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논문검색

Information Center for Mathematical Science

논문검색

Duke Mathematical Journal
( Vol. 116 NO.3 / (2003))
Tensor Product Varieties and Crystals : the Ade Case
Anton Malkin,
Pages. 477-524
Abstract
Contents 0. Introduction
1. Crystals
1.1 Weights and roots
1.2 Definition of $\mathfrak{g}$-crystals
1.3 Tensor product of $\mathfrak{g}$-crystals
1.4 Highest-weight crystals and closed families
1.5 Uniqueness theorem
2. Nakajima varieties and tensor product varieties
2.1 Oriented graphs and path algebras
2.2 Quiver varieties
2.3 Stability and $G_V$-action
2.4 The variety $\mathfrak{M}^s(d, v_0, v)$
2.5 The set $\mathscr{M}(d, v_0, v)$
2.6 Tensor product varieties and multiplicity varieties
2.7 Inductive construction of tensor product varieties
2.8 Dimensions of tensor product and multiplicity varieties
2.9 Irreducible components of tensor product varieties
2.10 The first bojection for a tensor product variety
2.11 The multiplicity variety for two multiples
2.12 The first bijection for a multiplicity variety
2.13 The second bijection
2.14 The tensor decomposition bijection
3. Levi restriction and the crystal structure on quiver varieties
3.1 A subquiver ${\it Q'$
3.2 The set $_{\it QQ'}\mathscr{M}(d, v_0, v)$
3.3 Levi restriction
3.4 Levi restriction and the second bijection for tensor product varieties
3.5 Levi restriction and the first bijection for tensor product varieties
3.6 Levi restriction and the tensor product decomposition
3.7 Digression : the ${\mathfrak sl}_2$-case
3.8 A crystal structure on $_{\it Q}\mathscr{M}(d, v_0)$
3.9 The main theorem
3.10 Corollary
3.11 The extended Lie algebra $\mathfrak{g'}$
A. Appendix. Another description of multiplicity varieties and the tensor product diagram
A.1 Lusztig's description of the variety $\mathfrak{M}^{s, *s}_{D,V}$
A.2 Multiplicity varieties
A.3 The tensor product diagram
Key words
Mathmatical Subject Classification 20G99