We define an equivalence φ between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1 ⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

In this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S⟶End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. [‘A unified approach to various generalizations of Armendariz rings’, Bull. Aust. Math. Soc.81 (2010), 361–397].

A radical ρ is called prime-like if for every prime ring A, the polynomial ring A[x] is ρ-semisimple. Let α be a radical satisfying the polynomial equation α(A[x])=(α(A))[x] for every ring A. A radical γ is called α-like if for every α-semisimple ring A, the polynomial ring A[x]is γ-semisimple. In this paper, we study properties of α-like radicals. We show that α-likeness is a generalization of prime-likeness and extend some results concerning prime-like radicals. This allows us easily to find distinct special radicals which coincide on simple rings and on polynomial rings, which answers a question put by Ferrero.

Let k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

Several characterizations are given of (Zelmanowitz) regular modules among the torsionless modules, the locally projective modules, the nonsingular modules, and modules where certain submodules are pure. Along the way, a version of the unimodular row lemma for torsionless modules is given, and it is shown that a regular ring is left self-injective if and only if every nonsingular left module is regular.

We give a moduli interpretation of the outer automorphism group Out of a finite-dimensional algebra similar to that of the Picard group of a scheme. We deduce that the connected component of Out is invariant under derived and stable equivalences. This allows us to transfer gradings between algebras and gives rise to conjectural homological constructions of interesting gradings on blocks of finite groups with abelian defect. We give applications to the lifting of stable equivalences to derived equivalences. We give a counterpart of the invariance result for smooth projective varieties: the product Pic0 × Aut0 is invariant under derived equivalence.

We discuss some basic properties of the graded centre of a triangulated category and compute examples arising in representation theory of finite-dimensional algebras.

We investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups.

A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert-scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson–Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.

In a recent paper, Iyama and Yoshino considered two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen–Macaulay modules in terms of linear algebra data. In this paper, we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov’s result on the graded singularity category.

In this paper, we review the parametrized strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrized strict deformation quantization. Rieffel’s basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrized strict deformation quantization of Hilbert C*-modules over C*-bundles with fibrewise torus action.

Real ideals of the ring ℜL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜL is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka’s theorem that characterizes realcompact spaces.

The main purpose of this paper is to give a new, elementary proof of Flanigan’s theorem, which says that a given ring A has a maximal essential extension ME(A) if and only if the two-sided annihilator of A is zero. Moreover, we discuss the problem of description of ME(A) for a given right ideal A of a ring with an identity.

Let 𝒦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1→E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 𝒦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 𝒦 with the class 𝒰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(ℂ) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

Let R be an integral domain and A a symmetric cellular algebra over R with a cellular basis {CλS,T∣λ∈Λ,S,T∈M(λ)}. We construct an ideal L(A) of the centre of A and prove that L(A) contains the so-called Higman ideal. When R is a field, we prove that the dimension of L(A)is not less than the number of nonisomorphic simple A-modules.

Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.

The holonomic rank of the A-hypergeometric system MA(β) is the degree of the toric ideal IA for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of non-generic parameters to construct a combinatorial formula for the rank jump of MA(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of MA(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a non-trivial weakly toric module M⊆ℂ[ℤA] . We also provide a direct proof of the main result in [M. Saito, Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323–338] regarding the isomorphism classes of MA (β) .