Derived Representation Theory and the Algebraic K
... theoretic model for the algebraic K-theory spectrum of the field F . It is the goal of this paper to propose such a homotopy theoretic model, and to verify it in some cases. We will construct a model which depends only on the complex representation theory (or twisted versions, when the roots of unit ...
... theoretic model for the algebraic K-theory spectrum of the field F . It is the goal of this paper to propose such a homotopy theoretic model, and to verify it in some cases. We will construct a model which depends only on the complex representation theory (or twisted versions, when the roots of unit ...
Lecture 8
... Lecture VIII - Categories and Functors Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qua ...
... Lecture VIII - Categories and Functors Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qua ...
Group Actions and Representations
... subspace, and let f1 , . . . fn be a basis of V . Writing O(V ) = V ⊕ * W and using again the formula (∗) one sees that µ∗ (V ) ⊆ O(G) ⊗ V , i.e. gfj = i hij (g−1 )fi for j = 1, . . . , n. Thus the representation of G on V is given by the map g $→ H(g) where H(g) = (hij (g−1 ))ij is an n × n matrix ...
... subspace, and let f1 , . . . fn be a basis of V . Writing O(V ) = V ⊕ * W and using again the formula (∗) one sees that µ∗ (V ) ⊆ O(G) ⊗ V , i.e. gfj = i hij (g−1 )fi for j = 1, . . . , n. Thus the representation of G on V is given by the map g $→ H(g) where H(g) = (hij (g−1 ))ij is an n × n matrix ...
Which spheres admit a topological group structure?
... (vi) if f : Sn → Sn is a continuous map without fixed points, then deg(f ) = (−1)n+1 . Hence from (iii), (v) and (vi) we get that every continuous map from the n-sphere in itself without fixed points is homotopic to the antipodal map. The converse statement is false, for if n is odd then (i), (iii) ...
... (vi) if f : Sn → Sn is a continuous map without fixed points, then deg(f ) = (−1)n+1 . Hence from (iii), (v) and (vi) we get that every continuous map from the n-sphere in itself without fixed points is homotopic to the antipodal map. The converse statement is false, for if n is odd then (i), (iii) ...
TOPOLOGICAL TRANSFORMATION GROUPS: SELECTED
... space X there exists a family of convex metrizable G-compactaQ{Kf }f ∈F such that |F | = w(X) and X possesses a G-embedding into the product f ∈F Kf . The following natural question of Antonyan remains open (even for τ = ℵ0 ). ? 1013 Question 4.5 (Antonyan [7, 8]). Let G be a uniformly Lindelöf gro ...
... space X there exists a family of convex metrizable G-compactaQ{Kf }f ∈F such that |F | = w(X) and X possesses a G-embedding into the product f ∈F Kf . The following natural question of Antonyan remains open (even for τ = ℵ0 ). ? 1013 Question 4.5 (Antonyan [7, 8]). Let G be a uniformly Lindelöf gro ...
lecture notes 5
... Definition 1.4. Let S be a subset of a group G, and let g ∈ G. By the conjugate of S by g, written g−1 S g, we mean the subset {g−1 sg : s ∈ S } of G. Given two subsets S , T of a group G, if there exists some element g ∈ G such that g−1 S g = T , then we say that S is conjugate to T . Definition 1. ...
... Definition 1.4. Let S be a subset of a group G, and let g ∈ G. By the conjugate of S by g, written g−1 S g, we mean the subset {g−1 sg : s ∈ S } of G. Given two subsets S , T of a group G, if there exists some element g ∈ G such that g−1 S g = T , then we say that S is conjugate to T . Definition 1. ...
STABLE COHOMOLOGY OF FINITE AND PROFINITE GROUPS 1
... (a) The image H i (BG, F ) → H i (X/G, F ) defines an ordering on spaces with G-action and provides with natural obstruction to the existence of G-maps between spaces with G-actions. (b) We could also replace Hom(∗, F ) by ⊗F in the beginning and get group homology. However, the cohomology groups h ...
... (a) The image H i (BG, F ) → H i (X/G, F ) defines an ordering on spaces with G-action and provides with natural obstruction to the existence of G-maps between spaces with G-actions. (b) We could also replace Hom(∗, F ) by ⊗F in the beginning and get group homology. However, the cohomology groups h ...
Category Theory: an abstract setting for analogy
... m = n = 1. Better still we might try some more complicated property of numbers such as ‘every whole number other than 1 can be written as a product of prime numbers’. The analogy works. There are knots that are ‘prime’ or ‘irreducible’: K is prime if given any equation L + M = K then either L or M m ...
... m = n = 1. Better still we might try some more complicated property of numbers such as ‘every whole number other than 1 can be written as a product of prime numbers’. The analogy works. There are knots that are ‘prime’ or ‘irreducible’: K is prime if given any equation L + M = K then either L or M m ...
Inverse semigroups and étale groupoids
... that a = aa−1a and a−1 = a−1aa−1. Example: the symmetric inverse monoid Let X be a set equipped with the discrete topology. Denote by I(X) the set of all partial bijections of X. This is an example of an inverse semigroup called the symmetric inverse monoid. Theorem [Vagner-Preston] Symmetric invers ...
... that a = aa−1a and a−1 = a−1aa−1. Example: the symmetric inverse monoid Let X be a set equipped with the discrete topology. Denote by I(X) the set of all partial bijections of X. This is an example of an inverse semigroup called the symmetric inverse monoid. Theorem [Vagner-Preston] Symmetric invers ...
The Type of the Classifying Space of a Topological Group for the
... In Section 3 we will reduce the case of a totally disconnected group to the one of a discrete group. Throughout the paper we will denote the discretization of a topological group G by Gd , i.e. the same group but now with the discrete topology. Given a family F of (closed) subgroups of G, denote by ...
... In Section 3 we will reduce the case of a totally disconnected group to the one of a discrete group. Throughout the paper we will denote the discretization of a topological group G by Gd , i.e. the same group but now with the discrete topology. Given a family F of (closed) subgroups of G, denote by ...
booklet of abstracts - DU Department of Computer Science Home
... In Group Theory there has been a lot of research on the properties of groups given the geometric and combinatorial properties of their Cayley graphs. More recently, thanks to the works of mathematicians like G. Sabidoussi, G. Gauyacq and E. Mwambené, it has been possible to define and study the Cay ...
... In Group Theory there has been a lot of research on the properties of groups given the geometric and combinatorial properties of their Cayley graphs. More recently, thanks to the works of mathematicians like G. Sabidoussi, G. Gauyacq and E. Mwambené, it has been possible to define and study the Cay ...
Subgroups of Finite Index in Profinite Groups
... obtain a central series for G/(H ∩ K) by taking a central series for G/H and extending it via intersecting a central series for HK/K in G/K with H. ...
... obtain a central series for G/(H ∩ K) by taking a central series for G/H and extending it via intersecting a central series for HK/K in G/K with H. ...
On the Universal Space for Group Actions with Compact Isotropy
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
... for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)H is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if the induced map f∗ : πn (X, x) → πn (Y, f (x)) is an isomo ...
M04/01
... modern applied mathematics. We also discuss right division in loops with the antiautomorphic inverse property. ...
... modern applied mathematics. We also discuss right division in loops with the antiautomorphic inverse property. ...
Topology in the 20th century
... properties of geometric figures and maps between them that are invariant under continuous deformations (or homotopies); here geometric figures are interpreted widely, to include any objects for which continuity makes sense. The aim of A.T. is, in principle, the complete enumeration of such properties; ...
... properties of geometric figures and maps between them that are invariant under continuous deformations (or homotopies); here geometric figures are interpreted widely, to include any objects for which continuity makes sense. The aim of A.T. is, in principle, the complete enumeration of such properties; ...
One-parameter subgroups and Hilbert`s fifth problem
... to analysis. On the analytical side of the channel, the stringency of the conditions leading to analytic structure have gradually been relaxed from requiring three times differentiable coordinates to certain rather strong Lipschitz conditions; but all conditions have been truly analytic in character ...
... to analysis. On the analytical side of the channel, the stringency of the conditions leading to analytic structure have gradually been relaxed from requiring three times differentiable coordinates to certain rather strong Lipschitz conditions; but all conditions have been truly analytic in character ...
Uniformities and uniformly continuous functions on locally
... right uniformly continuous (and vice versa). Rather surprisingly, it is still unknown if the converse holds true. OPEN QUESTION. (Itzkowitz, [5]) Is a topological group G SIN whenever every left uniformly continuous real-valued function on G is right uniformly continuous? Firstly, Itzkowitz [4] has ...
... right uniformly continuous (and vice versa). Rather surprisingly, it is still unknown if the converse holds true. OPEN QUESTION. (Itzkowitz, [5]) Is a topological group G SIN whenever every left uniformly continuous real-valued function on G is right uniformly continuous? Firstly, Itzkowitz [4] has ...
GROUPS WITH FINITELY MANY COUNTABLE MODELS Dejan Ilić
... I(ℵ0 , ThL (L)) = I ℵ0 , ThLG (G(M)) and if ThL (L) is an Ehrenfeucht theory, then ThLG (G(M)) is an Ehrenfeucht theory, too. Since Ehrenfeucht’s example can be turned into a densely ordered relational structure, we obtain: Theorem 1.1. There is an Abelian group (with additional structure) whose the ...
... I(ℵ0 , ThL (L)) = I ℵ0 , ThLG (G(M)) and if ThL (L) is an Ehrenfeucht theory, then ThLG (G(M)) is an Ehrenfeucht theory, too. Since Ehrenfeucht’s example can be turned into a densely ordered relational structure, we obtain: Theorem 1.1. There is an Abelian group (with additional structure) whose the ...
Symmetry in the World of Man and Nature -RE-S-O-N-A-N-C
... Theorem 1. An isometry that fixes two distinct points fixes the entire line passing through them. An isometry that fixes three non-collinear points is the identity map. Proof. Let f be an isometry that fixes two points A, B. Then, any point P on the line AB is uniquely specified by the two distances ...
... Theorem 1. An isometry that fixes two distinct points fixes the entire line passing through them. An isometry that fixes three non-collinear points is the identity map. Proof. Let f be an isometry that fixes two points A, B. Then, any point P on the line AB is uniquely specified by the two distances ...
Chapter 10. Abstract algebra
... Groups and subgroups are algebraic structures. They are the ones that allow solving equations like x +x =a ⇒x = ...
... Groups and subgroups are algebraic structures. They are the ones that allow solving equations like x +x =a ⇒x = ...
C3.4b Lie Groups, HT2015 Homework 4. You
... 1Recall the centre of a group is Z(G) = {g ∈ G : hg = gh for all h ∈ G} = {g ∈ G : hgh−1 = g for all h ∈ G}. 2meaning continuous loops can always be continuously deformed to a point. ...
... 1Recall the centre of a group is Z(G) = {g ∈ G : hg = gh for all h ∈ G} = {g ∈ G : hgh−1 = g for all h ∈ G}. 2meaning continuous loops can always be continuously deformed to a point. ...
On Schwartz groups - Instytut Matematyczny PAN
... We start Section 3 by defining Schwartz groups. Then we obtain permanence properties for this class with respect to subgroups, Hausdorff quotients, products and local isomorphisms, and show that bounded subsets of locally quasi-convex Schwartz groups are precompact. We finish this section by proving ...
... We start Section 3 by defining Schwartz groups. Then we obtain permanence properties for this class with respect to subgroups, Hausdorff quotients, products and local isomorphisms, and show that bounded subsets of locally quasi-convex Schwartz groups are precompact. We finish this section by proving ...
algebra part of MT2002 - MacTutor History of Mathematics
... that the following hold: Closure: For all x, y ∈ G we have x ∗ y ∈ G. Associativity: For all x, y, z ∈ G we have (x ∗ y) ∗ z = x ∗ (y ∗ z). Identity: There exists a distinguished element e such that for all x ∈ G we have x ∗ e = e ∗ x = x. Inverses: For every x ∈ G there is a distinguished element x ...
... that the following hold: Closure: For all x, y ∈ G we have x ∗ y ∈ G. Associativity: For all x, y, z ∈ G we have (x ∗ y) ∗ z = x ∗ (y ∗ z). Identity: There exists a distinguished element e such that for all x ∈ G we have x ∗ e = e ∗ x = x. Inverses: For every x ∈ G there is a distinguished element x ...
A Brief Summary of the Statements of Class Field Theory
... group, the Pontryagin dual of the profinite group Gal(K ab /K). It follows that the problem of classifying finite abelian extensions of K is more or less the same as the problem of describing all these characters. The Langlands program is an attempt to understand Gal(K s /K) more completely by descr ...
... group, the Pontryagin dual of the profinite group Gal(K ab /K). It follows that the problem of classifying finite abelian extensions of K is more or less the same as the problem of describing all these characters. The Langlands program is an attempt to understand Gal(K s /K) more completely by descr ...
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.