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Showing 1 to 20 of 32 for “"algebraic group"”.
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Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic
Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\Phi$, and a set of simple roots $\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of …
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Bounding cohomology for low rank algebraic groups
Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.
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The Grothendieck-Cousin complex on G/B x G/B
… to an induced representation of a reductive algebraic group G. His technique uses the geometry of the homogeneous space G/B, B being a Borel subgroup of G. The complex gives a resolution by B-modules, which easily yields the Weyl character formula.
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Fourier transforms of Nilpotent Orbits, limit formulas for reductive lie groups, and wave front cycles of tempered representations
… orbit for GL(n, R). If G is a real, reductive algebraic group, and O C g* = Lie(G)* is a nilpotent coadjoint orbit, a necessary condition is given for 0 to appear in the wave front cycle of a tempered representation. In addition, the coefficients of the wave front cycle of a tempered …
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Computing the Lusztig-Vogan bijection
Let G be a connected complex reductive algebraic group with Lie algebra g. The Lusztig-Vogan bijection relates two bases for the bounded derived category of G-equivariant coherent sheaves on the nilpotent cone 11 of g. One basis is indexed by ..., the set of dominant weights of G, and the other by …
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Tilting sheaves for real groups and Koszul duality
… of real analytic varieties with the real Lie group action we define a tstructure on the category of equivariant-monodromic sheaves and develop the theory of tilting sheaves. In case of a quasi-split real form of an algebraic group acting on the flag variety we construct an analog of a Soergel …
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Rigid Analytic Quantum Groups
… introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions $\wideparen{U_q(\mathfrak{g})}$ and $\wideparen{\mathcal{O}_q(G)}$ of the quantized enveloping algebra of a semisimple Lie algebra $\mathfrak{g}$ and the quantized coordinate ring of the corresponding …
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A motivic norm structure on equivariant algebraic K-theory
… theory is a homotopy theory of schemes with algebraic group actions. This thesis is mainly divided into two parts. In the first part, we define four model categories of motivic spectra that present the $\infty$-category $\SH^G(S)$. We use the model categorical setup to reproduce the motivic …
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Geometric Langlands in prime characteristic
Let C be a smooth projective curve over an algebraically closed field k of sufficiently large characteristic. Let G be a semisimple algebraic group over k and let GV be its Langlands dual group over k. Denote by BunG the moduli stack of G-bundles on C and LocSysGv the moduli stack of Gv-local …
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On the Inverse Problem in Differential Galois Theory
… equation, and the differential Galois group is the group of automorphisms of this extension which fix the base field and commute with the derivation. Differential Galois groups are linear algebraic groups over the field of constants of the base field. In analogy to the classical …
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Connections on conformal blocks
For an algebraic group G and a projective curve X, we study the category of D-modules on the moduli space Bung of principal G-bundles on X using ideas from conformal field theory. We describe this category in terms of the action of infinitesimal Hecke functors on the category of quasi-coherent …
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Irreducible components of the restricted nilpotent commuting variety of G2, F4 and E6 in good characteristic
… nullcone of the Lie algebra g of a simple algebraic group in characteristic p>0, i.e. the set of x∈g such that x|p| = 0. For representatives e1,...,en of the nilpotent orbits of g we find the irreducible components of gei∩N1 for g = G2 and F4 in good characteristic p. We do the same for g = …
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The Space of Left Orders on a Group
The study of orderable groups is a topic that is all too often overlooked as a topic in algebra. The subject of orderable groups is a field of study which is directly associated with algebraic group theory, algebraic topology, and set theory. This paper will act as a guide into the world of …
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Advances in signatures, encryption, and E-Cash from bilinear groups
… Our constructions are based on a special type of algebraic group called bilinear groups. 1. Re-Signatures: We present the first public key signature scheme where a semi-trusted proxy, given special information, can translate Alice's signature on a message into Bob's signature on the same message. …
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On mobs with certain group-like properties
Topological groupoids with"approximate" inverses are studied. In the compact case, these"approximate" inverses turn out to be true inverses. Examples of groupoids wL:h"approximate" inverses are given in the section dealing with function spaces. Using the classical construction of Haar as a guide, …
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Unipotent elements in algebraic groups
… the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin …
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On the Isomorphy Classes of Involutions over SO(2n, k)
… The study of symmetric spaces involves group theory, ï¬ eld theory, linear algebra, and Lie algebras, as well as involving the related disciplines of topology, manifold theory, and analysis. The notion of symmetric space was generalized in the 1980’s to groups deï¬ ned over arbitrary …
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Privacy Enhancing Protocols using Pairing Based Cryptography
… Our constructions are based on a special type of algebraic group called bilinear groups. We present existing cryptographic constructions which use bilinear pairings, namely Identity-Based Encryption (IBE). We define a desirable property of digital signatures, blindness, and present new IBE …
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Trace Diagrams, Representations, and Low-Dimensional Topology
… a surface. Let G be a connected reductive linear algebraic group, and let S be a surface whose fundamental group pi is a free group. Then the coordinate ring C[Hom(pi,G)] of the homomorphisms from pi to G is isomorphic to C[G^r]=C[G]^{tensor r} for some r>=0. The coordinate ring C[G] may be …
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Line Bundles on Projective Homogeneous Spaces
… homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $>$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $>$ 0 there are an infinite number of subgroup schemes containing B--the reduced ones are the ordinary parabolic subgroups P …
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