TY - UNPB

T1 - Complete reducibility for Lie subalgebras and semisimplification

AU - Bate, Michael

AU - Böhm, Sören

AU - Martin, Benjamin

AU - Roehrle, Gerhard

AU - Voggesberger, Laura

N1 - Acknowledgments: The research of this work was supported in part by the DFG (Grant #RO 1072/22-1 (project number: 498503969) to G. R¨ohrle).

PY - 2023/5/1

Y1 - 2023/5/1

N2 - Let $G$ be a connected reductive linear algebraic group over a field $k$. Using ideas from geometric invariant theory, we study the notion of $G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the Lie algebra $\mathfrak g = Lie(G)$ of $G$ and prove some results when $\mathfrak h$ is solvable or $char(k)= 0$. We introduce the concept of a $k$-semisimplification $\mathfrak h'$ of $\mathfrak h$; $\mathfrak h'$ is a Lie subalgebra of $\mathfrak g$ associated to $\mathfrak h$ which is $G$-completely reducible over $k$. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $\mathfrak h'$ is unique up to $Ad(G(k))$-conjugacy in $\mathfrak g$. Moreover, we prove that the two concepts are compatible: for $H$ a closed subgroup of $G$ and $H'$ a $k$-semisimplification of $H$, the Lie algebra $Lie(H')$ is a $k$-semisimplification of $Lie(H)$.

AB - Let $G$ be a connected reductive linear algebraic group over a field $k$. Using ideas from geometric invariant theory, we study the notion of $G$-complete reducibility over $k$ for a Lie subalgebra $\mathfrak h$ of the Lie algebra $\mathfrak g = Lie(G)$ of $G$ and prove some results when $\mathfrak h$ is solvable or $char(k)= 0$. We introduce the concept of a $k$-semisimplification $\mathfrak h'$ of $\mathfrak h$; $\mathfrak h'$ is a Lie subalgebra of $\mathfrak g$ associated to $\mathfrak h$ which is $G$-completely reducible over $k$. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $\mathfrak h'$ is unique up to $Ad(G(k))$-conjugacy in $\mathfrak g$. Moreover, we prove that the two concepts are compatible: for $H$ a closed subgroup of $G$ and $H'$ a $k$-semisimplification of $H$, the Lie algebra $Lie(H')$ is a $k$-semisimplification of $Lie(H)$.

KW - Semisimplification

KW - G-complete reducibility

KW - geometric invariant theory

KW - rationality

KW - cocharacter-closed orbits

KW - degeneration of G-orbits

M3 - Preprint

BT - Complete reducibility for Lie subalgebras and semisimplification

PB - ArXiv

ER -