has not been solved. Logarithmic conformal field theory At central charge , the Virasoro algebra can be extended by a triplet of generators of dimension , thus forming a W-algebra with the set of dimensions . Then it is possible to build a rational conformal field theory based on this W-algebra, which is logarithmic. The simplest case is obtained for , has central charge , and has been particularly well studied, including in the presence of a boundary. Related concepts Classical W-algebras Finite W-algebras Finite W-algebras are certain associative algebras associated to nilpotent elements of semisimple Lie algebras. The original definition, provided by Alexander Premet, starts with a pair consisting of a reductive Lie algebra over the complex numbers and a nilpotent element e. By the Jacobson-Morozov theorem, e is part of a sl2 triple (e, h, f). The eigenspace decomposition of ad(h) induces a -grading on : Define a character (i.e. a homomorphism from to the trivial 1-dimensional Lie algebra) by the rule , where denotes the Killing form. This induces a non-degenerate anti-symmetric bilinear form on the −1 graded piece by the rule: After choosing any Lagrangian subspace , we may define the following nilpotent subalgebra which acts on the universal enveloping algebra by the adjoint action. The left ideal of the universal enveloping algebra generated by is invariant under this action. It follows from a short calculation that the invariants in under ad inherit the associative algebra structure from . The invariant subspace is called the finite W-algebra constructed from , and is usually denoted . References Further reading Conformal field theory Integrable systems Representation theory