In the present contribution, I report on certain { \it non-linear} and { \it non-local} extensions of the conformal (Virasoro) algebra. These so-called $V$-algebras are matrix generalizations of $W$-algebras. First, in the context of two-dimensional field theory, I discuss the non-abelian Toda model which possesses three conserved (chiral) ``currents". The Poisson brackets of these ``currents" give the simplest example of a $V$-algebra. The classical solutions of this model provide a free-field realization of the $V$-algebra. Then I show that this $V$-algebra is identical to the second Gelfand-Dikii symplectic structure on the manifold of $2\times 2$-matrix Schr\"odinger operators $L=-\d^2+U$ (with $\tr\sigma_3 U=0$). This provides a relation with matrix KdV-hierarchies and allows me to obtain an infinite family of conserved charges (Hamiltonians in involution). Finally, I work out the general $V_{ n,m}$-algebras as symplectic structures based on $n\times n$-matrix $m^{ \rm th}$-order differential operators $L=-\d^m +U_2\d^{ m-2}+U_3 \d^{ m-3}+\ldots +U_m$. It is the absence of $U_1$, together with the non-commutativity of matrices that leads to the non-local terms in the $V_{ n,m}$-algebras. I show that the conformal properties are similar to those of $W_m$-algebras, while the complete $V_{ n,m}$-algebras are much more complicated, as is shown on the explicit example of $V_{ n,3}$.
Categories: hep-th