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In our paper [KR] we began a systematic study of representations of the universal central extension[InlineEquation not available: see fulltext.] of the Lie algebra of differential operators on the circle.
      
We express them in terms of generatorsEij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities.
      
In the case of 4-dimensional anticommutative algebras a construction is given that links the associated cubic surface and the 27 lines on it with the structure of subalgebras of the algebra.
      
Reductive group actions on affine quadrics with 1-dimensional quotient: Linearization when a linear model exists
      
We study reductive group actions on complex affine quadrics.
      
We prove that if a reductive group action on an affine quadric with a 1-dimensional quotient has a linear model, then the action is linearizable.
      
Methods are developed for the calssification of homogeneous Riemannian hypersurfaces and the classification of linear transitive reductive algebraic group actions on pseudo-Riemannian hypersurfaces.
      
Whenever the action of a maximal torus on the coneCλ* has some nice properties, we obtain simple closed formulas for all weight multiplicities and theirq-analogs in the representationsVnλ,n∈?.
      
If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1).
      
We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety.
      
On the other hand, there is a locally trivialGa-action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).
      
As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genusg is C-M.
      
We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations.
      
Xi defined a partition ofWf into canonical right cells and the right order ≤R on the set of cells.
      
In this paper, we prove three types of rigidity results related to CAT(-1) spaces, namely the rigidity of the isometric actions on CAT(-1) spaces under the commensurability subgroups, the higher rank lattices and certain ergodic cocycles.
      
The main idea for our approach relies on a study of the boundary theory we established for the general CAT(-1) spaces.
      
There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators.
      
We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure.
      
In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsnt in an affine Lie algebra.
      
The quantum toroidal action on the Fock space depends on a certain parameter κ.
      
 

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