AN INTRODUCTION TO KK-THEORY These are the lecture notes of
... Proof. We only prove the last assertion here. The map T → T ⊗ 1 is linear and contractive from L(E1 , E2 ) to L(E1 ⊗ F, E2 ⊗ F ). So it suffices to consider T of the form θe2 ,e1 with e1 ∈ E1 and e2 ∈ E2 . Because E2 = E2 · B, it suffices to consider θe2 b,e1 with b ∈ B. Now for all e01 ⊗ f ∈ E1 ⊗ F ...
... Proof. We only prove the last assertion here. The map T → T ⊗ 1 is linear and contractive from L(E1 , E2 ) to L(E1 ⊗ F, E2 ⊗ F ). So it suffices to consider T of the form θe2 ,e1 with e1 ∈ E1 and e2 ∈ E2 . Because E2 = E2 · B, it suffices to consider θe2 b,e1 with b ∈ B. Now for all e01 ⊗ f ∈ E1 ⊗ F ...
- Journal of Linear and Topological Algebra
... the existence of module virtual (approximate) diagonals in an appropriate sense. Also the main example in [7] asserts that for a discrete abelian group G, Lp (G) is △-amenable as an L1 (G)-module if and only if G is amenable. In this paper we shall focus on an especial kind of Banach algebras which ...
... the existence of module virtual (approximate) diagonals in an appropriate sense. Also the main example in [7] asserts that for a discrete abelian group G, Lp (G) is △-amenable as an L1 (G)-module if and only if G is amenable. In this paper we shall focus on an especial kind of Banach algebras which ...
Frobenius monads and pseudomonoids
... to the categorical notions just mentioned and a few others we require. Of course, the use in physics of (classical) groups and their representations goes back many score years. A lot of information about a group G is contained in its characters. Characters are group morphisms from G into the multipl ...
... to the categorical notions just mentioned and a few others we require. Of course, the use in physics of (classical) groups and their representations goes back many score years. A lot of information about a group G is contained in its characters. Characters are group morphisms from G into the multipl ...
Elliptic spectra, the Witten genus, and the theorem of the cube
... In §2.5 we give in more detail the argument sketched in the introduction that there is a unique cubical structure on any elliptic curve. We give an argument with explicit formulae which works when the elliptic curves in question are allowed to degenerate to singular cubics (“generalized elliptic cur ...
... In §2.5 we give in more detail the argument sketched in the introduction that there is a unique cubical structure on any elliptic curve. We give an argument with explicit formulae which works when the elliptic curves in question are allowed to degenerate to singular cubics (“generalized elliptic cur ...
Diffeological Levi-Civita connections
... be compatible if for all y ∈ Y and for any two compatible (in the sense of Definition 1.4) pairs (ω1 , ω2 ) and (µ1 , µ2 ), where ω1 , µ1 ∈ (π1Λ )−1 (y) and ω2 , µ2 ∈ (π2Λ )−1 (f (y)) we have g1Λ (y)(ω1 , µ1 ) = g2Λ (f (y))(ω2 , µ2 ). Then the following is true. Theorem 1.8. Let X1 and X2 be two dif ...
... be compatible if for all y ∈ Y and for any two compatible (in the sense of Definition 1.4) pairs (ω1 , ω2 ) and (µ1 , µ2 ), where ω1 , µ1 ∈ (π1Λ )−1 (y) and ω2 , µ2 ∈ (π2Λ )−1 (f (y)) we have g1Λ (y)(ω1 , µ1 ) = g2Λ (f (y))(ω2 , µ2 ). Then the following is true. Theorem 1.8. Let X1 and X2 be two dif ...
1. Group actions and other topics in group theory
... turns out that it plays an important role in the structure theory of a large class of interesting groups (not just the general linear groups), and enters into algebraic geometry and topology via “Schubert varieties” and “Schubert cells”. For example, when F = R or F = C then the orbit set GLn F/B is ...
... turns out that it plays an important role in the structure theory of a large class of interesting groups (not just the general linear groups), and enters into algebraic geometry and topology via “Schubert varieties” and “Schubert cells”. For example, when F = R or F = C then the orbit set GLn F/B is ...
Quotient Modules in Depth
... how weak a notion of normality is this?” For example, for subgroups and their corresponding group algebra pairs, the answers to these questions are in [7] and in [11, 5], respectively, where it is also noted that subgroup depth of any finitedimensional pair of group algebras is finite. The challenge ...
... how weak a notion of normality is this?” For example, for subgroups and their corresponding group algebra pairs, the answers to these questions are in [7] and in [11, 5], respectively, where it is also noted that subgroup depth of any finitedimensional pair of group algebras is finite. The challenge ...
Subject: Mathematics Lesson: Isomorphism and Theorems on
... An isomorphism from a group G to a group G is one to one mapping from G to G that preserved the group operation. Value Addition: Note Two groups G and G are called isomorphic, written as G1 ~ G2. If there is an isomorphism from Gonto G' i.e. two groups g and g are isomorphic if there exist a ma ...
... An isomorphism from a group G to a group G is one to one mapping from G to G that preserved the group operation. Value Addition: Note Two groups G and G are called isomorphic, written as G1 ~ G2. If there is an isomorphism from Gonto G' i.e. two groups g and g are isomorphic if there exist a ma ...
THE COHOMOLOGY RING OF FREE LOOP SPACES 1. Introduction
... is also a differential graded Hopf algebra up to homotopy (notion defined page 14). In section 5, using the Perturbation Lemma, we construct a small algebra up to homotopy whose homology is isomorphic to HH ∗ (A). In section 6, we make explicit this algebra up to homotopy in the following particular ...
... is also a differential graded Hopf algebra up to homotopy (notion defined page 14). In section 5, using the Perturbation Lemma, we construct a small algebra up to homotopy whose homology is isomorphic to HH ∗ (A). In section 6, we make explicit this algebra up to homotopy in the following particular ...
Algebra 3(written by Ngo Bao Chau)
... as a Q-vector space. We define TrL/Q (α) as the trace of this transformation and NmL/Q (α) as its determinant, the subscript L/Q can be dropped if no confusion is possible. The Q-linear transformation induces by the multiplication by α in L has a characteristic polynomial ch(α) = x n − c 1 x n−1 + · ...
... as a Q-vector space. We define TrL/Q (α) as the trace of this transformation and NmL/Q (α) as its determinant, the subscript L/Q can be dropped if no confusion is possible. The Q-linear transformation induces by the multiplication by α in L has a characteristic polynomial ch(α) = x n − c 1 x n−1 + · ...
Homework assignments
... 2 (x + y) ∈ X. An element a ∈ X is said to be an extremal point of X if the only solution (x, y) ∈ X × X of the equation a = 21 (x + y) is x = y = a. Find all extremal points of X. 5. The algebra H of Quaternions is defined as a 4-dimensional R-algebra with basis {1, i, j, k} and the following multi ...
... 2 (x + y) ∈ X. An element a ∈ X is said to be an extremal point of X if the only solution (x, y) ∈ X × X of the equation a = 21 (x + y) is x = y = a. Find all extremal points of X. 5. The algebra H of Quaternions is defined as a 4-dimensional R-algebra with basis {1, i, j, k} and the following multi ...
On finite congruence
... Vandiver [4]. Though the concept of a semiring might seem a bit strange and unmotivated, additively commutative semirings arise naturally as the endomorphisms of commutative semigroups. Furthermore, every such semiring is isomorphic to a sub-semiring of such endomorphisms [2]. For a more thorough in ...
... Vandiver [4]. Though the concept of a semiring might seem a bit strange and unmotivated, additively commutative semirings arise naturally as the endomorphisms of commutative semigroups. Furthermore, every such semiring is isomorphic to a sub-semiring of such endomorphisms [2]. For a more thorough in ...
Methods of Mathematical Physics II
... Thus fν = Aµν fµ0 and the fµ components transform in the same way as the basis. They are therefore said to transform covariantly. Given a basis eµ of V , we can define a dual basis for V ∗ as the set of covectors e∗µ ∈ V ∗ such that e∗µ (eν ) = δνµ . It is clear that this is a basis for V ∗ , and th ...
... Thus fν = Aµν fµ0 and the fµ components transform in the same way as the basis. They are therefore said to transform covariantly. Given a basis eµ of V , we can define a dual basis for V ∗ as the set of covectors e∗µ ∈ V ∗ such that e∗µ (eν ) = δνµ . It is clear that this is a basis for V ∗ , and th ...
LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS 7. Hochschild cohomology and deformations
... A0 . Definitely, if A is a graded deformation of A0 over S(P ), then A/AP 2 is a 1st order deformation of A0 . So understanding 1st order deformation is the 1st step in understanding deformations over S(P ). In what follows we will always assume that the degree of P is 2. We denote the product on A0 ...
... A0 . Definitely, if A is a graded deformation of A0 over S(P ), then A/AP 2 is a 1st order deformation of A0 . So understanding 1st order deformation is the 1st step in understanding deformations over S(P ). In what follows we will always assume that the degree of P is 2. We denote the product on A0 ...
A MONOIDAL STRUCTURE ON THE CATEGORY OF
... B ⊗ A → A, which is not assumed to be associative or unital from the beginning. These additional structures (coalgebra and B-action) on A allow us to define a right A-action on the tensor product of two relative Hopf modules, and on the unit object 1 of C. Left B-coactions on these objects are suppl ...
... B ⊗ A → A, which is not assumed to be associative or unital from the beginning. These additional structures (coalgebra and B-action) on A allow us to define a right A-action on the tensor product of two relative Hopf modules, and on the unit object 1 of C. Left B-coactions on these objects are suppl ...
A Relative Spectral Sequence for Topological Hochschild Homology
... the unit map k → R is a cofibration in the Quillen model category of kalgebras (see section VII.4 of [EKMM]); similarly R is called a q-cofibrant commutative k-algebra if its unit map is a cofibration in the model category of commutative k-algebras. Lemma VII.5.8 in [EKMM] shows that for any k-algeb ...
... the unit map k → R is a cofibration in the Quillen model category of kalgebras (see section VII.4 of [EKMM]); similarly R is called a q-cofibrant commutative k-algebra if its unit map is a cofibration in the model category of commutative k-algebras. Lemma VII.5.8 in [EKMM] shows that for any k-algeb ...
Shimura.pdf
... to this quotient and then to prove that this quotient “is” indeed a quasi-projective normal variety over C (more precisely, that it can be endowed with a canonical embedding into a complex projective space with an image whose closure is a projective normal variety). 1.3. Torsion points and level str ...
... to this quotient and then to prove that this quotient “is” indeed a quasi-projective normal variety over C (more precisely, that it can be endowed with a canonical embedding into a complex projective space with an image whose closure is a projective normal variety). 1.3. Torsion points and level str ...
Basic Arithmetic Geometry Lucien Szpiro
... module A(M ×N ) /R and by x ⊗ y the image of (x, y) in the quotient. Then, for each A-module P and each A-bilinear map ϕ : M × N → P , there exists a unique A-linear map ψ : M ⊗A N → P such that for each (x, y) ∈ M × N , ϕ ((x, y)) = ψ (x ⊗ y). The proof is straightforward. It is clear from the prop ...
... module A(M ×N ) /R and by x ⊗ y the image of (x, y) in the quotient. Then, for each A-module P and each A-bilinear map ϕ : M × N → P , there exists a unique A-linear map ψ : M ⊗A N → P such that for each (x, y) ∈ M × N , ϕ ((x, y)) = ψ (x ⊗ y). The proof is straightforward. It is clear from the prop ...
Dualizing DG modules and Gorenstein DG algebras
... algebras and DG modules; if this is not the case, then they may consult, for instance, [8]. Moreover, in what follows, a few well known results in this subject are used; for these we quote from [4] whenever possible, in the interest of uniformity. A few words about notation: Most graded objects that ...
... algebras and DG modules; if this is not the case, then they may consult, for instance, [8]. Moreover, in what follows, a few well known results in this subject are used; for these we quote from [4] whenever possible, in the interest of uniformity. A few words about notation: Most graded objects that ...
F1.3YE2/F1.3YK3 ALGEBRA AND ANALYSIS Part 2: ALGEBRA
... result a − b is not always a natural number. (It may be a negative integer.) If we allow a, b to be arbitrary integers, we can add, multiply and subtract them and the result will also be an integer. We can also divide a by b (provided b 6= 0), but the result will not always be an integer. If a, b ar ...
... result a − b is not always a natural number. (It may be a negative integer.) If we allow a, b to be arbitrary integers, we can add, multiply and subtract them and the result will also be an integer. We can also divide a by b (provided b 6= 0), but the result will not always be an integer. If a, b ar ...
Lie Theory, Universal Enveloping Algebras, and the Poincar้
... Proof. Since Φ is a continuous group homomorphism and etX is also continuous, we know Φ(etX ) will be a one-parameter subgroup of H. By Theorem 3.8, we know there is a unique matrix Z so that Φ(etX ) = etZ for all t ∈ R. Furthermore, we know Z ∈ h, since etZ = Φ(etX ) ∈ H for all t. Now we simply de ...
... Proof. Since Φ is a continuous group homomorphism and etX is also continuous, we know Φ(etX ) will be a one-parameter subgroup of H. By Theorem 3.8, we know there is a unique matrix Z so that Φ(etX ) = etZ for all t ∈ R. Furthermore, we know Z ∈ h, since etZ = Φ(etX ) ∈ H for all t. Now we simply de ...
Moduli Problems for Ring Spectra - International Mathematical Union
... The situation is dramatically simpler if we wish to study not arbitrary ncategories, but n-groupoids. An n-category C is called an n-groupoid if every k-morphism in C is invertible. If X is any topological space, then the n-category π≤n X is an example of an n-groupoid: for example, the 1-morphisms ...
... The situation is dramatically simpler if we wish to study not arbitrary ncategories, but n-groupoids. An n-category C is called an n-groupoid if every k-morphism in C is invertible. If X is any topological space, then the n-category π≤n X is an example of an n-groupoid: for example, the 1-morphisms ...
Independence Theorem and Flat Base Change
... Let I 0 be an injective R0 -module. Then Γa (I 0R ) is a quasi-divisible R-module. Proof. By 3.14 ΓaR0 (I 0 ) is injective and thus by Remark 6 a quasi-divisible R0 -module. So, by Lemma 8, ΓaR0 (I 0 ) R is a quasi-divisible R-module, and by Remark and Exercise 7 we have ΓaR0 (I 0 )R = Γa (I 0R ...
... Let I 0 be an injective R0 -module. Then Γa (I 0R ) is a quasi-divisible R-module. Proof. By 3.14 ΓaR0 (I 0 ) is injective and thus by Remark 6 a quasi-divisible R0 -module. So, by Lemma 8, ΓaR0 (I 0 ) R is a quasi-divisible R-module, and by Remark and Exercise 7 we have ΓaR0 (I 0 )R = Γa (I 0R ...
Quotient Morphisms, Compositions, and Fredholm Index
... form X/X0 , where X is a vector space and X0 is a vector subspace of X. As X/X0 is itself a vector space, a legitimate question is why to complicate the matter instead of simply using vector spaces. The answer to this question is that this framework allows us to introduce a composition of morphisms ...
... form X/X0 , where X is a vector space and X0 is a vector subspace of X. As X/X0 is itself a vector space, a legitimate question is why to complicate the matter instead of simply using vector spaces. The answer to this question is that this framework allows us to introduce a composition of morphisms ...
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.