MODULES 1. Modules Let A be a ring. A left module M over A
... (If A is a field, recall that a module homomorphism is called a linear function or linear transformation.) Let A be a ring, M a left A-module, and N a submodule. The factor group M/N (as additive abelian group) may be made into an A-module by defining a(x + N ) = ax + N for a coset x + N ∈ M/N . The ...
... (If A is a field, recall that a module homomorphism is called a linear function or linear transformation.) Let A be a ring, M a left A-module, and N a submodule. The factor group M/N (as additive abelian group) may be made into an A-module by defining a(x + N ) = ax + N for a coset x + N ∈ M/N . The ...
Spring 2016 Math 285 Past Exam II Solutions 3-13-16
... f g , h ( f ( x) g ( x))h( x)dx ( f ( x)h( x) g ( x)h( x))dx ...
... f g , h ( f ( x) g ( x))h( x)dx ( f ( x)h( x) g ( x)h( x))dx ...
Ch 3
... There is a final observation to be made about the condition (3-48) for linear dependence. The determinant of a 3x3 matrix can be interpreted as the cross product of the vectors forming its rows or columns. So, if the determinant is zero, it means that ...
... There is a final observation to be made about the condition (3-48) for linear dependence. The determinant of a 3x3 matrix can be interpreted as the cross product of the vectors forming its rows or columns. So, if the determinant is zero, it means that ...
Symmeric self-adjoint Hopf categories and a categorical Heisenberg double June 17, 2014
... Examples of Hopf algebras with adjoint maps of multiplication and comultiplication arise naturally in representation theory. A principal L example which we consider throughout the article is the Hopf algebra Λ := n K(Rep(Sn )) with multiplication and comultiplication given by induction and restrict ...
... Examples of Hopf algebras with adjoint maps of multiplication and comultiplication arise naturally in representation theory. A principal L example which we consider throughout the article is the Hopf algebra Λ := n K(Rep(Sn )) with multiplication and comultiplication given by induction and restrict ...
HOMOLOGY OF LIE ALGEBRAS WITH Λ/qΛ COEFFICIENTS AND
... Proof. First note that Ln V2 (α, γ) = Ln V2 (h0 , h1 ), n ≥ 0. Then using Proposition 3.1 it is easy to get the following natural long exact sequence (compare [El1, Lemma 31]) · · · → Ln V(P ) → Ln V(M/M ∩ N ) ⊕ Ln V(N/M ∩ N ) → Ln−1 V2 (α, γ) → · · · → L0 V2 (α, γ) → L0 V(P ) → L0 V(M/M ∩ N ) ⊕ L0 ...
... Proof. First note that Ln V2 (α, γ) = Ln V2 (h0 , h1 ), n ≥ 0. Then using Proposition 3.1 it is easy to get the following natural long exact sequence (compare [El1, Lemma 31]) · · · → Ln V(P ) → Ln V(M/M ∩ N ) ⊕ Ln V(N/M ∩ N ) → Ln−1 V2 (α, γ) → · · · → L0 V2 (α, γ) → L0 V(P ) → L0 V(M/M ∩ N ) ⊕ L0 ...
On a classic example in the nonnegative inverse eigenvalue problem
... in an unpublished work that the smallest t for which τ (t) is realizable by a symmetric nonnegative matrix is t = 1. The smallest t for which τ (t) is the spectrum of a symmetric nonnegative matrix is t = 1. This was first shown by Loewy and a different proof can be found in [12]. The case when we a ...
... in an unpublished work that the smallest t for which τ (t) is realizable by a symmetric nonnegative matrix is t = 1. The smallest t for which τ (t) is the spectrum of a symmetric nonnegative matrix is t = 1. This was first shown by Loewy and a different proof can be found in [12]. The case when we a ...
A FEW REMARKS ON THE TUBE ALGEBRA OF A MONOIDAL...
... It is independent of the choice of standard solutions (RX , R̄X ). For X, Y ∈ C and a choice of standard solutions (RX , R̄X ) and (RY , R̄Y ), we can define a linear antimultiplicative map C(X, Y ) → C(Ȳ , X̄), denoted by T 7→ T ∨ , which is characterized by (T ⊗ ι)R̄X = (ι ⊗ T ∨ )R̄Y . This map ...
... It is independent of the choice of standard solutions (RX , R̄X ). For X, Y ∈ C and a choice of standard solutions (RX , R̄X ) and (RY , R̄Y ), we can define a linear antimultiplicative map C(X, Y ) → C(Ȳ , X̄), denoted by T 7→ T ∨ , which is characterized by (T ⊗ ι)R̄X = (ι ⊗ T ∨ )R̄Y . This map ...
1 VECTOR SPACES AND SUBSPACES
... • A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . In general, all ten vector space axioms must be verified to show that a set W with addition and scalar multiplication forms a vector space. However, if ...
... • A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V . In general, all ten vector space axioms must be verified to show that a set W with addition and scalar multiplication forms a vector space. However, if ...
(pdf)
... Lie groups were initially introduced as a tool to solve or simplify ordinary and partial differential equations. The most important example of a Lie group (and it turns out, one which encapsulate almost the entirety of the theory) is that of a matrix group, i.e., GLn (R) and SLn (R). First, we disco ...
... Lie groups were initially introduced as a tool to solve or simplify ordinary and partial differential equations. The most important example of a Lie group (and it turns out, one which encapsulate almost the entirety of the theory) is that of a matrix group, i.e., GLn (R) and SLn (R). First, we disco ...
QUANTUM GROUPS AND HADAMARD MATRICES Introduction A
... n−1/2 h is a unitary matrix with all entries having the same absolute value. The basic example is the Fourier n × n matrix, given by F ij = wij , where w = 2πi/n e . There are many other examples, but there is no other known family of complex Hadamard matrices that exist for every n. These matrices ...
... n−1/2 h is a unitary matrix with all entries having the same absolute value. The basic example is the Fourier n × n matrix, given by F ij = wij , where w = 2πi/n e . There are many other examples, but there is no other known family of complex Hadamard matrices that exist for every n. These matrices ...
Lie algebra cohomology and Macdonald`s conjectures
... Proof. If π(g)v = v ∀g ∈ G then by equation 1.1 dπ(X)v = 0 ∀X ∈ g. On the other hand suppose that v ∈ V g . By our definition of a representation the map πv : G → V : g → π(g)v is analytic, and all its partial derivatives (at e ∈ G) are 0. Since G is connected, πv is a constant map and πv (g) = πv ( ...
... Proof. If π(g)v = v ∀g ∈ G then by equation 1.1 dπ(X)v = 0 ∀X ∈ g. On the other hand suppose that v ∈ V g . By our definition of a representation the map πv : G → V : g → π(g)v is analytic, and all its partial derivatives (at e ∈ G) are 0. Since G is connected, πv is a constant map and πv (g) = πv ( ...
2-3
... Write the function 2x – y = 9 in slope-intercept form. Then graph the function. Solve for y first. 2x – y = 9 –2x –2x Add –2x to both sides. –y = –2x + 9 y = 2x – 9 Multiply both sides by –1. The line has y-intercept –9 and slope 2, which is . Plot the point (0, –9). Then move up 2 and right 1 to fi ...
... Write the function 2x – y = 9 in slope-intercept form. Then graph the function. Solve for y first. 2x – y = 9 –2x –2x Add –2x to both sides. –y = –2x + 9 y = 2x – 9 Multiply both sides by –1. The line has y-intercept –9 and slope 2, which is . Plot the point (0, –9). Then move up 2 and right 1 to fi ...
A2C02L03
... Write the function 2x – y = 9 in slope-intercept form. Then graph the function. Solve for y first. 2x – y = 9 –2x –2x Add –2x to both sides. –y = –2x + 9 y = 2x – 9 Multiply both sides by –1. The line has y-intercept –9 and slope 2, which is . Plot the point (0, –9). Then move up 2 and right 1 to fi ...
... Write the function 2x – y = 9 in slope-intercept form. Then graph the function. Solve for y first. 2x – y = 9 –2x –2x Add –2x to both sides. –y = –2x + 9 y = 2x – 9 Multiply both sides by –1. The line has y-intercept –9 and slope 2, which is . Plot the point (0, –9). Then move up 2 and right 1 to fi ...
Groups with exponents I. Fundamentals of the theory and tensor
... P r o p o s i t i o n 11. Let an A-group_ G be a residually-B group with respect to a h o m o m o r p h i s m #. Then the h o m o m o r p h i s m A : G ---* G B is an embedding. [] Let 1 ~ g E G and let Cg : G --~ H be an A-homomorphism consistent with # and such that ~g(g) ~ 1. There exists a homom ...
... P r o p o s i t i o n 11. Let an A-group_ G be a residually-B group with respect to a h o m o m o r p h i s m #. Then the h o m o m o r p h i s m A : G ---* G B is an embedding. [] Let 1 ~ g E G and let Cg : G --~ H be an A-homomorphism consistent with # and such that ~g(g) ~ 1. There exists a homom ...
3. Abstract Boolean Algebras 3.1. Abstract Boolean Algebra.
... As these examples illustrate, the names for addition and multiplication in a particular Boolean algebra may be idiomatic to that example. Addition may be called sum, union, join, or disjunction; whereas, multiplication may be called product, intersection, meet, or conjunction. Because the addition a ...
... As these examples illustrate, the names for addition and multiplication in a particular Boolean algebra may be idiomatic to that example. Addition may be called sum, union, join, or disjunction; whereas, multiplication may be called product, intersection, meet, or conjunction. Because the addition a ...
STRING CONES AND TORIC VARIETIES Contents 1. Convex
... This fact is called complete reducibility. In particular, if W is a finite dimensional representation of GLn , then there exist irreducible subrepresentations W1 , . . . , Wr (not necessarily distinct, nor unique) such that W = W1 ⊕ · · · ⊕ Wr . Therefore, in order to understand the (finite dimensio ...
... This fact is called complete reducibility. In particular, if W is a finite dimensional representation of GLn , then there exist irreducible subrepresentations W1 , . . . , Wr (not necessarily distinct, nor unique) such that W = W1 ⊕ · · · ⊕ Wr . Therefore, in order to understand the (finite dimensio ...
Homological algebra
... MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to do this first so that grad students will be more familiar with the ideas when they are applied to a ...
... MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to do this first so that grad students will be more familiar with the ideas when they are applied to a ...
Full Text (PDF format)
... Proof. By part 8 of Proposition 2.2, we may assume that k is algebraically closed. Now, it is straightforward to check that the theorem holds for (k[G], 1 ⊗1) where G is a finite group. But by [EG1, Theorem 2.1], there exist a finite group G and a twist J ∈ k[G] ⊗ k[G] such that H ∼ = k[G]J as Hopf al ...
... Proof. By part 8 of Proposition 2.2, we may assume that k is algebraically closed. Now, it is straightforward to check that the theorem holds for (k[G], 1 ⊗1) where G is a finite group. But by [EG1, Theorem 2.1], there exist a finite group G and a twist J ∈ k[G] ⊗ k[G] such that H ∼ = k[G]J as Hopf al ...
Linear Algebra Review and Reference Contents Zico Kolter (updated by Chuong Do)
... It may seem like overkill to dissect matrix multiplication to such a large degree, especially when all these viewpoints follow immediately from the initial definition we gave (in about a line of math) at the beginning of this section. However, virtually all of linear algebra deals with matrix multi ...
... It may seem like overkill to dissect matrix multiplication to such a large degree, especially when all these viewpoints follow immediately from the initial definition we gave (in about a line of math) at the beginning of this section. However, virtually all of linear algebra deals with matrix multi ...
Geometry - Beck
... Geometry - Beck November 7th to November 11th Monday: Triangle Sum Assign: Triangle Sum with Algebra Handout Tuesday: Exterior Angles Theorem Assign: Exterior Angles Theorem with Algebra Handout Wednesday: Congruent Triangles – Day 1 Assign: Congruent Triangles Handout Thursday: Congruent Triangles ...
... Geometry - Beck November 7th to November 11th Monday: Triangle Sum Assign: Triangle Sum with Algebra Handout Tuesday: Exterior Angles Theorem Assign: Exterior Angles Theorem with Algebra Handout Wednesday: Congruent Triangles – Day 1 Assign: Congruent Triangles Handout Thursday: Congruent Triangles ...
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a space known as the kth exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.