@article{mainellis_2022, title={Multipliers and unicentral Leibniz algebras}, volume={21}, ISSN={["1793-6829"]}, DOI={10.1142/S0219498823500081}, abstractNote={ In this paper, we prove Leibniz analogues of results found in Peggy Batten’s 1993 dissertation. We first construct a Hochschild–Serre-type spectral sequence of low dimension, which is used to characterize the multiplier in terms of the second cohomology group with coefficients in the field. The sequence is then extended by a term and a Ganea sequence is constructed for Leibniz algebras. The maps involved with these exact sequences, as well as a characterization of the multiplier, are used to establish criteria for when a central ideal is contained in a certain set seen in the definition of unicentral Leibniz algebras. These criteria are then specialized, and we obtain conditions for when the center of the cover maps onto the center of the algebra. }, number={12}, journal={JOURNAL OF ALGEBRA AND ITS APPLICATIONS}, author={Mainellis, Erik}, year={2022}, month={Dec} }
 @article{mainellis_2022, title={Multipliers and unicentral diassociative algebras}, ISSN={["1793-6829"]}, DOI={10.1142/S0219498823501074}, abstractNote={ This paper details the diassociative analogue of results concerning the Schur multiplier and other extension-theoretic concepts that originate in group theory. We first prove that covers of diassociative algebras are unique. Second, we show that the multiplier of a diassociative algebra is characterized by the second cohomology group with coefficients in the field. Third, we establish criteria for when the center of a cover maps onto the center of the algebra. Along the way, we obtain a collection of exact sequences, characterizations, and a brief theory of unicentral diassociative algebras and stem extensions. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras. }, journal={JOURNAL OF ALGEBRA AND ITS APPLICATIONS}, author={Mainellis, Erik}, year={2022}, month={Feb} }
 @article{mainellis_2021, title={Nonabelian extensions and factor systems for the algebras of Loday}, ISSN={["1532-4125"]}, DOI={10.1080/00927872.2021.1939044}, abstractNote={Abstract Factor systems are a tool for working on the extension problem of algebraic structures such as groups, Lie algebras, and associative algebras. Their applications are numerous and well-known in these common settings. We construct algebra analogues to a series of results from W. R. Scott’s Group Theory, which gives an explicit theory of factor systems for the group case. Here ranges over Leibniz, Zinbiel, diassociative, and dendriform algebras, which we dub “the algebras of Loday,” as well as over Lie, associative, and commutative algebras. Fixing a pair of algebras, we develop a correspondence between factor systems and extensions. This correspondence is strengthened by the fact that equivalence classes of factor systems correspond to those of extensions. Under this correspondence, central extensions give rise to 2-cocycles while split extensions give rise to (nonabelian) 2-coboundaries.}, journal={COMMUNICATIONS IN ALGEBRA}, author={Mainellis, Erik}, year={2021}, month={Jun} }