Nilpotent cone
From HandWiki
In mathematics, the nilpotent cone [math]\displaystyle{ \mathcal{N} }[/math] of a finite-dimensional semisimple Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is the set of elements that act nilpotently in all representations of [math]\displaystyle{ \mathfrak{g}. }[/math] In other words,
- [math]\displaystyle{ \mathcal{N}=\{ a\in \mathfrak{g}: \rho(a) \mbox{ is nilpotent for all representations } \rho:\mathfrak{g}\to \operatorname{End}(V)\}. }[/math]
The nilpotent cone is an irreducible subvariety of [math]\displaystyle{ \mathfrak{g} }[/math] (considered as a vector space).
Example
The nilpotent cone of [math]\displaystyle{ \operatorname{sl}_2 }[/math], 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 [math]\displaystyle{ 1. }[/math]
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, https://books.google.com/books?id=oqx-p6N2XSEC&pg=PA173.
- Anker, Jean-Philippe; Orsted, Bent (2006), Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics, 229, Birkhäuser, p. 166, ISBN 9780817644307, https://books.google.com/books?id=nc__yHsnHFQC&pg=PA166.
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