Nilpotent cone explained
In mathematics, the nilpotent cone
of a finite-dimensional
semisimple Lie algebra
is the set of elements that act nilpotently in all
representations of
In other words,
l{N}=\{a\inak{g}:\rho(a)isnilpotentforallrepresentations\rho:ak{g}\to\operatorname{End}(V)\}.
The nilpotent cone is an irreducible subvariety of
(considered as a
vector space).
Example
The nilpotent cone of
, 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
References
