‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

This article being a continuation of the first part [1] is addressed to specialists in representation theory, infinite dimensional geometry, quantum algebra, mathematical physics and informatics of interactive systems. The interrelations of the infinite dimensional geometry of the homogeneous Kähler manifold M = Diff+(S )/S (see e.g. [2] and numerous refs wherein), its quotient M1 = Diff+(S )/P...

A central topic in representation theory is to understand the irreducible modules. In this paper we are concerned with nite dimensional irreducible modules of quantized hy-peralgebras and of hyperalgebras. Recently, big progress has been made on the characters of the irreducible modules, namely the Lusztig's conjectural formula for the characters is true for quantized hyperalgebras and is true ...

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We give a geometric categorification of the Verma modules Mpλq for quantum sl2. CONTENTS