bio | website | tau.ac.il/~borovoi |
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location | Tel Aviv, Israel | |
age | ||
visits | member for | 5 years, 4 months |
seen | 20 hours ago | |
stats | profile views | 2,213 |
I am a professor at Tel Aviv University, interested in linear algebraic groups, homogeneous spaces, number theory and nonabelian cohomology.
Jun 26 |
accepted | Permutation covering of a $G$-lattice |
Jun 26 |
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Permutation covering of a $G$-lattice
@FriederLadisch: You write: "It is, however, well known that the group ring of a $p$-group over a field of characteristic $p$ or over a local ring with residue field of characteristic $p$ is local." Could you please give a reference? Many thanks in advance, |
Jun 26 |
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Permutation covering of a $G$-lattice
Excellent!Thank you! |
Jun 25 |
revised |
Permutation covering of a $G$-lattice
edited tags |
Jun 25 |
revised |
Permutation covering of a $G$-lattice
edited tags |
Jun 25 |
asked | Permutation covering of a $G$-lattice |
Jun 24 |
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Compact form of symplectic groups defined over the rationals
@YCor: Moreover, any $\mathbb{Q}$-form of ${\rm Sp}_{2m}$ splits at almost all primes $p$ (because it is an inner form of a split $\mathbb{Q}$-group). |
Jun 24 |
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Compact form of symplectic groups defined over the rationals
I guess that the standard quaternion algebra $H=\mathbb{Q}(i,j)$ with $i^2=-1,\ j^2=-1,\ ji=-ij$ ramifies exactly at $\infty$ and 2. It follows that the unitary group $G=SU(H^n,F)$ of the hermitian form $F(x)=x_1 \bar{x}_1+\dots+x_n\bar{x}_n$ from YCor's first comment splits at every prime $p$ except $\infty$ and maybe at $p=2$. It does not split at 2 because it cannot be nonsplit at one place only by the reciprocity law (I mean the Hasse-Brauer-Noether theorem). |
Jun 2 |
asked | Minimal rank of a permutation resolution of a $G$-lattice |
May 28 |
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A question on an set of 8 matrices related to the SU(3) generators
Maybe you could type your formulas in LaTeX.... |
May 27 |
awarded | Civic Duty |
May 24 |
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Localized at $p$ integral representations of finite elementary $p$-groups
@GeoffRobinson You are right! Thank you! |
May 24 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |