A new class of infinite-dimensional Lie algebras: an analytical continuation of the arbitrary finite-dimensional semisimple Lie algebra
The authors construct for any semisimple finite-dimensional Lie algebra \(\mathfrak g\) a unique infinite-dimensional Lie algebra \(\text{AC}(\mathfrak g)\) which is an analytic continuation of \(\mathfrak g\) from its root system to the root lattice. They show that each finite-dimensional irreducible representation of \(\mathfrak g\) has a unique analytic continuation to an infinite-dimensional representation of \(\text{AC}(\mathfrak g)\). As special cases, they give the Poisson bracket realizations of \(\text{AC}(\mathfrak g)\) for the simple finite-dimensional Lie algebras of the classical series, the analytic continuation of the Lie superalgebra \(\text{osp}(1|2n)\), and higher spin algebras. The case \(\mathfrak g=\text{sp}_2\) gives the Virasoro algebra. The authors believe that the proposed theory may be an algebraic basis for exactly solvable \(D\)-dimensional quantum models.
scientific article; zbMATH DE number 1630713