2,865 reputation
1123
bio website tau.ac.il/~borovoi
location Tel Aviv, Israel
age
visits member for 5 years, 3 months
seen 4 hours ago

I am a professor at Tel Aviv University, interested in linear algebraic groups, homogeneous spaces, number theory and nonabelian cohomology.


2d
comment Localized at $p$ integral representations of finite elementary $p$-groups
@GeoffRobinson You are right! Thank you!
May
24
comment Correspondence between real forms and real structures on complex Lie groups
Explanation: by an anti-holomorphic map I mean a semi-algebraic anti-holomorphic map, i.e., given by polynomials in $\bar g_{i,j}$.
May
24
comment Localized at $p$ integral representations of finite elementary $p$-groups
There are infinitely many isomorphism classes for $n>2$. Anyway, thank you for the link, I did not know it.
May
24
comment Correspondence between real forms and real structures on complex Lie groups
Serre proves the existence of descent (the map from (B) to (A) ) also for quasi-projective varieties (not necessarily affine) and for all algebraic groups (not necessarily affine), see Corollary 2 in Serre's book. For not quasi-projective varieties descent is not always possible, even for $\mathbb{C}/\mathbb{R}$.
May
24
comment Correspondence between real forms and real structures on complex Lie groups
This is called "Galois descent", a reference is Serre's book "Algebraic Groups and Class Fields", Ch. V-20, Prop. 12 (page 142 of Russian edition). This is a reference for any Galois extension $k_1/k$. You should take $k=\mathbb{R}$, $k_1=\mathbb{C}$.
May
24
comment Correspondence between real forms and real structures on complex Lie groups
This is a bijection between the isomorphism classes of: (A) affine real algebraic groups $G_0$ (resp. affine real varieties) and (B) affine complex algebraic groups $G$ (resp. varieties) endowed with a anti-holomorphic involutive automorphism $S$. The map from (A) to (B): $G_0\mapsto G_0\times_R C$. The map from (B) to (A): you take the ring ($\mathbb{C}$-algebra) of regular functions $R=\mathbb{C}[G]$ on $G$, consider the ring ($\mathbb{R}$-algebra) of $S$-invariants $R^S$, and set $G_0={\rm Spec}\,R^S$.
May
23
comment Correspondence between real forms and real structures on complex Lie groups
2. Conversely, if $G_0$ is a compact Lie group, then it is a real algebraic group, i.e., it is the set of real zeros in ${\rm GL}(n,\mathbb{R})$ of a finite set polynomials with real coefficients in the $n^2$ matrix elements $g_{i,j}$. Let $G$ denote the complex Lie group of complex zeros of these polynomials in ${\rm GL}(n,\mathbb{C})$, it has a canonical real structure $S\colon g\mapsto \bar g$ (the complex conjugation on the matix elements). This is the desired complexification of a real algebraic group $G_0$.
May
22
comment Correspondence between real forms and real structures on complex Lie groups
1. If $G$ is a complex Lie group, and $S\colon G\to G$ is an anti-holomorphic involutive (i.e., $S^2=1$) automorphism, then the subgroup of fixed points $G^S$ of $S$ in $G$ is the desired real Lie group (the real form corresponding to the real structure $S$).
May
20
asked Localized at $p$ integral representations of finite elementary $p$-groups
May
14
accepted Group schemes, adeles, double cosets, and étale cohomology
May
14
comment Group schemes, adeles, double cosets, and étale cohomology
@DanielLitt: Thank you! But this explanation should be included into your answer...
May
14
comment Group schemes, adeles, double cosets, and étale cohomology
@DanielLitt: Please kindly also explain the sentence: "Now descent data is given by a choice of element of $G(R_{\mathfrak{p}})$ for each ${\mathfrak{p}}\in U$ and an element of $G(K_{\mathfrak{p}})$ for each ${\mathfrak{p}}|N$".
May
14
comment Group schemes, adeles, double cosets, and étale cohomology
@DanielLitt: Thank you for the clarifying paragraph. Still I have questions. Please kindly explain the following sentence: "Now the descent data boils down to choosing elements of $G({\rm Frac}(K_{\mathfrak{p}}))$ for each $\mathfrak{p}|N$".
May
14
comment Group schemes, adeles, double cosets, and étale cohomology
@DanielLitt Could you please describe the construction more clearly? We start from an element $g\in G(\mathbf{A}^f)$. How do we construct the corresponding torsor? From your exposition is seems that you start from a torsor....
May
14
revised Group schemes, adeles, double cosets, and étale cohomology
added 14 characters in body
May
14
asked Group schemes, adeles, double cosets, and étale cohomology
May
11
comment Etale Fundamental group of an algebraic group
@AllyMath: Prop. 1.11 gives the topological fundamental group $\mathbb{Z}$, then you should take the profinite completion in order to get the étale fundamental group $\widehat{\mathbb{Z}}$.
May
7
comment Largest dimensional Lie subgroup of $SU(N)$
The answer of Victor Ostrik (the accepted answer) gives you all what you need assuming that you know the irreducible representation of least dimension of each simple group (Lie algebra). You can find this, for example, in Table 1 in: Onishchik, A. L.; Vinberg, È. B. Lie groups and algebraic groups. Springer-Verlag, Berlin, 1990. See also Table 5 in this book.
May
5
answered Etale Fundamental group of an algebraic group
Apr
27
asked A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group