Shifted Yangians and finite W-algebras

By Jonathan Brundan, Alexander S. Kleshchev. 2004

174 citations recorded by the source corpus.

We give a presentation for the finite W-algebra associated to a nilpotent matrix in the general linear Lie algebra over C. In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l, the presentation is that of the Yangian of level l associated to gln, as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b60c908fd0529d3e3df7b295f70f6c876e1d4f60

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In Algebra and Number Theory

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Citations in the corpus

Commutative quotients of finite W-algebras (2008, background): “Furthermore, if D(l, e0) is the open decomposition class of S(e) and Y is any irreducible component of X, then dim Y = dim z(l).”
The Coulomb Branch of 3d N = 4 Theories (2015, background): “Part of this work was developed during visits of TD (28)Shifted Yangians were first introduced and studied in [79, 80].”
The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras (2010, methodology): “We refer the reader to [7,8] for a description of the structure of the algebra W(π), which includes an analog of the Poincaré–Birkhoff–Witt theorem as well as a construction of algebraically independent generators of the center.”
Finite vs affine W-algebras (2005, background): “More recently there has been a revival of interest in them in connection to geometry and representation theory of simple finite-dimensional Lie algebras, see [M, P1, P2, GG, BrK ].”
On the structure of the category O for W-algebras (2008, background): “It was studied extensively during the last decade starting from Premet’s paper [Pr1], see also [BGK],[ BK1 ],[BK2],[GG],[Gi],[Lo1],[Lo2], [Pr2]-[Pr4].”
Primitive ideals, non-restricted representations and finite W-algebras (2006, background): “The action of m on Cχ = C1χ is given by x(1χ) = χ(x)1χ for all x ∈ m; see [18], [11], [6], [20] for more detail.”
On symmetric invariants of centralisers in reductive Lie algebras (2006, methodology): “The approach in [2] is different from ours; it relies on the earlier work of Brundan–Kleshchev [ 3 ] and employs an argument in the spirit of [24].”
Good grading polytopes (2005, methodology): “Moreover, every integral good grading is adjacent to an even good grading, as was noted already in the introduction of [3].”
Quantized symplectic actions and W -algebras (2007, background): “Brundan and Kleshchev, [ BrKl1 ], [BrKl2], studied the algebras U (g ,e ) for g = sln.”