L infty algebra

attachments/Pasted%20image%2020220326235547.png

As an alternative to \(Q{\hbox{-}}\)manifolds:

attachments/Pasted%20image%2020220213203336.png

The role of zero locus of \(Q\) is played by the space of solutions of Maurer-Cartan (MC) equation: \begin{align*} \sum_{n} \frac{1}{n !} \mu_{n}(a, \ldots, a)=0 . \end{align*}
- For A_infty algebras, the role of equations of motion is played by the MC equation \begin{align*} \sum_{n \geq 1} \mu_{n}(a, \ldots, a)=0 \end{align*}
An \(L_{\infty}\) algebra specifies a classical system and \(L_{\infty}\) algebra with invariant odd inner product specifies a Lagrangian classical system.
A \(\mathbb{Z}\)-graded \(L_{\infty}\)-algebra in BV formalism corresponds to the case when the fields are classified according to the ghost number.
An \(\mathrm{L}_{\infty}\) algebra where all operations \(\mu_{n}\) with \(n \geq 3\) vanish can be identified with differential graded Lie algebra (the operation \(\mu_{1}\) is the differential, \(\mu_{2}\) is the bracket).
An \(\mathrm{L}_{\infty}\) algebra corresponding to Lie algebra with zero differential is \(\mathbb{Z}\)-graded.
Considering gauge theories for all \({\operatorname{U}}_n\) at the same time: it is more convenient to work with \(\mathrm{A}_{\infty}\) instead of \(\mathrm{L}_{\infty}\) algebras. See A_infty.

Derived brackets

attachments/Pasted%20image%2020220327000218.png