bio | website | tau.ac.il/~borovoi |
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location | Tel Aviv, Israel | |
age | ||
visits | member for | 4 years, 10 months |
seen | Dec 16 at 19:38 | |
stats | profile views | 1,889 |
I am a professor at Tel Aviv University, interested in linear algebraic groups, homogeneous spaces, number theory and nonabelian cohomology.
Nov 22 |
comment |
Reducible reductive Lie subalgebras of so(p,q)
@user49908: Corrected! |
Nov 22 |
revised |
Reducible reductive Lie subalgebras of so(p,q)
added 2 characters in body |
Nov 21 |
awarded | Explainer |
Nov 21 |
revised |
Reducible reductive Lie subalgebras of so(p,q)
Tags added, grammar corrected |
Nov 21 |
answered | Reducible reductive Lie subalgebras of so(p,q) |
Nov 20 |
comment |
Reductive subgroup and its derived subgroup with an irreducible represenation
@user49908: If you ask a new question titled, say, "Irreducible reductive Lie subalgebras of $so(p,q)$", I will answer it. |
Nov 20 |
comment |
Reductive subgroup and its derived subgroup with an irreducible represenation
@user49908: Yes, it is true that the only connected non-semisimple reductive $\mathbb{R}$-subgroup $H$ of full rank of the group $G=SO(p,q)$, that satisfies the $\mathbb{R}$-irreducibility property, is, up to conjugacy, $U(p/2,q/2)$. This follows from Dynkin's classification of maximal reductive $\mathbb{C}$-Lie-subalgebras of simple $\mathbb{C}$-Lie-algebras. Dynkin was our contemporary, he passed away on November 14, 2014. |
Oct 17 |
answered | Witt index of the sum of 24 squares |
Oct 17 |
asked | Witt index of the sum of 24 squares |
Sep 24 |
awarded | Autobiographer |
Jul 26 |
comment |
Rational structures on the flag variety over a finite field
@user148212: I continue. Since the action of the Frobenius $F$ on any point $x\in X$ is uniquely determined, and our variety $X$ is reduced, the action of $F$ on the structure sheaf of $X$ is uniquely determined, thus the rational structure on $X$ is uniquely determined - if it exists. The existence of a rational structure can be proved by an explicit construction, see my previous comment. |
Jul 26 |
comment |
Rational structures on the flag variety over a finite field
@user148212: I add details. Write $x_0=eB\in X$. Since it is a rational point, we have $F(x_0)=x_0$, where $F$ denotes the Frobenius. Now consider any point $x=gB\in G/B$ where $g\in G$, then $x=g x_0$. Since we want the action of $G$ on $X$ to be rational, we have $F(x)=F(g x_0)=F(g) F(x_0)=F(g) x_0$. Since the rational structure on $G$ is given, the element $F(g)\in G$ is uniquely determined. Since the $G$-action on $X$ is given, the product $F(g) x_0$ is uniquely determined. Thus $F(x)$ is uniquely determined. |
Jul 26 |
comment |
Rational structures on the flag variety over a finite field
@user148212: Yes, the choice of a rational point and of a G-structure on a homogeneous space implies the choice of a rational structure. |
Jul 25 |
comment |
Rational structures on the flag variety over a finite field
@user148212: Note that the reference to Theorem 12.2.1 in Springer's book works when the Frobenius in $G$ comes from a structure of an algebraic $\mathbb{F}_q$-groups (not in the case when you deal with Suzuki groups or Ree groups). |
Jul 24 |
comment |
Rational structures on the flag variety over a finite field
@user148212: On your variety $X=G/B$ you already have a canonical structure of $G$-variety (over $\overline{F}_q$) and a canonical rational point $eB=B$, the image of the unit element $e$ of $G$. Therefore, the $F_q$-rational structure on $X$ is unique, if exists. What is not evident is the existence of an $F_q$-rational structure. This can be proved by an explicit construction of a closed (in your case) subvariety in a projective space, see the proof of Corollary 5.5.4 in Springer's book mentioned by Daniel Loughran. |
Jul 4 |
awarded | Inquisitive |
Jul 3 |
comment |
What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
@user52824: Thank you! As I expected, my questions reduce to linear algebra. Now what is the induced quadratic (resp., Hermitian) form on $\det(V)$? Could you please either explain or give references? I would be grateful if you could write an answer based on your comment. I will accept it immediately. |
Jul 3 |
revised |
What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
edited title |
Jul 3 |
asked | What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$? |
Jul 2 |
awarded | Curious |