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In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L ∞ -algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context. [ABSTRACT FROM AUTHOR]

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L , we are interested in finding the pairs (ϕ , ψ) ∈ Aut (V) × Aut (L) , which are inducible by an automorphism in Aut (L ˜). We connect the inducibility problem to the (2 , 3) -cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut (V) × Aut (L) to be inducible lies in the (2 , 3) -cohomology group H (2 , 3) (L , V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2 , 3) -cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2. [ABSTRACT FROM AUTHOR]

The notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [19] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by Bardakov and Gubarev. In this paper, we study extensions of Rota-Baxter groups by constructing suitable cohomology theories. Among others, we find relations with the extensions of skew braces. Given an extension of Rota-Baxter groups, we also construct a short exact sequence connecting various automorphism groups, which generalizes the Wells short exact sequence. [ABSTRACT FROM AUTHOR]

Let Γ be a congruence subgroup of level N in GL n (Z). Let P be a maximal Q -parabolic subgroup of GL n / Q , with unipotent radical U , and let Q = (P ∩ Γ) / (U ∩ Γ). Let p > dim Q ⁡ (U (Q)) + 1 be a prime number that does not divide N. Let M be a (U , p) -admissible Γ-module. Consider the Lyndon-Hochschild-Serre spectral sequence arising from the exact sequence 1 → U ∩ Γ → P ∩ Γ → Q → 1 , which abuts to H ⁎ (P ∩ Γ , M). We show that if M is a trivial U ∩ Γ -module, then certain classes in the E 2 page survive to E ∞. We use this to obtain information about classes in H ⁎ (P ∩ Γ , M) even if M is not a trivial U ∩ Γ -module. This information will be used in future work to prove a Serre-type conjecture for sums of two irreducible Galois representations. [ABSTRACT FROM AUTHOR]

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

An R -module M is called quasi-morphic if for any f ∈ End R (M) , there exist g , h ∈ End R (M) such that Im f = Ker g and Ker f = Im h. In addition, M R is said to be morphic whenever g = h in the above definition. The main objective of this paper is investigating quasi-morphic property for several classes of modules. First we obtain general properties of quasi-morphic modules via exact sequence approach. Moreover, we investigate conditions under which a finite length quasi-morphic module is morphic. As a result, we show that for uniserial finite length modules, the notions of morphic and quasi-morphic coincide. Over a principal ideal domain R , direct sums of cyclic modules which are (quasi-)morphic are characterized. Among applications of our results, nonsingular extending (quasi-)morphic modules are characterized completely. We also prove that over a commutative Noetherian domain R which is not a field, quasi-morphic nonsingular extending modules are precisely direct sums of copies of Q (the quotient field of R). (Quasi-)Morphic singular extending abelian groups are also characterized. [ABSTRACT FROM AUTHOR]

We extend [13] to the nonsemisimple case. We define and study exact factorizations B = A • C of a finite tensor category B into a product of two tensor subcategories A , C , and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories [8]. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category [14] by another one. We also provide several examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this note (R , m) denotes a complete regular local ring and B mostly denotes its absolute integral closure. The four objectives of this paper are the following: i) to determine the highest non-vanishing local cohomology of Ω B / R in equicharacteristic 0, ii) to establish a connection between each of Ω B / R and Ω B / V and pull-back of Ω A / V via a short exact sequence together with new observations on corresponding local cohomologies in mixed characteristic where V is the coefficient ring of R and A is its absolute integral closure, iii) to demonstrate that Ω B / R can be mapped onto a cohomologically Cohen-Macaulay module and iv) to study torsion-free property for Ω C / V and Ω C / k along with their respective completions where C is an integral domain and a module finite extension of R. In this connection an extension of Suzuki's theorem on normality of complete intersections to the formal set-up in all characteristics is accomplished. [ABSTRACT FROM AUTHOR]

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε -th component with ε the identity of the grade group. For Z -graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras. [ABSTRACT FROM AUTHOR]

Let B ⊂ A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we give a structure result for extensions of an algebra given by a quiver and admissible relations, and examples of non split left or right bounded extensions. [ABSTRACT FROM AUTHOR]

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles. [ABSTRACT FROM AUTHOR]