Bimultiplications and Annihilators of Crossed Modules in Associative Algebras
Abstract
In this paper, we present a generalization of the concept of the bimultiplication algebra by defining the bimultiplication of crossed modules in associative algebras. Using this structure, we construct the actor of a crossed module in associative algebras, and we obtain annihilators of crossed modules as the kernel of a crossed module morphism from a crossed module to its actor. Moreover, we construct a link between this morphism and a crossed square, two-dimensional crossed module.
Keywords
Bimultiplication crossed module, actor crossed module, annihilator of crossed module, crossed square
References
- S. M. Lane, Extensions and Obstructions for Rings, Illinois Journal of Mathematics 121 (1958) 316-345.
- J. H. C. Whitehead, Combinatorial Homotopy I, Bulletin of the American Mathematical Society 55 (1949) 231-245.
- J. H. C. Whitehead, Combinatorial Homotopy II, Bulletin of the American Mathematical Society 55 (1949) 453-456.
- S. Lichtenbaum, M. Schlessinger, The Cotangent Complex of a Morphism, Transactions of the American Mathematical Society 128(1) (1967) 41-70.
- M. Gerstenhaber, On the Deformation of Rings and Algebras: II, Annals of Mathematics 84(1) (1966) 1-19.
- P. Dedecker, A. S. T. Lue, A Nonabelian Two-Dimensional Cohomology for Associative Algebras, Bulletin of the American Mathematical Society 72(6) (1966) 1044-1050. K. J. Norrie, Actions and Automorphisms of Crossed Modules, Bulletin de la Société Mathématique de France 118 (1990) 129-146.
- A. S. T. Lue, Semi-Complete Crossed Modules and Holomorp hs of Groups, Bulletin of the London Mathematical Society 11 (1979) 8-16.
- Z. Arvasi, U. Ege, Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures 11 (2003) 487-506.
- S. Hürmetli, Bimultiplication of Crossed Modules, PhD Dissertation, Eskişehir Osmangazi University, (2016) Eskişehir, Turkey.
- Y. Boyacı, J. M. Casas, T. Datuashvili, E. Ö. Uslu, Actions in Modified Categories of Interest with Application Crossed Modules, Theory and Applications of Categories 30(25) (2015) 882-908.