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Maximal compact subgroups in the o-minimal setting
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 ...
An Introduction to Topological Groups
An Introduction to Topological Groups

... Let G be a group and τ a topology on G which is Čech–complete (e.g. complete and metrizable). If all left and right translations are continuous, then (G, τ) is a topological group. ...
On skew Heyting algebras - ars mathematica contemporanea
On skew Heyting algebras - ars mathematica contemporanea

... While Boolean algebras provide algebraic models of classical logic, Heyting algebras provide algebraic models of intuitionistic logic. In the present paper we introduce the notion of a skew Heyting algebra. In passing to the non-commutative setting one needs to sacrifice either the top or the bottom ...
PDF file - Library
PDF file - Library

... The Brauer group is something like a mathematical chameleon, it assumes the characteristics of its environment. For example, if you look at it from the point of view of representation theory you seem to be dealing with classes of noncommutative algebras appearing in the representation theory of fini ...
Iterated Bar Complexes of E-infinity Algebras and Homology
Iterated Bar Complexes of E-infinity Algebras and Homology

... spectral sequence of [1] which is defined with Goodwillie’s calculus of functors. On one hand, one might gain quantitative information on the cohomology of iterated loop spaces from the study of such spectral sequences, arising from filtrations of iterated bar complexes. On the other hand, the conne ...
Some structure theorems for algebraic groups
Some structure theorems for algebraic groups

... setting of smooth connected algebraic groups again, Theorem 2 was announced by Chevalley in the early 1950’s. But he published his proof in 1960 only (see [17]), as he had first to build up a theory of Picard and Albanese varieties. Meanwhile, proofs of Chevalley’s theorem had been published by Bars ...
Lecture Notes
Lecture Notes

... with M.n; R/ and to identify the Lie group GL.Rn / with GL.n; R/: We shall now discuss an important criterion for a subgroup of a Lie group G to be a Lie group. In particular this criterion will have useful applications for G D GL.V /: We start with a result that illustrates the idea of homogeneity. ...
Lie groups, lecture notes
Lie groups, lecture notes

... with M.n; R/ and to identify the Lie group GL.Rn / with GL.n; R/: We shall now discuss an important criterion for a subgroup of a Lie group G to be a Lie group. In particular this criterion will have useful applications for G D GL.V /: We start with a result that illustrates the idea of homogeneity. ...
Quaternion algebras and quadratic forms
Quaternion algebras and quadratic forms

... We will often denote quaternion algebras by B, and use the letter A for a non-necessarily quaternion algebra. (In number theory it’s common to use B, or sometimes D, for quaternion algebras. I believe this is because A often was used for an abelian ring, so B was used for something nonabelian (belia ...
SEMIDEFINITE DESCRIPTIONS OF THE CONVEX HULL OF
SEMIDEFINITE DESCRIPTIONS OF THE CONVEX HULL OF

... Throughout much of the paper, we consider only spectrahedral representations, confining our discussion of PSD lifts to section 5.2. Doubly spectrahedral convex sets. In this paper, we are interested in both SO(n)◦ and conv SO(n), and so we study both from the point of view of semidefinite programming. ...
Amenability for dual Banach algebras
Amenability for dual Banach algebras

... n=1 Mn , the proof of non-amenability requires that amenability implies nuclearity. • If G is a locally compact group, then M (G) is amenable if and only if G is discrete and amenable ([D–G–H]). • The only Banach spaces E for which L(E) is known to be amenable are the finite-dimensional ones, and th ...
Glanon groupoids - Dr. Madeleine Jotz Lean
Glanon groupoids - Dr. Madeleine Jotz Lean

... in the sense of Hitchin [15] on a manifold M is a smooth bundle map J : T M ⊕ T ∗ M → T M ⊕ T ∗ M such that J 2 = − idT M ⊕T ∗ M , J J ∗ = idT M ⊕T ∗ M , and the +i-eigenbundle of J is involutive with respect to the Courant bracket (or, equivalently, the Nijenhuis tensor of J vanishes). A generalize ...
Cloak and dagger
Cloak and dagger

... Frobenius law in physics Quantum field theory: replace particles by fields; state space varies over space-time I ...
Frobenius monads and pseudomonoids
Frobenius monads and pseudomonoids

... many score years. A lot of information about a group G is contained in its characters. Characters are group morphisms from G into the multiplicative monoid of an appropriate field k. In other words, we find a category (in this case the category of monoids) where G and k both live as objects so that ...
On fusion categories - Annals of Mathematics
On fusion categories - Annals of Mathematics

... PAVEL ETINGOF, DMITRI NIKSHYCH, AND VIKTOR OSTRIK ...
Lecture Notes on C -algebras
Lecture Notes on C -algebras

... As for showing the spectrum is non-empty, the basic idea is as follows. If a is in A and λ1 − a is invertible, for all λ in C, then we may take a non-zero linear functional φ and look at φ((λ1 − a)−1 ). One first shows this function is analytic. Then by using the formula from the last paragraph for ...
slides
slides

... The existence of such products for each ordered pair of objects (and also what is called a terminal object) gives rise to a monoidal structure on C which is referred to as a cartesian monoidal category. For instance the category of sets with the cartesian product is a cartesian monoidal category, so ...
Algebraic D-groups and differential Galois theory
Algebraic D-groups and differential Galois theory

... will see that such a generalized logarithmic derivative is essentially equivalent to an algebraic D-group structure on G (in the sense of Buium [3]). Our resulting exposition of the generalized differential Galois theory will be equivalent to that in [8] when the base field K is algebraically closed. ...
A REDUCTION TO THE COMPACT CASE FOR GROUPS
A REDUCTION TO THE COMPACT CASE FOR GROUPS

... The subgroup H can be decomposed as a product of definably connected subgroups H = A · N , with A abelian and N nilpotent. The previous theorem and the work on torsion-free definable groups made in [PeSta], yields a reduction to the compact case in analogy to the real case Fact 1.1: Theorem 1.3. Eve ...
[math.QA] 23 Feb 2004 Quantum groupoids and
[math.QA] 23 Feb 2004 Quantum groupoids and

... atmosphere, and excellent research conditions. ...
An introduction to matrix groups and their applications
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 ...
Classification of 7-Dimensional Unital Commutative Algebras
Classification of 7-Dimensional Unital Commutative Algebras

... It follows that the following linear system (with q11 , q12 , q21 , and q22 as the unknowns) has a nonzero solution: β1 q11 + β1 ασ(1) q12 − q21 − ασ(1) q22 = 0, β2 q11 + β2 ασ(2) q12 − q21 − ασ(2) q22 = 0, β3 q11 + β3 ασ(3) q12 − q21 − ασ(3) q22 = 0, β4 q11 + β4 ασ(4) q12 − q21 − ασ(4) q22 = 0. ...
Nondegenerate Solutions to the Classical Yang
Nondegenerate Solutions to the Classical Yang

... He developed the theory of “finite and continuous groups” and in 1873, outlined a general theory of not necessarily commutative groups. In principle, Lie’s theory reduced problems on Lie groups, of an analytic nature to algebraic problems on Lie algebras, and this led to an intrinsic study of Lie al ...
From Poisson algebras to Gerstenhaber algebras
From Poisson algebras to Gerstenhaber algebras

... Frolicher-Nijenhuis bracket on cochains on associative (resp., Lie) algebras is closely related to the derived bracket constructed from the composition (resp., Nijenhuis-Richardson) bracket and the Hochschild (resp., ChevalleyEilenberg) differential on an associative algebra (resp., a Lie algebra). ...
A MONOIDAL STRUCTURE ON THE CATEGORY OF
A MONOIDAL STRUCTURE ON THE CATEGORY OF

... have been discussed in [5], in the particular case where C is the category of vector spaces over a field k. A monoidal structure on B CA can be constructed if A is a bialgebra and two additional compatibility conditions are satisfied. The aim of this paper is to present a more general result in the ...
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Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.A feature of representation theory is its pervasiveness in mathematics. There are two sides to this. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory: illuminates and generalizes Fourier analysis via harmonic analysis, is connected to geometry via invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlands program.The second aspect is the diversity of approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.A representation should not be confused with a presentation.
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