Non-associative Algebras of Cubic Matrices and their Gauge Theories

Motivated by M-theory, we define a new type of non-associative algebra involving usual and cubic matrices at the same time. The resulting algebra can be regarded as a two-term truncated $L_\infty$ algebra giving rise to a fundamental identity between the two- and the three-bracket. We provide a simple class of concrete examples of such algebras based on the structure constants of a Lie algebra. Connecting to previous results on higher structures, we generalize the construction of Yang-Mills theories, topological BF theory and generalized IKKT models and point out some appearing issues.