Nilpotent cone

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

${\displaystyle {\mathcal {N}}=\{a\in {\mathfrak {g}}:\rho (a){\mbox{ is nilpotent for all representations }}\rho :{\mathfrak {g}}\to \operatorname {End} (V)\}.}$

The nilpotent cone is an irreducible subvariety of ${\mathfrak {g}}$ (considered as a $k$-vector space).

Example

The nilpotent cone of ${\displaystyle \operatorname {sl} _{2}}$, 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 $1.$

References

• Aoki, T.; Majima, H.; Takei, Y.; Tose, N. (2009), Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics, Springer, p. 173, ISBN 9784431732402.
• Anker, Jean-Philippe; Orsted, Bent (2006), Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics, 229, Birkhäuser, p. 166, ISBN 9780817644307.

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