The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all $H$-reduced $\hat{ \cal G}_{ r}$-current algebras belong to one of the following basic algebraic structures: local quadratic $W$-algebras, rational $U$-algebras, nonlocal $V$-algebras, nonlocal quadratic $WV$-algebras and rational nonlocal $UV$-algebras. The main new features of the quantum $V$-algebras and their heighest weight representations are demonstrated on the example of the quantum $V_{ 3}^{ (1,1)}$-algebra.
Categories: hep-th