arXiv subjects
Publications and source records attributed to Mikhail Borovoi.
Following an idea of Demarche and using computer calculations of the authors and ideas of the LLM Claude Fable 5, we construct an explicit 10-dimensional complex two-step nilpotent Lie algebra that is isomorphic to its complex conjugate but cannot be defined over the field of real numbers R.
In 1999 Labesse introduced quasi-connected reductive groups and investigated their abelian Galois cohomology over local and global fields of characteristic 0. We (1) generalize some of the constructions of Labesse from quasi-connected reductive groups to arbitrary reductive groups, not necessarily connected or quasi-connected; (2) generalize results of Labesse on the abelian Galois cohomology of quasi-connected reductive groups to the case of local and global fields of arbitrary characteristic; and (3) investigate the functoriality properties of the abelian Galois cohomology. In particular, we introduce the notion of a principal homomorphism of quasi-connected reductive groups, and show that if G is a quasi-connected reductive group over a local or global field k of *positive* characteristic, then the first Galois cohomology set H^1(k,G) has a canonical structure of abelian group, which is functorial with respect to *principal* homomorphisms.
In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T).
We develop a hypercohomology theory of a group with coefficients in a crossed module, and apply it to define abelianization maps for Galois cohomology of reductive algebraic groups.
In this note, following Dedecker and Debremaeker, we extend the cohomology exact sequence for nonabelian $H^1$, using nonabelian $H^2$ with coefficients in crossed modules.
For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(ξ,n)\mapsto ξ^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set (this operation is new when $K$ is a number field). We show that this power operation has many good properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $ξ^{\Diamond n}$ then coincides with the $n$-th power of $ξ$ in this group. On the other hand, we show that a power operation on $H^1(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using this power operation, for a cohomology class $ξ$ in $H^1(K,G)$ over local or global field, we define the period ${\rm per}(ξ)$ to be the least integer $n\ge 1$ such that $ξ^{\Diamond n}=1$. We define the index ${\rm ind}(ξ)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $ξ$. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ defined over a local or global field $K$, we show that ${\rm per}(ξ)$ divides ${\rm ind}(ξ)$ and that ${\rm ind}(ξ)$ may be strictly greater than ${\rm per}(ξ)$, but they always have the same prime factors.
We compute the defect of weak approximation for a reductive group G over a global field K in terms of the algebraic fundamental group of G.
Let $K$ be a local or global field. For a connected reductive group $G$ over $K$, in another preprint [5] we defined a power operation $$(ξ,n)\mapsto ξ^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set $H^1(K,G)$. In this paper, for a cohomology class $ξ$ in $H^1(K,G)$, we compare the period ${\rm per}(ξ)$ defined to be the least integer $n\ge 1$ such that $ξ^{\Diamond n}=1$, and the index ${\rm ind}(ξ)$ defined to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $ξ$. These period and index generalize the period and index a central simple algebra over $K$. For an arbitrary reductive $K$-group $G$, we proved in [5] that ${\rm per}(ξ)$ divides ${\rm ind}(ξ)$. In this paper we show that the index may be strictly greater than the period. In [5] we proved that for any $K$, $G$, and $ξ\in H^1(K,G)$ as above, the index ${\rm ind}(ξ)$ divides ${\rm per}(ξ)^d$ for some positive integer $d$, and we gave upper bounds for $d$ in the local case and in the case of a number field. Here we give a characteristic-free proof of the fact that ${\rm ind}(ξ)$ divides ${\rm per}(ξ)^d$ for some positive integer $d$ in the global field case, and our proof gives an upper bound for $d$ that is valid also in the case of a function field.
We give closed formulas for the abelian Galois cohomology groups H^1_{ab}(F,G) and H^2_{ab}(F,G) of a connected reductive group G over a global field F in terms of the algebraic fundamental group π_1(G) introduced earlier by one of us (M.B.). We further give closed formulas for the effects of restriction, corestriction, and localization, in terms of these formulas and the analogous known formulas in the case of local fields. Building on this, we give formulas, suitable for computer computations, for the first nonabelian Galois cohomology set H^1(F,G) of G and for the second Galois cohomology group H^2(F,T) of an F-torus T. As a preparation for the derivation of our formulas, we review the interpretation of Tate cohomology of a finite group in terms of the stable derived category of Z[Γ]-modules due to Buchweitz, and relate it to the explicit definition via cochains due to Kottwitz-Shelstad. We use this to construct the Tate-Nakayama isomorphisms for bounded complexes of tori over local and global fields, whose specialization to complexes of length 2 is then applied to obtain the desired formulas.
Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is "Yes" when K has no real embeddings. We show that otherwise the answer is "No". Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets H^1(K,G) for all reductive K-groups G in a functorial way.
Let G be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers R. We describe a method, implemented on computer, to find the first Galois cohomology set H^1(R,G). The output is a list of 1-cocycles in G. Moreover, we have an implemented algorithm that, given a 1-cocycle z in Z^1(R,G), finds the cocycle in the computed list to which z is equivalent, together with an element of G(C) realizing the equivalence.
Any complex-analytic vector bundle $\mathbb E$ admits naturally defined homotheties $ϕ_α$, $α\in \mathbb C^*$, i.e. $ϕ_α$ is the multiplication of a local section by a complex number $α$. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to $\mathbb E$. Further, we compute the automorphism supergroup of a $Π$-symmetric super-Grassmannian $Π\!\operatorname{Gr}_{n,k}$, and, using Galois cohomology, we classify the real structures on $Π\!\operatorname{Gr}_{n,k}$ and compute the corresponding supermanifolds of real points.
We classify the non-degenerate two-step nilpotent Lie algebras of dimension 8 over the field of real numbers, using known results over complex numbers. We write explicit structure constants for these real Lie algebras.
Let $X$ be a homogeneous space of a connected linear algebraic group $G$ defined over the field of complex numbers $\mathbb C$. Let $x\in X({\mathbb C})$ be a point. We denote by $H$ the stabilizer of $x$ in $G$. When $H$ is connected, we compute the topological fundamental group $π_1^{\rm top}(X({\mathbb C}),x)$. Moreover, we compute the second homotopy group $π_2^{\rm top}(X({\mathbb C}),x)$.
We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$ of a connected reductive algebraic group $G$ over the field $\mathbb R$ of real numbers in terms of a certain action of the Weyl group on the real points of order dividing 2 of the maximal torus containing a maximal compact torus. This result was announced with a sketch of proof in the author's 1988 note. Here we give a detailed proof.
Let $G$ be a connected reductive group over a number field $F$, and let $S$ be a set (finite or infinite) of places of $F$. We give a necessary and sufficient condition for the surjectivity of the localization map from $H^1(F,G)$ to the "direct sum" of the sets $H^1(F_v,G)$ where $v$ runs over $S$. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.
Using results of Cartan, Matsumoto, and Casselman, we give a short proof of Timashev's theorem computing the real component group π_0 G(R) of a connected reductive real algebraic group G in terms of a maximal torus of G containing a maximal split torus.
Let G be a connected reductive group over the field of real numbers R. Using results of our previous joint paper, we compute combinatorially the first Galois cohomology set H^1(R,G) in terms of reductive Kac labelings. Moreover, we compute the group of connected components π_0 G(R) of the real Lie group G(R) and the maps in exact sequences containing π_0 G(R) and H^1(R,G).