Infinite-dimensional Lie bialgebras and Manin pairs

Abstract: This PhD thesis is devoted to the theory of infinite-dimensional Lie bialgebra structures as well as their close relatives such as r-matrices and Manin pairs. The thesis is based on three papers. Paper I. The standard structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. We obtain a full classification of the induced twisted Lie bialgebra structures in terms of Belavin-Drinfeld quadruples. First, we prove that the induced structures are pseudo quasi-triangular. Then, using the algebro-geometric theory of the classical Yang-Baxter equation (CYBE), we reduce the problem of classification to the well-known Belavin-Drinfeld list of trigonometric solutions. Paper II. We classify topological Lie bialgebra structures on the Lie algebra of Taylor series g[[x]], where g is a simple Lie algebra over an algebraically closed field F of characteristic 0. We formalize the notion of a topological Lie bialgebra and introduce topological analogues of Manin pairs, Manin triples, Drinfeld doubles and twists. By relating topological Manin pairs with trace extension of F[[x]] we obtain their complete classification. The classification of topological doubles, which was known before, becomes a special case of the classification of Manin pairs. The classification of doubles tells us that there are only three non-trivial doubles over g[[x]], namely g((x))×(g[x]/x^n g[x]), n ∈ {0, 1, 2}. We prove that topological Lie bialgebra structures on g[[x]] are in one-to-one correspondence with Lagrangian Lie subalgebras of these doubles complementary to the diagonal embedding Δ of g[[x]]. The classification of topological Lie bialgebra structures is then obtained by associating the corresponding Lagrangian subalgebras with algebro-geometric datum. When the underlying field F is the field of complex numbers, the classification becomes explicit. Paper III. In this paper we associate arbitrary subspaces of g((x))×(g[x]/x^n g[x]) complementary to Δ with so-called series of type (n, s). We prove that skew-symmetric (n, s)-type series are in bijection with Lagrangian subspaces and topological quasi-Lie bialgebra structures on g[[x]]. We classify all quasi-Lie bialgebra structures using the classification of Manin pairs from Paper II. We show that series of type (n, s), solving the generalized CYBE, correspond to Lie subalgebras.