2,397 reputation
1022
bio website tau.ac.il/~borovoi
location Tel Aviv, Israel
age
visits member for 4 years, 10 months
seen Dec 16 at 19:38

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