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A diagrammatic approach to categorification of quantum groups I
, 2009
"... To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph. ..."
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Cited by 182 (18 self)
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To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the KacMoody Lie algebra associated with the graph.
Special transverse slices and their enveloping algebras
, 2002
"... Let G be a simple, simply connected algebraic group over C, g = LieG, N (g) the nilpotent cone in g, and (E,H,F) an sl2triple in g. Let S = E+Ker adF, the special transverse slice to the adjoint orbit Ω of E, and S0 = S ∩ N (g). The coordinate ring C[S0] is naturally graded (see [35]). Let Z(g) be ..."
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Cited by 83 (8 self)
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Let G be a simple, simply connected algebraic group over C, g = LieG, N (g) the nilpotent cone in g, and (E,H,F) an sl2triple in g. Let S = E+Ker adF, the special transverse slice to the adjoint orbit Ω of E, and S0 = S ∩ N (g). The coordinate ring C[S0] is naturally graded (see [35]). Let Z(g) be the centre of the enveloping algebra U(g) and η: Z(g) → C an algebra homomorphism. Identify g with g ∗ via a Killing isomorphism and let χ denote the linear function on g corresponding to E. Following [32] we attach to χ a nilpotent subalgebra mχ ⊂ g of dimension (dim Ω)/2 and a 1dimensional mχmodule Cχ. Let H̃χ denote the algebra opposite to Endg(U(g) ⊗U(mχ) Cχ) and H̃χ,η = H̃χ ⊗Z(g) Cη. It is proved in the paper that the algebra H̃χ,η has a natural filtration such that gr(H̃χ,η), the associated graded algebra, is isomorphic to C[S0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of g.
A categorification of quantum sl(2)
 ADV. MATH
, 2008
"... We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs betwee ..."
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Cited by 67 (9 self)
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We categorify Lusztig’s ˙U – a version of the quantized enveloping algebra Uq(sl2). Using a graphical calculus a 2category ˙ U is constructed whose Grothendieck ring is isomorphic to the algebra ˙ U. The indecomposable morphisms of this 2category lift Lusztig’s canonical basis, and the Homs between 1morphisms are graded lifts of a semilinear form defined on ˙U. Graded lifts of various homomorphisms and antihomomorphisms of U ˙ arise naturally in the context of our graphical calculus. For each positive integer N a representation of U˙ is constructed using iterated flag varieties that categorifies the irreducible (N + 1)dimensional representation of ˙ U.
Formulas For Lagrangian And Orthogonal Degeneracy Loci: The QPolynomials Approach
 COMPOSITIO MATH
, 1996
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THE CLASSIFICATION OF pCOMPACT GROUPS FOR p ODD
, 2003
"... A pcompact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined plocal analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we fi ..."
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Cited by 39 (16 self)
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A pcompact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined plocal analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a onetoone correspondence between connected pcompact groups and finite reflection groups over the padic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as pcompact groups by their Weyl groups seen as finite reflection groups over the padic integers. Our approach in fact gives a largely selfcontained proof of the entire
The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras
 Math. Z
, 2001
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Coadjoint action of a semisimple algebraic group and the center of the enveloping algebra in characteristic p
 Indag. Math
, 1976
"... The aim of the present note is to extend to fields of arbitrary nonzero characteristic the theorem on the connection between characters of the center of the universal enveloping algebra of a classical Lie algebra and weights of irreducible representations (Theorem 2). Unlike known precedents we wo ..."
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Cited by 30 (0 self)
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The aim of the present note is to extend to fields of arbitrary nonzero characteristic the theorem on the connection between characters of the center of the universal enveloping algebra of a classical Lie algebra and weights of irreducible representations (Theorem 2). Unlike known precedents we work completely in characteristic p (and do not use reduction modulo p from characteristic zero, compare [2], [4], [lo]). Hence for characteristics between 2 and the Coxeter number our results are new, at least for algebras of type different from A,. At the same time our approach is not constructive. Three points should be emphasized in the present paper. First, we consider our Lie algebras as Lie algebras of algebraic groups. The action of the corresponding algebraic group gives to the object under study the desired rigidity. Second, our argument, at least at crucial points, is local in the sense that it uses only Lie subalgebras and algebraic subgroups of type Al, normalized by some maximal torus. Third, most assertions below are standard and the only points where a result is obtained by an argument which seems to be not completely standard are Lemmas 4.2, 4.3, 5.2. Let k be an algebraically closed field of characteristic p> 0, 59 be a connected almostsimple algebraic kgroup. For an algebraic group X we denote by H or by Lie 2 the Lie algebra of Z, endowed with the poperation x + z@l. Let now 3? (resp. M+, x, Y) be a Bore1 subgroup of 53 (resp. the maximal unipotent subgroup of 39, a maximal unipotent subgroup opposite to g’, the maximal torus normalizing 93 and M). We suppose (as we can) that all these groups are defined over Hr. Put B=Lie a’, N+=Lie M+, N=Lie Jy^, T=Lie Y. Let W be the Weyl group of B (with respect to Y), and let X(Y) be the character group
INVARIANTS, TORSION INDICES AND ORIENTED COHOMOLOGY OF COMPLETE FLAGS
"... Part 1. Invariants, torsion indices and formal group laws. 4 2. Formal group rings 4 3. Differential operators ∆α and Cα 7 ..."
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Cited by 23 (9 self)
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Part 1. Invariants, torsion indices and formal group laws. 4 2. Formal group rings 4 3. Differential operators ∆α and Cα 7
Descent representations and multivariate statistics
 Trans. Amer. Math. Soc
"... Abstract. Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the GarsiaStanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of ..."
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Cited by 22 (5 self)
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Abstract. Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the GarsiaStanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of the descent basis to type B, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine wellknown decomposition rules of the coinvariant algebra and to generalize various identities. 1.