Nilpotent cone

In mathematics, the nilpotent cone N {displaystyle {mathcal {N}}} of a finite-dimensional semisimple Lie algebra g {displaystyle {mathfrak {g}}} is the set of elements that act nilpotently in all representations of g . {displaystyle {mathfrak {g}}.} In other words, In mathematics, the nilpotent cone N {displaystyle {mathcal {N}}} of a finite-dimensional semisimple Lie algebra g {displaystyle {mathfrak {g}}} is the set of elements that act nilpotently in all representations of g . {displaystyle {mathfrak {g}}.} In other words, The nilpotent cone is an irreducible subvariety of g {displaystyle {mathfrak {g}}} (considered as a k {displaystyle k} -vector space). The nilpotent cone of sl 2 {displaystyle operatorname {sl} _{2}} , the Lie algebra of 2×2 matrices with vanishing trace, is the variety of all 2×2 traceless matrices with rank less than or equal to 1. {displaystyle 1.} This article incorporates material from Nilpotent cone on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.