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.

Algebraic structures relative to a semigroup Ω (also called family algebraic structures) first appeared in the work of renormalization in quantum field theory. In this paper, we first consider Ω-diassociative algebras and diassociative family algebras as family analogues of diassociative algebras. We define the cohomologies of these algebras generalizing the cohomology of diassociative algebras introduced by Frabetti. We also introduce (relative) averaging family operators as the family analogue of averaging operators and show that they induce diassociative family algebras and Ω-diassociative algebras. Next, we define the cohomology of a (relative) averaging family operator as the cohomology of the induced Ω-diassociative algebra with coefficients in a suitable representation. Finally, as an application of our cohomology, we study formal one-parameter deformations of relative averaging family operators.

In this article, we study Nijenhuis operators on Hom-Lie algebras. We construct a graded Lie algebra (via the Hom-analog of the Frölicher-Nijenhuis bracket) whose Maurer-Cartan elements are given by Nijenhuis operators. This allows us to define the cohomology associated to a Nijenhuis operator. As an application, we study formal deformations of Nijenhuis operators that are generated by the above-defined cohomology. Finally, we introduce Hom-NS-Lie algebras as an algebraic structure behind Nijenhuis operators on Hom-Lie algebras. We provide various examples of Hom-NS-Lie algebras.

In [1], D. Daigle has proved a few results on “tameness” of a degree function on an integral domain B containing Q, which ensure that, under certain hypotheses, a derivation on B gives rise to a homogeneous derivation on an associated graded ring of B induced by the degree function. In this paper, we extend Daigle's results to arbitrary integral domains not necessarily containing Q.

During recent decades, Ga-actions, especially certain invariants of Ga-actions, have been important tools in the study of affine varieties. The Ga-actions are usually studied through locally nilpotent derivations in characteristic zero and exponential maps (see Definition 1.1) in arbitrary characteristic. The “Makar-Limanov invariant” of locally nilpotent derivations played a pivotal role in solving the linearization conjecture in the 1990s, while invariants of exponential maps were central to N. Gupta’s resolution of the Zariski cancellation problem in positive characteristic. In the study of locally nilpotent derivations on commutative algebras containing Q, Freudenburg and Moser-Jauslin (Mich Math J 62:227–258, (2013), Theorem 6.1) have introduced a new invariant called “rigid core” and used it to formulate an alternative version of Mason’s theorem and to prove a well-known analogue of Fermat’s last theorem for rational functions (Freudenburg and Moser-Jauslin (2013), Corollary 6.1). In this note, we consider the concept of the rigid core in the framework of exponential maps on commutative algebras over an algebraically closed field k of arbitrary characteristic. We observe that for any factorial k-domain B with tr.deg k(B) = 2 , the concept of rigid core coincides with the Makar-Limanov invariant. We also show that over any affine two-dimensional normal k-domain B, its rigid core is a stable invariant.

Extensive studies are being made on the family of Danielewski surfaces — they provide counter-examples to the Cancellation Problem. In [2], the authors investigated another family of non-cancellative surfaces which were named “double Danielewski surfaces”. In this note, we determine all the locally nilpotent derivations of a double Danielewski surface.

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.