Maximal compact subgroups in the o-minimal setting

... The canonical projection π : G → G/N (G) induces an isomorphism between K and the maximal definable definably compact subgroup of G/N (G). Also P = A(G) × K is the smallest definable subgroup of G containing K, and [P, P ] = [K, K] is a maximal definable semisimple definably compact subgroup of G. C ...

... The canonical projection π : G → G/N (G) induces an isomorphism between K and the maximal definable definably compact subgroup of G/N (G). Also P = A(G) × K is the smallest definable subgroup of G containing K, and [P, P ] = [K, K] is a maximal definable semisimple definably compact subgroup of G. C ...

Locally analytic vectors in representations of locally p

... of G. In fact, we prove a slightly more general result, which implies not only that Dla (H, K) is a Fréchet-Stein algebra, but also that the completed tensor product ˆ K Dla (H, K) is Fréchet-Stein, for a fairly general class of commutative FréchetA⊗ Stein K-algebras A, namely, those possessing a ...

... of G. In fact, we prove a slightly more general result, which implies not only that Dla (H, K) is a Fréchet-Stein algebra, but also that the completed tensor product ˆ K Dla (H, K) is Fréchet-Stein, for a fairly general class of commutative FréchetA⊗ Stein K-algebras A, namely, those possessing a ...

MODEL-CATEGORIES OF COALGEBRAS OVER OPERADS JUSTIN R. SMITH

... We develop the general theory for unbounded coalgebras, and derive the bounded results by applying a truncation functor. In § 2, we define operads and coalgebras over operads. We also give a basic condition (see 4.3) on the operad under consideration that we assume to hold throughout the paper. Cofi ...

... We develop the general theory for unbounded coalgebras, and derive the bounded results by applying a truncation functor. In § 2, we define operads and coalgebras over operads. We also give a basic condition (see 4.3) on the operad under consideration that we assume to hold throughout the paper. Cofi ...

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 Point-Set Topology

... then x and y will not differ in any entry, in which case d(x, y ) = 0. However, if x 6= y , then x and y will differ in at least one entry, so d(x, y ) > 0. We have the Triangle Inequality left to prove. Exercise. What if our alphabet has more than 2 letters (say, 26)? Dan Swenson, Black Hills Sta ...

... then x and y will not differ in any entry, in which case d(x, y ) = 0. However, if x 6= y , then x and y will differ in at least one entry, so d(x, y ) > 0. We have the Triangle Inequality left to prove. Exercise. What if our alphabet has more than 2 letters (say, 26)? Dan Swenson, Black Hills Sta ...

Properties of Algebraic Spaces

... In other words, the small Zariski site XZar of X is canonically identified with a site associated to the topological space |X| (see Sites, Example 6.4). Proof. In order to prove (1) let us construct the inverse of the rule. Namely, suppose that W ⊂ |X| is open. Choose a presentation X = U/R correspo ...

... In other words, the small Zariski site XZar of X is canonically identified with a site associated to the topological space |X| (see Sites, Example 6.4). Proof. In order to prove (1) let us construct the inverse of the rule. Namely, suppose that W ⊂ |X| is open. Choose a presentation X = U/R correspo ...

Motivic Homotopy Theory

... More concretely, let’s endow the category Sch/k of schemes of finite type over an algebraically closed field k with a homotopy theory. The first observation is that from a categorical point of view, the category Sch/k is intractable since it does not contain all colimits. A good (universal) way to ...

... More concretely, let’s endow the category Sch/k of schemes of finite type over an algebraically closed field k with a homotopy theory. The first observation is that from a categorical point of view, the category Sch/k is intractable since it does not contain all colimits. A good (universal) way to ...

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 ...

# Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.