Supriya Ghosh (Editor)

Nilpotent cone

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In mathematics, the nilpotent cone N of a finite-dimensional semisimple Lie algebra g is the set of elements that act nilpotently in all representations of g . In other words,

N = { a ∈ g : ρ ( a )  is nilpotent for all representations  ρ : g → End ⁡ ( V ) } .

The nilpotent cone is an irreducible subvariety of g (considered as a k -vector space).

Example

The nilpotent cone of 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.

References

Nilpotent cone Wikipedia
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