A Linear Algebraic Group? Skip Garibaldi
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Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields. -
Finite Resolutions of Modules for Reductive Algebraic Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA I@473488 (1986) Finite Resolutions of Modules for Reductive Algebraic Groups S. DONKIN School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England Communicafed by D. A. Buchsbaum Received January 22, 1985 INTRODUCTION A popular theme in recent work of Akin and Buchsbaum is the construc- tion of a finite left resolution of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a field or the integers. The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successivequotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic group theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
Unitary Group - Wikipedia
Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1. -
An Overview of Topological Groups: Yesterday, Today, Tomorrow
axioms Editorial An Overview of Topological Groups: Yesterday, Today, Tomorrow Sidney A. Morris 1,2 1 Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia; [email protected]; Tel.: +61-41-7771178 2 Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia Academic Editor: Humberto Bustince Received: 18 April 2016; Accepted: 20 April 2016; Published: 5 May 2016 It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin’s book “Fourier Analysis on Groups” [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is “Introduction to Topological Groups” [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book “Topological Groups” [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: “This book belongs to that rare category of mathematical works that can truly be called classical - books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians”. The final book I mention from my graduate studies days is “Topological Transformation Groups” [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert’s fifth problem as well as a structure theory for locally compact non-abelian groups. -
1 Affine Varieties
1 Affine Varieties We will begin following Kempf's Algebraic Varieties, and eventually will do things more like in Hartshorne. We will also use various sources for commutative algebra. What is algebraic geometry? Classically, it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k-algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k ! R). P ⊆ R an ideal is prime iff R=P is an integral domain. Algebraic Sets n n We define affine n-space, A = k = f(a1; : : : ; an): ai 2 kg. n Any f = f(x1; : : : ; xn) 2 k[x1; : : : ; xn] defines a function f : A ! k : (a1; : : : ; an) 7! f(a1; : : : ; an). Exercise If f; g 2 k[x1; : : : ; xn] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x1; : : : ; xn] be any subset. Then V (S) = fa 2 An : f(a) = 0 for all f 2 Sg. A subset of An is called algebraic if it is of this form. e.g., a point f(a1; : : : ; an)g = V (x1 − a1; : : : ; xn − an). Exercises 1. I = (S) is the ideal generated by S. Then V (S) = V (I). 2. I ⊆ J ) V (J) ⊆ V (I). P 3. V ([αIα) = V ( Iα) = \V (Iα). 4. V (I \ J) = V (I · J) = V (I) [ V (J). Definition 1.2 (Zariski Topology). We can define a topology on An by defining the closed subsets to be the algebraic subsets. -
Some Finiteness Results for Groups of Automorphisms of Manifolds 3
SOME FINITENESS RESULTS FOR GROUPS OF AUTOMORPHISMS OF MANIFOLDS ALEXANDER KUPERS Abstract. We prove that in dimension =6 4, 5, 7 the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of Rn and various types of automorphisms of 2-connected manifolds. Contents 1. Introduction 1 2. Homologically and homotopically finite type spaces 4 3. Self-embeddings 11 4. The Weiss fiber sequence 19 5. Proofs of main results 29 References 38 1. Introduction Inspired by work of Weiss on Pontryagin classes of topological manifolds [Wei15], we use several recent advances in the study of high-dimensional manifolds to prove a structural result about diffeomorphism groups. We prove the classifying spaces of such groups are often “small” in one of the following two algebro-topological senses: Definition 1.1. Let X be a path-connected space. arXiv:1612.09475v3 [math.AT] 1 Sep 2019 · X is said to be of homologically finite type if for all Z[π1(X)]-modules M that are finitely generated as abelian groups, H∗(X; M) is finitely generated in each degree. · X is said to be of finite type if π1(X) is finite and πi(X) is finitely generated for i ≥ 2. Being of finite type implies being of homologically finite type, see Lemma 2.15. Date: September 4, 2019. Alexander Kupers was partially supported by a William R. -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Bott Periodicity for the Unitary Group
Bott Periodicity for the Unitary Group Carlos Salinas March 7, 2018 Abstract We will present a condensed proof of the Bott Periodicity Theorem for the unitary group U following John Milnor’s classic Morse Theory. There are many documents on the internet which already purport to do this (and do so very well in my estimation), but I nevertheless will attempt to give a summary of the result. Contents 1 The Basics 2 2 Fiber Bundles 3 2.1 First fiber bundle . .4 2.2 Second Fiber Bundle . .5 2.3 Third Fiber Bundle . .5 2.4 Fourth Fiber Bundle . .5 3 Proof of the Periodicity Theorem 6 3.1 The first equivalence . .7 3.2 The second equality . .8 4 The Homotopy Groups of U 8 1 The Basics The original proof of the Periodicity Theorem relies on a deep result of Marston Morse’s calculus of variations, the (Morse) Index Theorem. The proof of this theorem, however, goes beyond the scope of this document, the reader is welcome to read the relevant section from Milnor or indeed Morse’s own paper titled The Index Theorem in the Calculus of Variations. Perhaps the first thing we should set about doing is introducing the main character of our story; this will be the unitary group. The unitary group of degree n (here denoted U(n)) is the set of all unitary matrices; that is, the set of all A ∈ GL(n, C) such that AA∗ = I where A∗ is the conjugate of the transpose of A (conjugate transpose for short). -
LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0. -
Underlying Varieties and Group Structures 3
UNDERLYING VARIETIES AND GROUP STRUCTURES VLADIMIR L. POPOV Abstract. Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. 1. Introduction. This paper arose from a short preprint [16] and is its extensive extension. As starting point of [14] served the question of B. Kunyavsky [10] about the validity of the statement, which is for- mulated below as Corollary of Theorem 1. This statement concerns the possibility to present the underlying variety of a connected reductive algebraic group in the form of a product of some special algebraic va- rieties. Sections 2, 3 make up the content of [16], where the possibility of such presentations is explored. For some of them, in Theorem 1 is proved their existence, and in Theorems 2–5, on the contrary, their non-existence. Theorem 1 shows that there are non-isomorphic reductive groups whose underlying varieties are isomorphic. In Sections 4–10, we explore the problem, naturally arising in connection with this, as to what ex- tent the underlying variety of an algebraic group determines its group structure. In Theorems 6–8, it is shown that some group properties (dimension of unipotent radical, reductivity, solvability, unipotency, arXiv:2105.12861v1 [math.AG] 26 May 2021 toroidality in the sense of Rosenlicht) are equivalent to certain geomet- ric properties of the underlying group variety. Theorem 8 generalizes to solvable groups M. -
Matrix Lie Groups and Their Lie Algebras
Matrix Lie groups and their Lie algebras Alen Alexanderian∗ Abstract We discuss matrix Lie groups and their corresponding Lie algebras. Some common examples are provided for purpose of illustration. 1 Introduction The goal of these brief note is to provide a quick introduction to matrix Lie groups which are a special class of abstract Lie groups. Study of matrix Lie groups is a fruitful endeavor which allows one an entry to theory of Lie groups without requiring knowl- edge of differential topology. After all, most interesting Lie groups turn out to be matrix groups anyway. An abstract Lie group is defined to be a group which is also a smooth manifold, where the group operations of multiplication and inversion are also smooth. We provide a much simple definition for a matrix Lie group in Section 4. Showing that a matrix Lie group is in fact a Lie group is discussed in standard texts such as [2]. We also discuss Lie algebras [1], and the computation of the Lie algebra of a Lie group in Section 5. We will compute the Lie algebras of several well known Lie groups in that section for the purpose of illustration. 2 Notation Let V be a vector space. We denote by gl(V) the space of all linear transformations on V. If V is a finite-dimensional vector space we may put an arbitrary basis on V and identify elements of gl(V) with their matrix representation. The following define various classes of matrices on Rn: ∗The University of Texas at Austin, USA. E-mail: [email protected] Last revised: July 12, 2013 Matrix Lie groups gl(n) : the space of n