On discretized generalizations of Krichever-Novikov algebras (Q2747957)

Starting from the meromorphic functions on the complex plane with poles at \(0\) and \(\infty\), and the meromorphic functions doubly-periodic under the lattice \(L=\langle 1,\tau \rangle\) with poles at most at the points lying in the orbit of \(0\) and \(1/2\) under \(L\), a nonassociative algebra is constructed. This is done with the help of a certain differential-difference operator \(T\), which depends on two parameters \(x\) and \(y\). The algebra structure is calculated in terms of certain exhibited basis elements. Subalgebras are studied. In general, the subalgebras obtained will not be Lie algebras. Certain subalgebras that are Lie algebras, are classified. Special cases are considered. Depending on the parameters, the Witt algebra of meromorphic vector fields on \(\mathbb P^1(\mathbb C)\) holomorphic outside \(0\) and \(\infty\), and the Krichever-Novikov algebra on the torus \(\mathbb C/L\), consisting or meromorphic vector fields holomorphic outside \({0}\bmod L\) and \({1/2}\bmod L\) show up again. (They were the basic starting objects). The results are obtained via explicit calculations.NEWLINENEWLINE Two small formal points should be mentioned. Firstly, the systematic use of the misnomer ``Virasoro-De-Witt'' algebra to denote the Witt algebra (or equivalently the Virasoro algebra without central extension). Secondly, that the list of references is not in accordance with the referencing scheme in the article.