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First cohomology group of rank two Witt algebra to its Larsson modules
Jinglian JIANG, Xiaoli KONG
Front Math Chin. 2011, 6 (4): 671-688.
https://doi.org/10.1007/s11464-011-0143-8
Let U=?2, Γ=?2, and let ?[x1±1,x2±1] be the ring of Laurent polynomials. The Witt algebra ? is the Lie algebra of derivations over ?[x1±1,x2±1], which is spanned by elements of the form D(u, r) = xr(u1d1 + u2d2), u=(u1,u2)∈U, r∈Γ, where d1 and d2 are the degree derivations of ?[x1±1,x2±1]. The image of gl2-module V under Larsson functor Fα, denoted by W=Fα(V), gives a class of ?-modules, often called the Larsson-modules of ?. In this paper, we study the derivations from the Witt algebra ? to its Larsson-modules W, and we determine the first cohomology group H1(?, W).
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Path realization of crystal B(∞)
Bin LI, Hechun ZHANG
Front Math Chin. 2011, 6 (4): 689-706.
https://doi.org/10.1007/s11464-010-0073-x
A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equivalence classes of paths. With these induced operators, an explicit realization of B(∞) is given in terms of equivalence classes of paths, where B(∞) is the crystal base of the negative part of a quantum group Uq(g). Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when g is of finite type.
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