Lie conformal algebra
# Lie conformal algebra

A Lie conformal algebra is in some sense a generalization of a Lie algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category. Lie conformal algebras are very closely related to vertex algebras and have many applications in other areas of algebra and integrable systems.

## Definition and relation to Lie algebras

A Lie algebra is defined to be a vector space with a skew symmetric bilinear multiplication which satisfies the Jacobi identity. More generally, a Lie algebra is an object, $L$ in the category of vector spaces (read: $\mathbb {C}$-modules) with a morphism

    $[\cdot ,\cdot ]:L\otimes L\rightarrow L$

that is skew-symmetric and satisfies the Jacobi identity. A Lie conformal algebra, then, is an object $R$ in the category of $\mathbb {C} [\partial ]$-modules with morphism

    $[\cdot _{\lambda }\cdot ]:R\otimes R\rightarrow \mathbb {C} [\lambda ]\otimes R$

called the lambda bracket, which satisfies modified versions of bilinearity, skew-symmetry and the Jacobi identity:

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

One can see that removing all the lambda's, mu's and partials from the brackets, one simply has the definition of a Lie algebra.

## Examples of Lie conformal algebras

A simple and very important example of a Lie conformal algebra is the Virasoro conformal algebra. Over $\mathbb {C} [\partial ]$ it is generated by a single element $L$ with lambda bracket given by

    $[L_{\lambda }L]=(2\lambda +\partial )L.\,$

In fact, it has been shown by Wakimoto that any Lie conformal algebra with lambda bracket satisfying the Jacobi identity on one generator is actually the Virasoro conformal algebra.

## Classification

It has been shown that any finitely generated (as a $\mathbb {C} [\partial ]$-module) simple Lie conformal algebra is isomorphic to either the Virasoro conformal algebra, a current conformal algebra or a semi-direct product of the two.

There are also partial classifications of infinite subalgebras of ${\mathfrak {gc}}_{n}$ and ${\mathfrak {cend}}_{n}$.

