We discuss supersymmetric biorthogonal systems, with emphasis given to the periodic solutions that occur at spectral singularities of PT symmetric models. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. We construct in detail some explicit examples for the supersymmetric pairs of potentials V± (z) = −U (z) 2 ±z d dz U (z) where U (z...

This is a review of our previous works [FFR, F1, F3] (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the cent...

The classical r-matrices with carriers gc containing the Borel subalgebra b(g) of simple Lie algebra g are studied. Using the graphical presentation of the dual Lie algebra g(r) we show that such solutions rech of the CYBE always exist. To obtain the explicit form of rech we find the dual coordinates in which the adjoint action of gc can be reduced. This gives us the detailed structure of the J...

We survey the recent investigations on (bounded, sequential) approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We will discuss the core problems concerning these notions and address the signicance of any solutions to them to the development of the eld.

We discuss the notion of generalised U-duality as a solution generating technique in supergravity. demonstrate method to take solutions type IIA supergravity on 3-sphere, with NSNS flux, new 11-dimensional supergravity, using exceptional geometry techniques. These are characterised by an underlying 3-algebra structure, and generalise features obtained non-abelian T-duality, which involve ordina...