A complex algebraic group G is in this note a subgroup of GL(n, C), the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials {P …
A complex algebraic group G is in this note a subgroup of GL(n, C), the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials {P M [Xn, • • • , X nn ]} in n 2 indeterminates.It is said to be defined over a field KQC if the polynomials can be chosen so as to have coefficients in K. Given a subring B of C, we denote by GB the subgroup of elements of G which have coefficients in JB, and whose determinant is a unit of B. Assume in particular G to be defined over Q.Then Gz is an "arithmetically defined discrete subgroup" of G Rl or, more briefly, an arithmetic subgroup of GR.A typical example is the group of units of a nondegenerate integral quadratic form, and as a matter of fact, the main results stated below generalize facts known in this case from reduction theory.The proofs will be published elsewhere.
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the …
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups defined over the fields which admit arbitrary cyclic extensions.
CONTENTS Introduction ??1. Arithmetic groups ??2. Ad?le groups ??3. Tamagawa numbers ??4. Approximations in algebraic groups ??5. Class numbers and class groups of algebraic groups ??6. The genus problem in …
CONTENTS Introduction ??1. Arithmetic groups ??2. Ad?le groups ??3. Tamagawa numbers ??4. Approximations in algebraic groups ??5. Class numbers and class groups of algebraic groups ??6. The genus problem in arithmetic groups ??7. Classification of maximal arithmetic subgroups ??8. The congruence problem ??9. Groups of rational points over global fields ??10. Galois cohomology and the Hasse principle ??11. Cohomology of arithmetic groups References
Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n"> <mml:semantics> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> …
Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n"> <mml:semantics> <mml:msub> <mml:mi>GL</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\operatorname {GL}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over a field of any characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a commutative group of order prime to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.
Keywords: algebraic groups ; parabolic subgroups Reference CTG-ARTICLE-2004-002doi:10.1093/qmath/hag042 Record created on 2008-12-16, modified on 2017-05-12
Keywords: algebraic groups ; parabolic subgroups Reference CTG-ARTICLE-2004-002doi:10.1093/qmath/hag042 Record created on 2008-12-16, modified on 2017-05-12
A closed subgroup of a semisimple algebraic group G is said to be G‐irreducible if it lies in no proper parabolic subgroup of G. We prove a number of results …
A closed subgroup of a semisimple algebraic group G is said to be G‐irreducible if it lies in no proper parabolic subgroup of G. We prove a number of results concerning such subgroups. Firstly they have only finitely many overgroups in G; secondly, with some specified exceptions, there exist G‐irreducible subgroups of type A1; and thirdly, we prove an embedding theorem for G‐irreducible subgroups.
Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring …
Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures, and they arise in many areas of study.
The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers …
The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers were Michel Brion, Jens Carsten Jantzen, and Raphaël Rouquier. During the last years, the subject of algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a “general audience” of algebraic group-theorists, and to stimulate contacts between participants. Each of the first four days was dedicated to one area of research that has recently seen decisive progress: The first three days started with survey talks that will help to make the subject accessible to the next generation. The talks on the last day introduced to several recent advances in different areas: arithmetic groups, eigenvalue problems, counting orbits over finite fields, quivers and reflection functors. In order to leave enough time for fruitful discussions, the number of talks (generally of one hour) was limited to four per day. Besides the scientific program, the participants enjoyed a piano recital on Thursday evening, by Harry Tamvakis. The workshop was attended by 53 participants, coming mainly from Europe and North America. This includes 6 PhD students, supported by the Marie Curie program of the European Union. The organizers are grateful to the EU for this support, and to the MFO for providing excellent working conditions.
Background material. ,Topics include reviews of Henselian fields, fields of dimension at most 1, tori, reductive groups, Chevalley systems and pinnings, integral models, the dynamic method.Some important definitions, such as …
Background material. ,Topics include reviews of Henselian fields, fields of dimension at most 1, tori, reductive groups, Chevalley systems and pinnings, integral models, the dynamic method.Some important definitions, such as of the subgroup $G(k)^0$ of $G(k)$, are also given.
We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of …
We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of integral canonical models of Shimura varieties of preabelian (resp. of abelian) type in mixed characteristic $(0,p)$ with $p\Ge 3$ (resp. with $p=2$) and with respect to hyperspecial subgroups; if $p=3$ (resp. if $p=2$) we restrict in this part I either to the $A_n$, $C_n$, $D_n^{\dbH}$ (resp. $A_n$ and $C_n$) types or to the $B_n$ and $D_n^{\dbR}$ (resp. $B_n$, $D_n^{\dbH}$ and $D_n^{\dbR}$) types which have compact factors over $\dbR$ (resp. which have compact factors over $\dbR$ in some $p$-compact sense). Though the second application is new just for $p\Le 3$, a great part of its proof is new even for $p\Ge 5$ and corrects [Va1, 6.4.11] in most of the cases. The second application forms progress towards the proof of a conjecture of Milne. It also provides in arbitrary mixed characteristic the very first examples of general nature of projective varieties over number fields which are not embeddable into abelian varieties and which have Néron models over certain local rings of rings of integers of number fields.
We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed …
We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesically immerse in non-compact right-angled reflection orbifolds, extending work of the third author from the compact case. The first result gives examples to which the second applies, and for these we give explicit bounds on geodesic residual finiteness growth.
Abstract The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a …
Abstract The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.
We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the …
We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the integers). Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).
We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution …
We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-\'etale covers of N\'eron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.
Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, …
Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with an absolute value at most s by f, as s tends to + ∞.
We consider locally symmetric manifolds with a fixed universal covering, and we construct for each such manifold M a simplicial complex ℝ whose size is proportional to the volume of …
We consider locally symmetric manifolds with a fixed universal covering, and we construct for each such manifold M a simplicial complex ℝ whose size is proportional to the volume of M. When M is noncompact, $\mathcal{R}$ is homotopically equivalent to M, while when M is compact, $\mathcal{R}$ is homotopically equivalent to M\N, where N is a finite union of submanifolds of relatively small dimension. This reflects how the volume controls the topological structure of M, and yields concrete bounds for various finiteness statements that previously had no quantitative proofs. For example, it gives an explicit upper bound for the possible number of locally symmetric manifolds of volume bounded by v>0, and it yields an estimate for the size of a minimal presentation for the fundamental group of a manifold in terms of its volume. It also yields a number of new finiteness results.
This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing …
This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudo-Riemannian manifolds with constant curvatures, and discuss what kind of problems we propose to pursue. For pseudo-Riemannian manifolds, isometric actions of discrete groups are not always properly discontinuous. The fundamental problem is to understand when discrete subgroups of Lie groups $G$ act properly discontinuously on homogeneous spaces $G/H$ for non-compact $H$. For this, we introduce the concepts from a group-theoretic perspective, including the `discontinuous dual' of $G/H$ that recovers $H$ in a sense. We then summarize recent results giving criteria for the existence of properly discontinuous subgroups, and the known results and conjectures on the existence of cocompact ones. The final section discusses the deformation theory and in particular rigidity results for cocompact properly discontinuous groups for pseudo-Riemannian symmetric spaces.
In this paper we discuss the use of dynamical and ergodic-theoretic ideas and methods to solve some long-standing problems originating from Lie groups and number theory. These problems arise from …
In this paper we discuss the use of dynamical and ergodic-theoretic ideas and methods to solve some long-standing problems originating from Lie groups and number theory. These problems arise from looking at actions of Lie groups on their homogeneous spaces. Such actions, viewed as dynamical systems, have long been interesting and rich objects of ergodic theory and geometry. Since the 1930s ergodic-theoretic methods have been applied to the study of geodesic and horocycle flows on unit tangent bundlesof compact surfaces of negative curvature. From the algebraic point of view the latter flows are examples of semisimple and unipotent actions on finite-volume homogeneous spaces of real Lie groups. It was established in the 1960s through the fundamental work of D. Ornstein that typical semisimple actions are all statistically the same due to their extremal randomness caused by exponential instability of orbits. Their algebraic nature has little to do with the isomorphism problem for such actions: they are measure-theoretically isomorphic as long as their entropies coincide.
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians …
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
We develop the theory of adequate moduli spaces in characteristic $p$ (and mixed characteristic) characterizing quotients by geometrically reductive group schemes.
We develop the theory of adequate moduli spaces in characteristic $p$ (and mixed characteristic) characterizing quotients by geometrically reductive group schemes.
This book is made of two parts. The first is concerned with the differential form spectrum of congruence hyperbolic manifolds. We prove Selberg type theorems on the first eigenvalue of …
This book is made of two parts. The first is concerned with the differential form spectrum of congruence hyperbolic manifolds. We prove Selberg type theorems on the first eigenvalue of the laplacian on differential forms. The method of proof is representation theoritic, we hope the different chapters may as well serve as an introduction to the modern theory of automorphic forms and its application to spectral questions. The second part of the book is of a more differential geometric flavor, a new kind of lifting of cohomology classes is proved.
Let M be a manifold, on which a real reductive Lie group G acts transitively. The action of a discrete subgroup on M is not always properly discontinuous. In this …
Let M be a manifold, on which a real reductive Lie group G acts transitively. The action of a discrete subgroup on M is not always properly discontinuous. In this paper, we give a criterion for properly dis- continuous actions, which generalizes our previous work (6) for an analogous problem in the continuous setting. Furthermore, we introduce the discon- tinuous dual t(H:G) of a subset H of G , and prove a duality theorem that each subset H of G is uniquely determined by its discontinuous dual up to multiplication by compact subsets.
Let $ 1< p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by U. Haagerup and J. Kraus, then the noncommutative …
Let $ 1< p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by U. Haagerup and J. Kraus, then the noncommutative $L\sb p(VN(G))$-space has the operator space approximation property. If, in addition, the group von Neumann algebra $VN(G)$ has the quotient weak expectation property (QWEP), that is, is a quotient of a $C\sp \ast$-algebra with Lance's weak expectation property, then $L\sb p(V N(G))$ actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on $L\sb p(V N(G))$. Finally, we show that if $G$ is a countable discrete group having the approximation property and $V N(G)$ has the QWEP, then $L\sb p(V N(G))$ has a very nice local structure; that is, it is a $\mathscr {COL}\sb p$-space and has a completely bounded Schauder basis.
We investigate the implications of Gromov’s theorem on boundedness of primary characteristic classes for the continuous bounded cohomology of a semisimple Lie group G. We deduce that the comparison map …
We investigate the implications of Gromov’s theorem on boundedness of primary characteristic classes for the continuous bounded cohomology of a semisimple Lie group G. We deduce that the comparison map from continuous bounded cohomology to continuous cohomology is surjective for a large class of semisimple Lie groups, including all Hermitian groups. Our proof is based on a geometric implementation of the canonical map from the cohomology of the classifying space of G to the continuous group cohomology of G. We obtain this implementation by establishing a variant of Kobayashi–Ono–Hirzebruch duality.
Let G be a semisimple Lie group and Γ a lattice in G. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of …
Let G be a semisimple Lie group and Γ a lattice in G. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous ...
The problem of classification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 times 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2 \times 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> indefinite Hermitian matrices …
The problem of classification of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 times 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×<!-- × --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2 \times 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> indefinite Hermitian matrices over orders in Clifford algebras is considered. The unit groups of these matrices are analogous to maximal arithmetic Fuchsian subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="PSL left-parenthesis 2 comma German o right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>PSL</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">o</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {PSL}}(2,\mathfrak {o})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German o"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">o</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {o}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an order in a quadratic number field.
Abstract We establish new results on the weak containment of quasi‐regular and Koopman representations of a second countable locally compact group associated with nonsingular ‐spaces. We deduce that any two …
Abstract We establish new results on the weak containment of quasi‐regular and Koopman representations of a second countable locally compact group associated with nonsingular ‐spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that nonamenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Ł. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
A. Soit G un groupe de Lie complexe et connexe et soit K un sous-groupe compact maximal de G . Soient g et les algèbres de Lie de G et …
A. Soit G un groupe de Lie complexe et connexe et soit K un sous-groupe compact maximal de G . Soient g et les algèbres de Lie de G et K respectivement. est une sous-algèbre réelle de l’algèbre complexe g. Soit ( i 2 = –1). Alors est une sous-algèbre complexe de g.
A complex algebraic group G is in this note a subgroup of GL(n, C), the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials {P …
A complex algebraic group G is in this note a subgroup of GL(n, C), the elements of which are all invertible matrices whose coefficients annihilate some set of polynomials {P M [Xn, • • • , X nn ]} in n 2 indeterminates.It is said to be defined over a field KQC if the polynomials can be chosen so as to have coefficients in K. Given a subring B of C, we denote by GB the subgroup of elements of G which have coefficients in JB, and whose determinant is a unit of B. Assume in particular G to be defined over Q.Then Gz is an "arithmetically defined discrete subgroup" of G Rl or, more briefly, an arithmetic subgroup of GR.A typical example is the group of units of a nondegenerate integral quadratic form, and as a matter of fact, the main results stated below generalize facts known in this case from reduction theory.The proofs will be published elsewhere.
W. W. Morozov [10] and [1i]1 has announced and indicated proofs of the following theorems. If 2 is a semi-simple Lie algebra over the field of complex numbers, then any …
W. W. Morozov [10] and [1i]1 has announced and indicated proofs of the following theorems. If 2 is a semi-simple Lie algebra over the field of complex numbers, then any element d of 2 such that Ad d is nilpotent can be imbedded in a three-dimensional simple subalgebra. If 9 is a semi-simple subalgebra of a semi-simple algebra 3 over the field of complex numbers, then the centralizer 9 of V in ? is a direct sum of a semi-simple algebra and its center. Moreover, the elements of the center have adjoint mappings that have simple elementary divisors. The proof of the first result appears to have a gap2 and the proof of the second result is sketched only for the case 9) a three-dimensional simple algebra. In the present note we shall give simple and complete proofs of these results for arbitrary base fields of characteristic 0. Moreover, we shall give these results what appears to be their natural setting, namely, the theory of completely reducible Lie algebras of linear transformations. We shall also extend Morozov's first result to Lie triple systems (Lemma 4) and we shall use this extension to obtain an analogue of this result for Jordan algebras (Theorem 8).
A real (or complex) algebraic variety V is a point set in real n-space R n (or complex n-space C n ) which is the set of common zeros of …
A real (or complex) algebraic variety V is a point set in real n-space R n (or complex n-space C n ) which is the set of common zeros of a set of polynomials. The general properties of a real V as a point set have not been the subject of much study recently (but see for instance [2], [3] and [4]); attention has turned more to the complex case, the complex projective case, and especially the abstract algebraic theory. Facts about the real case are sometimes needed in the applications; proofs are commonly very difficult to locate.
If Fr^0 one can prove that there is only a finite number of possibilities for Tr (see [6(h), Theorem 3]).Lemma 1 (Mostow).Let fjo be a Cartan subalgebra of g0 and …
If Fr^0 one can prove that there is only a finite number of possibilities for Tr (see [6(h), Theorem 3]).Lemma 1 (Mostow).Let fjo be a Cartan subalgebra of g0 and fj the complexification of h0 in g.Then there exists a compact real form u of g with the following two properties:1. »j(u)=U, 2. hf^u is a Cartan subalgebra of u.Now let u be a fixed compact real form of g such that ij(u) =u and let
The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received …
The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.
Let Q be the field of rational numbers and k an extension of Q of finite degree. The multiplicative group k* of k, considered as a group of linear transformations …
Let Q be the field of rational numbers and k an extension of Q of finite degree. The multiplicative group k* of k, considered as a group of linear transformations of the vector space k over Q, forms an algebraic group, i.e., a Q-torus in the sense of A. Borel. As is well known in the algebraic number theory, the properties of k* and of its related structures, in particular that of the group Jk of id6les of k, play important roles. On the other hand, let f be a quadratic form on a vector space V over Q of finite dimension. The orthogonal group O( V, f ) composed of all linear transformations of V leaving invariant the form f forms an algebraic group. The properties of the group O( V, f) have essential relations to the arithmetic of the quadratic form f, and the study of these relations has been one of the principal themes in M. Eichler's book Quadratische Formen und Orthogonale Gruppen . Recently the theory of algebraic groups of linear transformations has been systematized by C. Chevalley on the basis of fundamental concepts of algebraic geometry, and the classical mechanism of the Lie theory (correspondence between groups and Lie algebras) has been generalized to the case where the basic field K is an arbitrary field of characteristic 0 (cf., Chevalley [2], [3]). By specializing K to Q, we may apply his methods and results to the study of arithmetic properties of algebraic groups. Thus it could be said that the above two theories, i. e., the arithmetic of k* and that of O( V, f) are two profiles of a kind of unified theory which we might call the arithmetic of algebraic groups. In the present paper, we shall formulate some fundamental concepts for algebraic groups from this point of view and prove some results which might possibly give us some suggestions for further developments in this direction. Thus, in Section 1, we shall introduce the notion of rational characters of algebraic groups and determine the structure of the group of rational characters for a special algebraic group, i. e., a Q-torus (Theorem 1). In Section 2, we shall introduce the notion of G-id6les for an algebraic group G which generalizes the usual notion of id6les of an algebraic number field k and define a subgroup J1(G) of the group J(G) of G-id6les in con266