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 ...

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. ...

... 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

... 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 ...

... 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

... 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 ...

... 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

... 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 ...

... 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

... 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 ...

... 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

... 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. ...

... 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

... 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 ...

... 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 ...

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. ...

... 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. ...

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. ...

... 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. ...

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 ...

... 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 ...

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 ...

... 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 ...

Cloak and dagger

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

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

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 ...

... 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

... PAVEL ETINGOF, DMITRI NIKSHYCH, AND VIKTOR OSTRIK ...

... PAVEL ETINGOF, DMITRI NIKSHYCH, AND VIKTOR OSTRIK ...

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 ...

... 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

... 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 ...

... 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

... 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. ...

... 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

... 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 ...

... 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 ...

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 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. ...

... 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

... 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 ...

... 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

... 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). ...

... 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

... 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 ...

... 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 ...