In this article, we classify all 2-term L∞-algebras up to isomorphism.
What are 2-term L∞-algebras?
Mathematics and physics are two vast areas of research and, since ages, they have been intertwined: physicists use mathematics to describe their models of their surrounding nature, and mathematicians draw their inspiration from the concepts physicists introduce. One of the places where intertwining occurs is classical mechanics.
In classical mechanics, physicists study how a point particle behaves under the influence of the forces of nature. This is described by Hamilton's equations of motion. These equations of motion are governed by the Poisson bracket on the functions of the given configuration space. In particular, this Poisson bracket is a Lie bracket too.
If we replace point particles by strings and try to repeat the above construction, the analogue of the former Poisson bracket needs not be a Lie bracket any more: it could fail to be antisymmetric, or it could fail to satisfy the Jacobi identity. This amount of failure therefore needs to be accounted for by adding extra spaces which contain the additional information. If the analogue of the former Poisson bracket is still antisymmetric, the new structure that describes the dynamics is no longer a Lie algebra, but rather a 2-term L∞-algebra [1].
Why do we want to classify them?
In order to understand 2-term L∞-algebras, it is useful to know which different 2-term L∞-algebras are isomorphic. This means that there is a structure-preserving map (which is called a morphism) from the one to the other and from the other to the one, such that both compositions yields identity morphisms. If two 2-term L∞-algebras are isomorphic, they share all defining characteristics and can be treated as the same.
Classifying 2-term L∞-algebras thus amounts to proving if between different 2-term L∞-algebras there is or there cannot be an isomorphism. If we know such a classification, it can simplify future research. Statements about 2-term L∞-algebras that are preserved under isomorphism can be proven by only considering one 2-term L∞-algebra from each isomorphism class. As the isomorphism classes are (at least here) purposefully designed to have a simple representative calculations should become much quicker.
The article is available on: https://arxiv.org/abs/2109.10202
[1] John C. Baez, Alexander E. Hoffnung, and Christopher L. Rogers. Categorified symplectic geometry and the classical string. Communications in Mathematical Physics, 293(3):701– 725, 2009.