- Source: Nilpotent cone
- Nilpotent cone
- Nilpotent orbit
- Split-quaternion
- Carnot group
- Glossary of Lie groups and Lie algebras
- Gromov–Hausdorff convergence
- Locally nilpotent derivation
- Quasi-isometry
- Hypercomplex number
- Smooth morphism
Artikel: Nilpotent cone GudangMovies21 Rebahinxxi
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,
N
=
{
a
∈
g
:
ρ
(
a
)
is nilpotent for all representations
ρ
:
g
→
End
(
V
)
}
.
{\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
g
{\displaystyle {\mathfrak {g}}}
(considered as a vector space).
Example
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.}
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, vol. 229, Birkhäuser, p. 166, ISBN 9780817644307.
This article incorporates material from Nilpotent cone on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.