Topological Field Theories
... • More importantly, M 7→ C ∗ (M ) − Mod does not respect the tensor structure! We will now concentrate on solving this last issue by using the topological category structures just defined on ] n and Catk . Cob ...
... • More importantly, M 7→ C ∗ (M ) − Mod does not respect the tensor structure! We will now concentrate on solving this last issue by using the topological category structures just defined on ] n and Catk . Cob ...
Algebras over a field
... G is a basis for F G: φ( ag g) = ag ψ(g). Since ψ is a group homomorphism, it follows immediately that φ is an F -algebra homomorphism. As is our custom, we will reformulate this proposition in two ways: Plain English version: If you want to define an F -algebra homomorphism F G−→R, it is enough (in ...
... G is a basis for F G: φ( ag g) = ag ψ(g). Since ψ is a group homomorphism, it follows immediately that φ is an F -algebra homomorphism. As is our custom, we will reformulate this proposition in two ways: Plain English version: If you want to define an F -algebra homomorphism F G−→R, it is enough (in ...
The Coinvariant Algebra in Positive Characteristic
... v ∈ V = V ⊗k k̄ is regular if and only if it is not in any reflecting hyperplane of any reflection in G, if and only if v is fixed by no reflection in G. An element c ∈ G is regular if and only if it has an eigenvector which is regular. Examples: (Springer, Invent. Math 25 (1974)) With Σn permuting ...
... v ∈ V = V ⊗k k̄ is regular if and only if it is not in any reflecting hyperplane of any reflection in G, if and only if v is fixed by no reflection in G. An element c ∈ G is regular if and only if it has an eigenvector which is regular. Examples: (Springer, Invent. Math 25 (1974)) With Σn permuting ...
THE LOWER ALGEBRAIC K-GROUPS 1. Introduction
... classes of finitely generated projective left R-modules. For a finitely generated projective left R-module M let M ∈ Proj(R) denote the isomorphism class of M . Remark 2.2. We consider arbitrary rings only in this subsection and the first subsection in the K1 -section for the sake of more generality ...
... classes of finitely generated projective left R-modules. For a finitely generated projective left R-module M let M ∈ Proj(R) denote the isomorphism class of M . Remark 2.2. We consider arbitrary rings only in this subsection and the first subsection in the K1 -section for the sake of more generality ...
Homework #3 Solutions (due 9/26/06)
... determined by a choice of image ϕ(1) ∈ Z. For n ∈ Z, let ϕn : Z → Z be the homomorphism with ϕn (1) = n, then for any a ∈ Z, ϕn (a) = n · a. Since multiplication of integers distributes over addition, we see that each ϕn is in fact a homomorphism, so the collection {ϕn : Z → Z : n ∈ Z} constitutes a ...
... determined by a choice of image ϕ(1) ∈ Z. For n ∈ Z, let ϕn : Z → Z be the homomorphism with ϕn (1) = n, then for any a ∈ Z, ϕn (a) = n · a. Since multiplication of integers distributes over addition, we see that each ϕn is in fact a homomorphism, so the collection {ϕn : Z → Z : n ∈ Z} constitutes a ...
Semidirect Products
... Proof. We identify K with the set of all (a, 1) where a ∈ K. This is a normal subgroup of K oθ Q: (∗, x)(c, 1)(∗, x−1 ) = (∗θx (c)∗, xx−1 ) ∈ K H = {(1, x) | x ∈ Q} is a subgroup complementary to K and isomorphic to Q. Thus K oθ Q is a semidirect product of K by Q. It is also clear that this realiz ...
... Proof. We identify K with the set of all (a, 1) where a ∈ K. This is a normal subgroup of K oθ Q: (∗, x)(c, 1)(∗, x−1 ) = (∗θx (c)∗, xx−1 ) ∈ K H = {(1, x) | x ∈ Q} is a subgroup complementary to K and isomorphic to Q. Thus K oθ Q is a semidirect product of K by Q. It is also clear that this realiz ...
Group Theory: Basic Concepts Contents 1 Definitions
... • If the order of a is infinite, hai = Z. ⋆ More generally, if T is any subset of G, hT i is defined to be the smallest subgroup containing T as a subset, the subgroup generated by T . If hT i = G, one says that T , or the elements in T , generate G. The {} denoting a set can be omitted; h{a, b}i ca ...
... • If the order of a is infinite, hai = Z. ⋆ More generally, if T is any subset of G, hT i is defined to be the smallest subgroup containing T as a subset, the subgroup generated by T . If hT i = G, one says that T , or the elements in T , generate G. The {} denoting a set can be omitted; h{a, b}i ca ...
Algebra Quals Fall 2012 1. This is an immediate consequence of the
... are a basis, then a − ci a = 0 gives an equation that a satisfies, which must divide f since a is a root of f . Consider the equation aq = b, let c = N (a), so cq = bd . Since gcd(q, d) = 1, there exist r, s such that sq + rd = 1, so b = bsq+rd = bsq crq = (bs cr )q , so b is a q − th power, and (a/ ...
... are a basis, then a − ci a = 0 gives an equation that a satisfies, which must divide f since a is a root of f . Consider the equation aq = b, let c = N (a), so cq = bd . Since gcd(q, d) = 1, there exist r, s such that sq + rd = 1, so b = bsq+rd = bsq crq = (bs cr )q , so b is a q − th power, and (a/ ...
UNIT-V - IndiaStudyChannel.com
... 18.Define Semi group and monoid. Give an example of a semi group which is not a monoid Definition : Semi group Let S be a non empty set and be a binary operation on S. The algebraic system (S, ) is called a semigroup if the operation is associative. In other words (S, ) is semi group if for any x,y, ...
... 18.Define Semi group and monoid. Give an example of a semi group which is not a monoid Definition : Semi group Let S be a non empty set and be a binary operation on S. The algebraic system (S, ) is called a semigroup if the operation is associative. In other words (S, ) is semi group if for any x,y, ...
Basic reference for the course - D-MATH
... by xy. Sometimes rings defined by (i)-(iv) are called unital rings since the existence of a multiplicative unit 1 is stipulated. The standard number systems all define rings: (Z, +, ·, 0, 1), (Q, +, ·, 0, 1), (R, +, ·, 0, 1), (C, +, ·, 0, 1). We can define multiplication · on the set Z/nZ by the usu ...
... by xy. Sometimes rings defined by (i)-(iv) are called unital rings since the existence of a multiplicative unit 1 is stipulated. The standard number systems all define rings: (Z, +, ·, 0, 1), (Q, +, ·, 0, 1), (R, +, ·, 0, 1), (C, +, ·, 0, 1). We can define multiplication · on the set Z/nZ by the usu ...
Math 3121 Lecture 9 ppt97
... • The map pi: S1 ×S2 × … ×Sn Si that takes the n-tuple s = (s1, s2, …, sn) to its ith component si is called the ith projection map. • In other words: For any s in S1 ×S2 × … ×Sn the result of applying the ith projection map to s is called the ith component of s and denoted by si . • Note: Two ele ...
... • The map pi: S1 ×S2 × … ×Sn Si that takes the n-tuple s = (s1, s2, …, sn) to its ith component si is called the ith projection map. • In other words: For any s in S1 ×S2 × … ×Sn the result of applying the ith projection map to s is called the ith component of s and denoted by si . • Note: Two ele ...
Module Fundamentals
... and scalar multiplication are defined exactly as for the direct product, so that the external direct sum coincides with the direct product when the index set I is finite. The R-module M is the internal direct sum of the submodules Mi if each x ∈ M can be expressed uniquely as xi1 + · · · + xin where 0 ...
... and scalar multiplication are defined exactly as for the direct product, so that the external direct sum coincides with the direct product when the index set I is finite. The R-module M is the internal direct sum of the submodules Mi if each x ∈ M can be expressed uniquely as xi1 + · · · + xin where 0 ...
Tensor products in the category of topological vector spaces are not
... is available, and it is well-known that an associative law holds for iterated projective tensor products; this is important for applications in topological algebra. For locally convex spaces over spherically complete ultrametric fields, the situation is similar (cf. [14]). Two-fold tensor products E ...
... is available, and it is well-known that an associative law holds for iterated projective tensor products; this is important for applications in topological algebra. For locally convex spaces over spherically complete ultrametric fields, the situation is similar (cf. [14]). Two-fold tensor products E ...
Finitely generated abelian groups, abelian categories
... Comment: One can give an abstract definition of an abelian category. See §II.1 of Hartshorne’s “Algebraic Geometry” or ”Homological Algebra” by Hilton and Stammbach. Our first definition of abelian categories then becomes a Theorem of Peter Freyd. One requires first that M orA(C, D) is an abelian g ...
... Comment: One can give an abstract definition of an abelian category. See §II.1 of Hartshorne’s “Algebraic Geometry” or ”Homological Algebra” by Hilton and Stammbach. Our first definition of abelian categories then becomes a Theorem of Peter Freyd. One requires first that M orA(C, D) is an abelian g ...
Fibre products
... tells us that products exist. Since Spec Z is the terminal object in the category of schemes. The product is X ⇥ Y = X ⇥Spec Z Y . Secondly, given a point s 2 S of a scheme, and a morphism f : X ! S, we want to put a natural scheme structure on the preimage f 1 s, which at moment is just a set. We p ...
... tells us that products exist. Since Spec Z is the terminal object in the category of schemes. The product is X ⇥ Y = X ⇥Spec Z Y . Secondly, given a point s 2 S of a scheme, and a morphism f : X ! S, we want to put a natural scheme structure on the preimage f 1 s, which at moment is just a set. We p ...
BABY VERMA MODULES FOR RATIONAL CHEREDNIK ALGEBRAS
... for x, x ∈ h and y, y ∈ h. Note that this definition does not depend on the choice of αs and αs∨ . This algebra is naturally Z-graded, setting deg W = 0, deg h∗ = 1, and deg h = −1. One may also view the parameters t, c as formal variables to obtain a universal Cherednik algebra H, of which Ht,c is ...
... for x, x ∈ h and y, y ∈ h. Note that this definition does not depend on the choice of αs and αs∨ . This algebra is naturally Z-graded, setting deg W = 0, deg h∗ = 1, and deg h = −1. One may also view the parameters t, c as formal variables to obtain a universal Cherednik algebra H, of which Ht,c is ...
What We Need to Know about Rings and Modules
... Moreover we have the following uniqueness. If a = q1 q2 · · · qm is anther expression of a as a product of primes, then m = n and after a reordering of q1 , q2 , . . . , qn there are units u1 , u2 , . . . , un so that qi = ui pi for i = 1, . . . , n. Problem 9 Prove this by induction on δ(a) in the ...
... Moreover we have the following uniqueness. If a = q1 q2 · · · qm is anther expression of a as a product of primes, then m = n and after a reordering of q1 , q2 , . . . , qn there are units u1 , u2 , . . . , un so that qi = ui pi for i = 1, . . . , n. Problem 9 Prove this by induction on δ(a) in the ...
solutions to HW#3
... 1.1.5 Multiplication of residue classes in Z/nZ cannot be the operation of a group because cancellation fails: 0 · 1 = 0 · 0 but 1 6= 0. 1.1.6 Determine which of the following sets are groups under addition. 1.1.6(a) The set of rational numbers in lowest terms whose denominators are odd: This set co ...
... 1.1.5 Multiplication of residue classes in Z/nZ cannot be the operation of a group because cancellation fails: 0 · 1 = 0 · 0 but 1 6= 0. 1.1.6 Determine which of the following sets are groups under addition. 1.1.6(a) The set of rational numbers in lowest terms whose denominators are odd: This set co ...
Math 5594 Homework 2, due Monday September 25, 2006 PJW
... (b)Give an example of a vector space V with three subspaces U1 , U2 and U3 such that V = U1 + U2 + U3 and U1 ∩ U2 = U1 ∩ U3 = U2 ∩ U3 = {0}, but V 6= U1 ⊕ U2 ⊕ U3 . 16. (page 29, no. 9) Suppose that θ is an endomorphism of the vector space V and θ 2 = 1V . Show that V = U ⊕ W , where U = {v ∈ V : vθ ...
... (b)Give an example of a vector space V with three subspaces U1 , U2 and U3 such that V = U1 + U2 + U3 and U1 ∩ U2 = U1 ∩ U3 = U2 ∩ U3 = {0}, but V 6= U1 ⊕ U2 ⊕ U3 . 16. (page 29, no. 9) Suppose that θ is an endomorphism of the vector space V and θ 2 = 1V . Show that V = U ⊕ W , where U = {v ∈ V : vθ ...
Lecture 11
... Given a category with enough injectives, define the right derived functors of an left exact functor F by fixing an injective resolution and then let Ri F (A) = hi (F (I • )). Now it is a well-known result that the category of modules over a ring has enough injectives. Proposition 11.4. Let (X, OX ) ...
... Given a category with enough injectives, define the right derived functors of an left exact functor F by fixing an injective resolution and then let Ri F (A) = hi (F (I • )). Now it is a well-known result that the category of modules over a ring has enough injectives. Proposition 11.4. Let (X, OX ) ...
Exploring Tensor Rank
... tensors B1 , . . . , Br such that A = i=1 Bi . This coincides with the definition of rank for matrices, Pk i.e. order two tensors. If a collection of simple tensors B1 , . . . , Bk satisfy i=1 Bi = A, then we call {B1 , . . . , Bk } a rank-k decomposition for A. Tensor rank is invariant to any permu ...
... tensors B1 , . . . , Br such that A = i=1 Bi . This coincides with the definition of rank for matrices, Pk i.e. order two tensors. If a collection of simple tensors B1 , . . . , Bk satisfy i=1 Bi = A, then we call {B1 , . . . , Bk } a rank-k decomposition for A. Tensor rank is invariant to any permu ...
Lecture 8 - Universal Enveloping Algebras and Related Concepts, II
... The Clifford algebras, on the other hand, are quotients by ideas generated by the nonhomogeneous elements x ⊗ y + y ⊗ x + 2 hx, yi, and so do not inherit a grading in the usual sense. Nevertheless they inherit what is called a Z2 -grading. We can decompose the tensor algebra into two parts T V = T 0 ...
... The Clifford algebras, on the other hand, are quotients by ideas generated by the nonhomogeneous elements x ⊗ y + y ⊗ x + 2 hx, yi, and so do not inherit a grading in the usual sense. Nevertheless they inherit what is called a Z2 -grading. We can decompose the tensor algebra into two parts T V = T 0 ...
C - GEOCITIES.ws
... also a right A-module, compatible with the complex algebra structure, equipped with an A -valued inner product ·, ·: E E → A which is C- and A -linear in its second variable and satisfies the following relations: 1. ξ , η* = η , ξ, for every ξ , η E; 2. ξ , ξ 0, for every ξ E; 3. ...
... also a right A-module, compatible with the complex algebra structure, equipped with an A -valued inner product ·, ·: E E → A which is C- and A -linear in its second variable and satisfies the following relations: 1. ξ , η* = η , ξ, for every ξ , η E; 2. ξ , ξ 0, for every ξ E; 3. ...
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.