Morphisms of Algebraic Stacks

... Lemma 3.5. Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent (1) f is separated, (2) ∆f is a closed immersion, (3) ∆f is proper, or (4) ∆f is universally closed. Proof. The statements “f is separated”, “∆f is a closed immersion”, “∆f ...

... Lemma 3.5. Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent (1) f is separated, (2) ∆f is a closed immersion, (3) ∆f is proper, or (4) ∆f is universally closed. Proof. The statements “f is separated”, “∆f is a closed immersion”, “∆f ...

Maximal compact subgroups in the o-minimal setting

... (i) G has maximal definable definably compact subgroups. (ii) G = K · N (G), for some definable definably compact subgroup K < G. (iii) A(G) = {e}. Proof. (i) ⇒ (iii): Take K to be a maximal definable definably compact subgroup, and T a 0-Sylow of G. Suppose, for a contradiction, that A(G) is infini ...

... (i) G has maximal definable definably compact subgroups. (ii) G = K · N (G), for some definable definably compact subgroup K < G. (iii) A(G) = {e}. Proof. (i) ⇒ (iii): Take K to be a maximal definable definably compact subgroup, and T a 0-Sylow of G. Suppose, for a contradiction, that A(G) is infini ...

Class Field Theory

... Hasse, and others. For nonabelian extensions, the first indication of the shape the theory should take is in a letter from Langlands to Weil in 1967. In recent years there has been much progress in the nonabelian local and function field cases6 , but less in the number field case. Beginning about 19 ...

... Hasse, and others. For nonabelian extensions, the first indication of the shape the theory should take is in a letter from Langlands to Weil in 1967. In recent years there has been much progress in the nonabelian local and function field cases6 , but less in the number field case. Beginning about 19 ...

INTEGRATING MORPHISMS OF LIE 2-ALGEBRAS 1. Introduction In

... measures the deviation of f from being an equivalence and it sits in a natural exact triangle which gives rise to a 7-term long exact sequence. The homotopy fiber comes with a rich structure consisting of various brackets and Jacobiators (see §4.1). Homotopy fibers of morphisms of 2-term L∞ -algebra ...

... measures the deviation of f from being an equivalence and it sits in a natural exact triangle which gives rise to a 7-term long exact sequence. The homotopy fiber comes with a rich structure consisting of various brackets and Jacobiators (see §4.1). Homotopy fibers of morphisms of 2-term L∞ -algebra ...

M14/13

... loops. Recall that for a loop Q, the middle nucleus is defined as Nµ (Q) = {y ∈ Q : (xy)z = x(yz) for every x, z ∈ Q}. Following [9], for an abelian group G and a bijection α of G, let G(α) be defined on Z2 × G by (0, u)(0, v) = (0, u + v), (0, u)(1, v) = (1, u + v) = (1, u)(0, v), (1, u)(1, v) = (0 ...

... loops. Recall that for a loop Q, the middle nucleus is defined as Nµ (Q) = {y ∈ Q : (xy)z = x(yz) for every x, z ∈ Q}. Following [9], for an abelian group G and a bijection α of G, let G(α) be defined on Z2 × G by (0, u)(0, v) = (0, u + v), (0, u)(1, v) = (1, u + v) = (1, u)(0, v), (1, u)(1, v) = (0 ...

Lecture 5 Message Authentication and Hash Functions

... Hardness of the discrete log In some groups it is easy: • in Zn it is easy because ae = e · a mod n • In Zp* (where p is prime) it is believed to be hard. • There exist also other groups where it is believed to be hard (e.g. based on the Elliptic curves). ...

... Hardness of the discrete log In some groups it is easy: • in Zn it is easy because ae = e · a mod n • In Zp* (where p is prime) it is believed to be hard. • There exist also other groups where it is believed to be hard (e.g. based on the Elliptic curves). ...

Introduction to Lie Groups

... These are notes for the course Introduction to Lie Groups (cross-listed as MAT 4144 and MAT 5158) at the University of Ottawa. At the title suggests, this in a first course in the theory of Lie groups. Students are expected to a have an undergraduate level background in group theory, ring theory and ...

... These are notes for the course Introduction to Lie Groups (cross-listed as MAT 4144 and MAT 5158) at the University of Ottawa. At the title suggests, this in a first course in the theory of Lie groups. Students are expected to a have an undergraduate level background in group theory, ring theory and ...

Three Types of Symmetry

... Bret Benesh College of St. Benedict/St. John’s University Department of Mathematics ...

... Bret Benesh College of St. Benedict/St. John’s University Department of Mathematics ...

A Bousfield-Kan algorithm for computing homotopy

... x0 , x1 , . . . , xk−1 , xk+1 , . . . , xq satisfying the compatibility condition does not imply it is always possible to determine it. We say that the Kan simplicial set K is a constructive Kan simplicial set if the desired x is given explicitly by an algorithm. It is not difficult to prove (see [1 ...

... x0 , x1 , . . . , xk−1 , xk+1 , . . . , xq satisfying the compatibility condition does not imply it is always possible to determine it. We say that the Kan simplicial set K is a constructive Kan simplicial set if the desired x is given explicitly by an algorithm. It is not difficult to prove (see [1 ...

Chapter 9 Lie Groups, Lie Algebras and the Exponential Map

... Let us also recall the definition of homomorphisms of Lie groups and Lie algebras. Definition 9.3. Given two Lie groups G1 and G2, a homomorphism (or map) of Lie groups is a function, f : G1 ! G2, that is a homomorphism of groups and a smooth map (between the manifolds G1 and G2). Given two Lie alge ...

... Let us also recall the definition of homomorphisms of Lie groups and Lie algebras. Definition 9.3. Given two Lie groups G1 and G2, a homomorphism (or map) of Lie groups is a function, f : G1 ! G2, that is a homomorphism of groups and a smooth map (between the manifolds G1 and G2). Given two Lie alge ...

Weyl Groups Associated with Affine Reflection Systems of Type

... Theorem 3.6]. Here we record some advances made on presentations of (hyperbolic) Weyl groups of certain subclasses of affine reflection systems. As the starting point in this direction, we can name the works of [Kr], [A2] and [A1] which consider certain presentationes for toroidal Weyl groups and ex ...

... Theorem 3.6]. Here we record some advances made on presentations of (hyperbolic) Weyl groups of certain subclasses of affine reflection systems. As the starting point in this direction, we can name the works of [Kr], [A2] and [A1] which consider certain presentationes for toroidal Weyl groups and ex ...

Abstract awakenings in algebra

... describe in footnote 3 below — is one of the great stories in the history of mathematics.3 Although the solution methods discovered in the 16th century are interesting in and of themselves, certain consequences of their discovery were of even greater importance to later developments in mathematics. ...

... describe in footnote 3 below — is one of the great stories in the history of mathematics.3 Although the solution methods discovered in the 16th century are interesting in and of themselves, certain consequences of their discovery were of even greater importance to later developments in mathematics. ...

A course on finite flat group schemes and p

... 1.2. . . . and what they are good for. Special properties of p-divisible groups are used in: (1) Analysis of the local p-adic Galois action on p-torsion points of elliptic curves, see Serre’s theorem on open image for non-CM elliptic curves over number fields [Se72], and more recently in modularity ...

... 1.2. . . . and what they are good for. Special properties of p-divisible groups are used in: (1) Analysis of the local p-adic Galois action on p-torsion points of elliptic curves, see Serre’s theorem on open image for non-CM elliptic curves over number fields [Se72], and more recently in modularity ...

elements of finite order for finite monadic church-rosser

... (modulo T). The congruence class [U)T of u is the set {v E L* Iv ~ t u}. It is well known that the set of congruence classes {[u)Tlu E L*} forms a monoid under the operation [u) T 0 [v) T = [uv) T with identity [A) T" This monoid is denoted as L* / ~ t, and the ordered pair (L; T) is called a presen ...

... (modulo T). The congruence class [U)T of u is the set {v E L* Iv ~ t u}. It is well known that the set of congruence classes {[u)Tlu E L*} forms a monoid under the operation [u) T 0 [v) T = [uv) T with identity [A) T" This monoid is denoted as L* / ~ t, and the ordered pair (L; T) is called a presen ...

Galois actions on homotopy groups of algebraic varieties

... in [37] applies to give a pro–Q` –algebraic homotopy type, which is a nonnilpotent generalisation of the Q` –homotopy type of Weil II (see Deligne [5]). Its homotopy groups are `–adic schematic homotopy groups, and Theorem 3.40 gives conditions for relating these to étale homotopy groups. Explicitly ...

... in [37] applies to give a pro–Q` –algebraic homotopy type, which is a nonnilpotent generalisation of the Q` –homotopy type of Weil II (see Deligne [5]). Its homotopy groups are `–adic schematic homotopy groups, and Theorem 3.40 gives conditions for relating these to étale homotopy groups. Explicitly ...

18 Divisible groups

... natural whereas the reduced subgroup is not. [Recall that all groups are abelian in this chapter.] Definition 18.1. A group G is called divisible if for every x ∈ G and every positive integer n there is a y ∈ G so that ny = x, i.e., every element of G is divisible by every positive integer. For exam ...

... natural whereas the reduced subgroup is not. [Recall that all groups are abelian in this chapter.] Definition 18.1. A group G is called divisible if for every x ∈ G and every positive integer n there is a y ∈ G so that ny = x, i.e., every element of G is divisible by every positive integer. For exam ...

HIGHER HOMOTOPY OF GROUPS DEFINABLE IN O

... Definable groups in o-minimal structures have been studied by several authors. The class of such groups includes all algebraic groups over either a real closed field or an algebraically closed field, but is not limited to such groups. Any compact Lie group is isomorphic to a real algebraic subgroup ...

... Definable groups in o-minimal structures have been studied by several authors. The class of such groups includes all algebraic groups over either a real closed field or an algebraically closed field, but is not limited to such groups. Any compact Lie group is isomorphic to a real algebraic subgroup ...

Coarse structures on groups - St. John`s University Unofficial faculty

... a Hausdorff topological group with a compact set of generators, then the asymptotic dimension of G with respect to the group-compact coarse structure is zero if and only if G is compact (Corollary 3.8). If H is a closed subgroup of G, then asdim H ≤ asdim G (Corollary 3.11). As a consequence, all di ...

... a Hausdorff topological group with a compact set of generators, then the asymptotic dimension of G with respect to the group-compact coarse structure is zero if and only if G is compact (Corollary 3.8). If H is a closed subgroup of G, then asdim H ≤ asdim G (Corollary 3.11). As a consequence, all di ...

Classification of Semisimple Lie Algebras

... raised considerable doubts about the validity of the proof, since there is no way any one person could check the proof from beginning to end. Since then, considerable effort has been devoted to the simplification of the proof, and since 2004, a third–generation of this effort is underway (good refer ...

... raised considerable doubts about the validity of the proof, since there is no way any one person could check the proof from beginning to end. Since then, considerable effort has been devoted to the simplification of the proof, and since 2004, a third–generation of this effort is underway (good refer ...

An introduction to matrix groups and their applications

... In Chapter 1 the general linear groups GLn (k) where k = R, the real numbers, or k = C, the complex numbers, are introduced and studied as both groups and topological spaces. Matrix groups are defined and a number of standard examples are discussed, including the unimodular groups SLn (k), orthogona ...

... In Chapter 1 the general linear groups GLn (k) where k = R, the real numbers, or k = C, the complex numbers, are introduced and studied as both groups and topological spaces. Matrix groups are defined and a number of standard examples are discussed, including the unimodular groups SLn (k), orthogona ...

CLASSIFICATION OF SEMISIMPLE ALGEBRAIC MONOIDS

... about algebraic varieties, algebraic groups and algebraic monoids. (2.1) Algebraic varieties. X, V, Z etc. will denote affine varieties over the algebraically closed field k and k[X] will denote the ring of regular functions on X. X is irreducible if k[X] is an integral domain and normal if further, ...

... about algebraic varieties, algebraic groups and algebraic monoids. (2.1) Algebraic varieties. X, V, Z etc. will denote affine varieties over the algebraically closed field k and k[X] will denote the ring of regular functions on X. X is irreducible if k[X] is an integral domain and normal if further, ...

670 notes - OSU Department of Mathematics

... (Z/nZ) As a set, define (Z/nZ) to be those congruence classes whose representatives are relatively prime to n. Here’s why we do this: Prop. An integer k has an inverse mod n k and n are relatively prime. Proof. One direction is easy. In the other direction, suppose that k and n are relatively pr ...

... (Z/nZ) As a set, define (Z/nZ) to be those congruence classes whose representatives are relatively prime to n. Here’s why we do this: Prop. An integer k has an inverse mod n k and n are relatively prime. Proof. One direction is easy. In the other direction, suppose that k and n are relatively pr ...

CLASSIFICATION OF PRINCIPAL BUNDLES AND LIE GROUPOIDS

... has been so useful - except in those cases (such as that of complex line bundles) where the group is actually abelian. Rather, this classification has provided one motivation for the study of nonabelian cohomology. Here we formulate the classification problem differently and obtain a description whi ...

... has been so useful - except in those cases (such as that of complex line bundles) where the group is actually abelian. Rather, this classification has provided one motivation for the study of nonabelian cohomology. Here we formulate the classification problem differently and obtain a description whi ...

Group Theory

... He was the first to use the term “group” in a technical sense, though to him it meant a collection of permutations closed under multiplication. Galois theory will be discussed much later in these notes. Galois was also motivated by the solvability of polynomial equations of degree n. From 1815 to 18 ...

... He was the first to use the term “group” in a technical sense, though to him it meant a collection of permutations closed under multiplication. Galois theory will be discussed much later in these notes. Galois was also motivated by the solvability of polynomial equations of degree n. From 1815 to 18 ...

Some structure theorems for algebraic groups

... To make the exposition more self-contained, we have gathered basic notions and results on group schemes over a field in Section 2, referring to the books [22] and [SGA3] for most proofs. Section 3 is devoted to the proof of Theorem 1, and Section 4 to that of Theorem 2. Although the statements of bo ...

... To make the exposition more self-contained, we have gathered basic notions and results on group schemes over a field in Section 2, referring to the books [22] and [SGA3] for most proofs. Section 3 is devoted to the proof of Theorem 1, and Section 4 to that of Theorem 2. Although the statements of bo ...

# Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.