scholarly journals ORBIT PARAMETRIZATIONS FOR K3 SURFACES

2016 ◽  
Vol 4 ◽  
Author(s):  
MANJUL BHARGAVA ◽  
WEI HO ◽  
ABHINAV KUMAR
Keyword(s):  
Fixed Point ◽  
Recent Work ◽  
Moduli Spaces ◽  
Natural Extension ◽  
Algebraic Groups ◽  
K3 Surfaces ◽  
Closed Field ◽  
Positive Entropy ◽  
Algebraically Closed Field

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron–Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

Classes of unipotent elements in simple algebraic groups. I

1976 ◽  
Vol 79 (3) ◽  
pp. 401-425 ◽  
Author(s):  
P. Bala ◽  
R. W. Carter
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Adjoint Action ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Elements

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

scholarly journals Invariant Hopf 2-Cocycles for Affine Algebraic Groups

2018 ◽  
Vol 2020 (2) ◽  
pp. 344-366
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Finite Groups ◽  
Cohomology Group ◽  
Algebraic Groups ◽  
Closed Field ◽  
Algebraically Closed Field

Abstract We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for finite groups G, developed in [3, 4, 14], to the case of affine algebraic groups G, using the methods of [9, 10, 12]. In particular, we show that for connected affine algebraic groups G over an algebraically closed field of characteristic 0, the map Θ from [14] is bijective (unlike for some finite groups, as shown in [14]). This allows us to compute $H^{2}_{\textrm{inv}}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [14]).

Moduli spaces of algebraic curves with rational maps

1975 ◽  
Vol 78 (2) ◽  
pp. 283-292 ◽  
Author(s):  
Herbert Lange
Keyword(s):  
Algebraic Variety ◽  
Moduli Spaces ◽  
Algebraic Curves ◽  
Rational Maps ◽  
Closed Field ◽  
Rational Map ◽  
Algebraically Closed Field

Let ℳg be the coarse moduli scheme of curves of genus g. For an algebraically closed field k define is a quasiprojective algebraic variety over k, its dimension being 3g – 3 for g ≥ 2, 1 for g = 1, and 0 for g = 0. It can be considered as the moduli variety for the classes of birationally equivalent curves of genus g over k. For 0 < g, g′ and n ≥ 1 let be the subset of those points of whose corresponding curves possess a rational map of degree n into a curve of genus g′ over k.

scholarly journals Modular finite W-algebras

2018 ◽  
Vol 2019 (18) ◽  
pp. 5811-5853 ◽  
Author(s):  
Simon M Goodwin ◽  
Lewis W Topley
Keyword(s):  
Algebraic Group ◽  
Recent Work ◽  
Nilpotent Element ◽  
Direct Approach ◽  
Closed Field ◽  
Characteristic P ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Equivalence Of Categories ◽  
Pbw Theorem

Abstract Let ${\mathbb{k}}$ be an algebraically closed field of characteristic p > 0 and let G be a connected reductive algebraic group over ${\mathbb{k}}$. Under some standard hypothesis on G, we give a direct approach to the finite W-algebra $U(\mathfrak{g},e)$ associated to a nilpotent element $e \in \mathfrak{g} = \textrm{Lie}\ G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the p-centre of $U(\mathfrak{g},e)$, which allows us to define reduced finite W-algebras $U_{\eta}(\mathfrak{g},e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin’s equivalence of categories, generalizing recent work of the second author.

Cup Products in Sheaf Cohomology

1986 ◽  
Vol 29 (4) ◽  
pp. 469-477 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Hopf Algebra ◽  
Prime Number ◽  
Algebraic Groups ◽  
Closed Field ◽  
Algebra Structure ◽  
Sheaf Cohomology ◽  
Algebraically Closed Field ◽  
Cup Product ◽  
Hopf Algebra Structure ◽  
Cup Products

AbstractLet k be an algebraically closed field, and let l be a prime number not equal to char(k). Let X be a locally fibrant simplicial sheaf on the big étale site for k, and let Y be a k scheme which is cohomologically proper. Then there is a Künneth-type isomorphismwhich is induced by an external cup-product pairing. Reductive algebraic groups G over k are cohomologically proper, by a result of Friedlander and Parshall. The resulting Hopf algebra structure on may be used together with the Lang isomorphism to give a new proof of the theorem of Friedlander-Mislin which avoids characteristic 0 theory. A vanishing criterion is established for the Friedlander-Quillen conjecture.

scholarly journals On a class of numbers generated by differencial equations related with algebraic groups

1994 ◽  
Vol 133 ◽  
pp. 1-55 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Number Theory ◽  
Differential Equations ◽  
Algebraic Groups ◽  
Complex Numbers ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Algebraically Closed Field ◽  
Algebraic Differential Equations

In this paper we propose a new category Qcl of complex numbers which contains π, e and the set of algebraic numbers. In fact this category contains most of the numbers studied so far in number theory. An element of the category is here called a classical number. The category of the classical numbers forms an algebraically closed field and consists of countably many numbers. The definition depends on algebraic differential equations related with algebraic groups. Throughout the paper unless otherwise stated, we deal with functions of one variable and a differential equation is an ordinary differential equation. We are inspired of the Leçons de Stockholm of Painlevé [P].

scholarly journals A note on the conjugacy of Cartan subalgebras

1979 ◽  
Vol 27 (2) ◽  
pp. 163-166
Author(s):  
David J. Winter
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Subalgebras ◽  
Cartan Subgroups

AbstractThe conjugacy of Cartan subalgebras of a Lie algebra L over an algebraically closed field under the connected automorphism group G of L is inherited by those G-stable ideals B for which B/Ci is restrictable for some hypercenter Ci of B. Concequently, if L is a restrictable Lie algebra such that L/Ci restrictable for some hypercenter Ci of L, and if the Lie algebra of Aut L contains ad L, then the Cartan subalgebras of L are conjugate under G. (The techniques here apply in particular to Lie algebras of characteristic 0 and classical Lie algebras, showing how the conjugacy of Cartan subgroups of algebraic groups leads quickly in these cases to the conjugacy of Cartan subalgebras.)

scholarly journals Unipotent elements forcing irreducibility in linear algebraic groups

2018 ◽  
Vol 21 (3) ◽  
pp. 365-396 ◽  
Author(s):  
Mikko Korhonen
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Algebraic Groups ◽  
Closed Field ◽  
Unipotent Element ◽  
Tilting Modules ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Linear Algebraic Groups ◽  
Highest Weight

Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

Smoothness of Quotients Associated With a Pair of Commuting Involutions

2004 ◽  
Vol 56 (5) ◽  
pp. 945-962 ◽  
Author(s):  
Aloysius G. Helminck ◽  
Gerald W. Schwarz
Keyword(s):  
Fixed Point ◽  
Algebraic Group ◽  
Symmetric Case ◽  
Closed Field ◽  
Categorical Quotient ◽  
Algebraically Closed Field ◽  
Semisimple Algebraic Group ◽  
Point Groups

AbstractLet σ, θ be commuting involutions of the connected semisimple algebraic group G where σ, θ and G are defined over an algebraically closed field , char = 0. Let H := Gσ and K := Gθ be the fixed point groups. We have an action (H × K) × G → G, where ((h, k), g) ⟼ hgk–1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K) denote the categorical quotient Spec (G)H×K. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where σ = θ and H = K.

Homomorphisms of Lie Algebras of Algebraic Groups and Analytic Groups

1995 ◽  
Vol 38 (3) ◽  
pp. 352-359
Author(s):  
Nazih Nahlus
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Groups ◽  
Algebra Homomorphism ◽  
Semisimple Element ◽  
Closed Field ◽  
Linear Complex ◽  
Algebraically Closed Field ◽  
Analytic Case ◽  
Equivalent Conditions

AbstractLet be a Lie algebra homomorphism from the Lie algebra of G to the Lie algebra of H in the following cases: (i) G and H are irreducible algebraic groups over an algebraically closed field of characteristic 0, or (ii) G and H are linear complex analytic groups. In this paper, we present some equivalent conditions for ϕ to be a differential in the above two cases. That is, ϕ is the differential of a morphism of algebraic groups or analytic groups as appropriate.In the algebraic case, for example, it is shown that ϕ is a differential if and only if ϕ preserves nilpotency, semisimplicity, and integrality of elements. In the analytic case, ϕ is a differential if and only if ϕ maps every integral semisimple element of into an integral semisimple element of , where G0 and H0 are the universal algebraic subgroups of G and H. Via rational elements, we also present some equivalent conditions for ϕ to be a differential up to coverings of G in the algebraic case, and for ϕ to be a differential up to finite coverings of G in the analytic case.

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