Lie conformal algebra

Definition

Let $\C[\partial]$ be the polynomial ring over $\C$ in one variable $\partial$, and let $R$ be a module over $\C[\partial]$. Let $\lambda$ and $\mu$ be further variables, and write $\C[\lambda] \otimes R$ for the polynomials in $\lambda$ with coefficients in $R$. A Lie conformal algebra structure on $R$ is a $\C$-bilinear map $R \times R \to \C[\lambda] \otimes R$, written $(a, b) \mapsto [a_\lambda b]$, such that for all $a, b, c \in R$:

1. $[\partial a_\lambda b] = -\lambda [a_\lambda b]$ and $[a_\lambda \partial b] = (\partial + \lambda) [a_\lambda b]$;
2. $[b_\lambda a] = -[a_{-\lambda - \partial} b]$;
3. $[a_\lambda [b_\mu c]\,] = [\,[a_\lambda b]_{\lambda + \mu} c] + [b_\mu [a_\lambda c]\,]$.

In clause 2, if $[a_\lambda b] = \sum_j \lambda^j c_j$ with $c_j \in R$, then $[a_{-\lambda - \partial} b]$ means $\sum_j (-\lambda - \partial)^j c_j$. Clause 3 is an identity of polynomials in $\lambda$ and $\mu$ with coefficients in $R$.