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comment A finiteness property for semi-simple algebraic groups
@YCor: A modern exposition of Dynkin's results on maximal subalgebras of semisimple Lie algebras is given in Chapter 6 of the survey: books.google.fr/books/about/…
Apr
25
comment Rationally connected spaces over non-algebraically-closed fields
You probably mean that $k$ is algebraically closed (rather than algebraically complete).
Apr
18
answered Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)
Apr
18
comment Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)
No. Take $G=\mathrm{G}_{m,\mathbb{Q}}\cdot R_{F/\mathbb{Q}}\mathrm{SL}_{2,F}$, where $F$ is a totally real extension of $\mathbb{Q}$. Here $g=[F:\mathbb{Q}]$.
Apr
12
comment Automorphism group of real orthogonal Lie groups
You should edit your answer rather than write a new answer and again a new answer....
Mar
23
comment Connected subgroups of $SL(2,C)$
There are three classes of connected complex algebraic subgroups, up to conjugacy: the maximal torus $T$ consisting of all diagonal matrices, the unipotent subgroup $U$ consisting of upper triangular matrices with 1 on the diagonal, and the Borel subgroup $B=TU$ of all upper triangular matrices.
Mar
6
comment Conjugation of group extensions
We wrote a joint note based on this answer: M. Borovoi and Y. Cornulier, Conjugate complex homogeneous spaces with non-isomorphic fundamental groups. It was published in C. R. Acad. Sci. Paris, Ser. I, 353 (2015) 1001–1005.
Mar
3
awarded  Quorum
Feb
23
comment Centralizer of a dense subgroup in a maximal subgroup of a reductive group
The centralizer of $H$ in $G$ is equal to the centralizer of $K$ in $G$ (because $H$ is dense in $K$), which is in turn is equal to the centralizer of of $G$ in $G$ (because $K$ is Zariski-dense in $G$, as @YCor has mentioned), that is, to the center of $G$. In general the center of $G$ is not equal to the centralizer of $H$ in $K$ (take $G=\mathbb{C}^*$ and let $H=K$ be the maxiamal compact subgroup of $\mathbb{C}^*$).
Feb
21
awarded  Yearling
Feb
20
comment Simple lie algebras, (almost-)simple groups of Lie type
@HAHelfgott: In any case you should try to specify what you mean by a subvariety (smooth subscheme?).
Feb
20
comment Simple lie algebras, (almost-)simple groups of Lie type
@HAHelfgott: Concerning the current version of the question, after "What I really need is the following": I am not expert, hopefully Jim Humphreys can help. Otherwise you will have to read Steinberg's proof and Hogeweij's counterexamples.
Feb
20
comment Simple lie algebras, (almost-)simple groups of Lie type
@HAHelfgott: This is not an extra dimension. $\mathrm{dim}_K\,\mathfrak{sl}(3,K)=8$ in any characteristic, in particular in characteristic 3.
Feb
19
comment Simple lie algebras, (almost-)simple groups of Lie type
@HAHelfgott: According to this review Steinberg proved that in characteristic $>3$ such a Lie algebra is simple modulo center.
Feb
19
comment Simple lie algebras, (almost-)simple groups of Lie type
@HAHelfgott: Yes, it still occurs! Consider the scalar matrix $\mathrm{diag}(1,1,1)$, in characteristic 3 it has trace 0, hence it is contained in the Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,K)$, and it is clearly central. Therefore, $\mathfrak{g}$ is not simple.
Feb
18
comment If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v?
Lemma 4.9 on page 18 of the following paper will partially help you with (1): Springer, T. A. Reductive groups. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 3–27, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.
Feb
18
comment Simple lie algebras, (almost-)simple groups of Lie type
@HAHelfgott: This is not true in characteristics 2 and 3, see the two papers by Hogeweij.
Feb
17
comment Simple lie algebras, (almost-)simple groups of Lie type
This is so simple!
Feb
16
comment Simple lie algebras, (almost-)simple groups of Lie type
You can find proofs in the book by Gunter Malle and Donna Testerman, Section 24.2.
Feb
16
revised Simple lie algebras, (almost-)simple groups of Lie type
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