We find sufficient conditions for principal toric bundles over compact Kähler manifolds to admit Calabi-Yau connections with torsion as well as conditions to admit strong Kähler connections with torsion. With the aid of a topological classification, we construct such geometry on (k − 1)(S 2 × S 4 )#k(S 3 × S 3 ) for all k ≥ 1.
We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional gl(m|n)-modules which are isomorphic to the parabolically induced (Kac) modules.
Preliminaries2.1. Weight modules. A Z 2 -graded vector space g = g0 ⊕ g1 with even bracket [•, •] : g ⊗ g → g is a Lie superalgebra iff the following conditions hold [a, b] = −(−1) p(a)p(b) [b, a]; Berkeley
Abstract. In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U q (q(n))-module V (λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted LittlewoodRichardson coefficients.
Abstract. In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra Uq(q(n)). The key ingredients are the triangular decomposition of Uq(q(n)) and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for Uq(q(n))-modules in the category O ≥0 q .
Abstract. In this paper, we develop the crystal basis theory for the quantum queer superalgebra U q (q(n)). We define the notion of crystal bases and prove the tensor product rule for U q (q(n))-modules in the category O
≥0int . Our main theorem shows that every U q (q(n))-module in the category O ≥0 int has a unique crystal basis.
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