Unitary representations of

$W$ infinity algebras.

*(English)* Zbl 0972.81554
Summary: We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of $W$ infinity algebras $({W}_{\infty},{W}_{1+\infty},{W}_{\infty}^{1,1},{W}_{\infty}^{M},{W}_{1+\infty}^{N},{W}_{\infty}^{M,N})$ with central charges $(2,1,3,2M,N,2M+N)$. The characters of these representations are computed. We contruct a new extended superalgebra ${W}_{\infty}^{M,N}$, whose bosonic sector is ${W}_{\infty}^{M}\otimes {W}_{1+\infty}^{N}$. Its representations obtained from a free field realization with central charge $2M+N$, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for $M=N$.

##### MSC:

81R10 | Infinite-dimensional groups and algebras motivated by physics |

17B68 | Virasoro and related algebras |

81T40 | Two-dimensional field theories, conformal field theories, etc. |