UNIVERSAL COVERING GROUPS OF MATRIX LIE GROUPS
... n n orthogonal matrices forms a group with respect to matrix multiplication, and it is denoted by O n . If we consider the set of all n n orthogonal matrices with determinant one only, it also forms a group with respect to matrix multiplication known as the Special orthogonal group, and it i ...
... n n orthogonal matrices forms a group with respect to matrix multiplication, and it is denoted by O n . If we consider the set of all n n orthogonal matrices with determinant one only, it also forms a group with respect to matrix multiplication known as the Special orthogonal group, and it i ...
Introduction to Lie groups - OpenSIUC
... gln , for some n. (Justifies assumption we are always in gln .) 2) Cambell-Hausdorff Formula: For U ⊂ g containing 0 and sufficiently small, exp : U → G is one-to-one and we can define an inverse, denoted log . Then log(exp(X) · exp(Y )) is given by a complicated formula involving the bracket [ , ]. ...
... gln , for some n. (Justifies assumption we are always in gln .) 2) Cambell-Hausdorff Formula: For U ⊂ g containing 0 and sufficiently small, exp : U → G is one-to-one and we can define an inverse, denoted log . Then log(exp(X) · exp(Y )) is given by a complicated formula involving the bracket [ , ]. ...
Hw #2 pg 109 1-13odd, pg 101 23,25,27,29
... 13. Suppose AB = AC, where B and C are n x p matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible? Since A is invertible to show we can multiply the equation by where we get so can be rewritten as which is equivalent to Pg. 101 23. Suppose CA = (the n x n ...
... 13. Suppose AB = AC, where B and C are n x p matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible? Since A is invertible to show we can multiply the equation by where we get so can be rewritten as which is equivalent to Pg. 101 23. Suppose CA = (the n x n ...
INTRODUCTION TO RATIONAL CHEREDNIK ALGEBRAS
... Proposition 2.4. If t 6= 0, then the polynomial representation of Ht,c is faithful and Zt,c = C. Proof. Faithfulness follows in the same way as in Corollary 2.3, where we note that the polynomial representation of Dt (V ) o C[W ] is faithful only when t 6= 0. Now, by faithfulness, the polynomial rep ...
... Proposition 2.4. If t 6= 0, then the polynomial representation of Ht,c is faithful and Zt,c = C. Proof. Faithfulness follows in the same way as in Corollary 2.3, where we note that the polynomial representation of Dt (V ) o C[W ] is faithful only when t 6= 0. Now, by faithfulness, the polynomial rep ...
Simple examples of Lie groups and Lie algebras
... 1. The simple example of the compact Lie group SO(2) A Lie group is a group whose group elements are functions of some continuously varying parameters. The parameters which specify the group elements form a smooth space, a differentiable manifold, called the group manifold with the property that the ...
... 1. The simple example of the compact Lie group SO(2) A Lie group is a group whose group elements are functions of some continuously varying parameters. The parameters which specify the group elements form a smooth space, a differentiable manifold, called the group manifold with the property that the ...
A note on linear codes and nonassociative algebras obtained from
... In recent years, several classes of linear codes were obtained from skew-polynomial rings (also called Ore rings). Using this approach, self-dual codes with a better minimal distance for certain lengths than previously known were constructed: while the classical cyclic codes of length m over a finit ...
... In recent years, several classes of linear codes were obtained from skew-polynomial rings (also called Ore rings). Using this approach, self-dual codes with a better minimal distance for certain lengths than previously known were constructed: while the classical cyclic codes of length m over a finit ...
MTH6140 Linear Algebra II
... (contains 0), and closed under vector addition and scalar multiplication, it is a subspace of V . 5. (a) D maps each non-constant polynomial to a polynomial of degree one lower, and each constant polynomial to 0. So D has only one eigenvalue, namely 0, with corresponding eigenvector 1 (the constant ...
... (contains 0), and closed under vector addition and scalar multiplication, it is a subspace of V . 5. (a) D maps each non-constant polynomial to a polynomial of degree one lower, and each constant polynomial to 0. So D has only one eigenvalue, namely 0, with corresponding eigenvector 1 (the constant ...
Linear Equations
... show this? How do you show a set is infinite? One way is to show that the set is described in a way that gets a different element of the set for each value of a parameter in the description. For example, the following set is infinite: { 3+5k | k is a real number }. Theorem: Every system of linear eq ...
... show this? How do you show a set is infinite? One way is to show that the set is described in a way that gets a different element of the set for each value of a parameter in the description. For example, the following set is infinite: { 3+5k | k is a real number }. Theorem: Every system of linear eq ...
Intrinsic differential operators 1.
... be conjugated (over the algebraic closure) to an upper-triangular matrix. But the assertion that a matrix x can be conjugated to an upper-triangular matrix is weaker than the assertion of Jordan normal form, only requiring that there is a basis v1 , . . . , vn for Cn such that x · vi ∈ Σj≤i C vj . T ...
... be conjugated (over the algebraic closure) to an upper-triangular matrix. But the assertion that a matrix x can be conjugated to an upper-triangular matrix is weaker than the assertion of Jordan normal form, only requiring that there is a basis v1 , . . . , vn for Cn such that x · vi ∈ Σj≤i C vj . T ...
Sample examinations Linear Algebra (201-NYC-05) Winter 2012
... so the general solution of Ax = 0 is the set of all vectors of the form ...
... so the general solution of Ax = 0 is the set of all vectors of the form ...
Algebras of Virasoro type, Riemann surfaces and structures of the
... a decomposition of type (2), where L0 = G + C, and also a theory of Verma-type modules. The starting point of our paper is the observation that the nontrivial Riemann surfaces generate, in the algebras of vector fields L(S 1 ), the current algebras G(S 1 ) and their central extensions, dense subalge ...
... a decomposition of type (2), where L0 = G + C, and also a theory of Verma-type modules. The starting point of our paper is the observation that the nontrivial Riemann surfaces generate, in the algebras of vector fields L(S 1 ), the current algebras G(S 1 ) and their central extensions, dense subalge ...
Revisions in Linear Algebra
... A linear function f is a mathematical function in which the variables appear only in the rst degree, are multiplied by constants, and are combined only by addition and subtraction. A linear equation is of the form f (x, y, · · · ) = 0 ...
... A linear function f is a mathematical function in which the variables appear only in the rst degree, are multiplied by constants, and are combined only by addition and subtraction. A linear equation is of the form f (x, y, · · · ) = 0 ...
Condition Number, LU, Cholesky
... start with the simplest unspecialized algorithm: Gaussian Elimination Assume the matrix is invertible, but otherwise nothing special known about it GE simply is row-reduction to upper triangular form, followed by backwards substitution ...
... start with the simplest unspecialized algorithm: Gaussian Elimination Assume the matrix is invertible, but otherwise nothing special known about it GE simply is row-reduction to upper triangular form, followed by backwards substitution ...
Let m and n be two positive integers. A rectangular array (of numbers)
... Matrices arise naturally as representation of linear transformations, but they can also considered as objects existing in their own right, without necessarily being connected to linear transformations. As such, they form another class of mathematical objects on which algebraic operations can be defi ...
... Matrices arise naturally as representation of linear transformations, but they can also considered as objects existing in their own right, without necessarily being connected to linear transformations. As such, they form another class of mathematical objects on which algebraic operations can be defi ...
![[2011 question paper]](http://s1.studyres.com/store/data/008843345_1-9a0802372adf6384e8c1e5127a999f79-300x300.png)
[2011 question paper]
... Entrance Examination, 2011 Part A For each of the statements given below, state whether it is True or False and give brief reasons in the sheets provided. Marks will be given only when reasons are provided. 1. (k!)2 is a factor of (2k + 2)! for any positive integer k. 2. Let p(x) be a polynomial of ...
... Entrance Examination, 2011 Part A For each of the statements given below, state whether it is True or False and give brief reasons in the sheets provided. Marks will be given only when reasons are provided. 1. (k!)2 is a factor of (2k + 2)! for any positive integer k. 2. Let p(x) be a polynomial of ...
Invariant differential operators 1. Derivatives of group actions: Lie
... g = Lie(G), by first defining (adx)(y) = [x, y] (this is the adjoint action of g on itself) and taking hx, yi = tr (adx◦ady). This is the Killing form, named after Wilhelm Killing (not because it kills anything). Up to a normalization, the trace-of-matrix definition we’ve given here is the same, tho ...
... g = Lie(G), by first defining (adx)(y) = [x, y] (this is the adjoint action of g on itself) and taking hx, yi = tr (adx◦ady). This is the Killing form, named after Wilhelm Killing (not because it kills anything). Up to a normalization, the trace-of-matrix definition we’ve given here is the same, tho ...
Pythagoreans quadruples on the future light cone
... the standard Pythagorean quadruples and also triples (when one of the four numbers is zero) belong to this set. We introduce the following generating set of primitive degenerate quadruples: q1 = (1, 0, 0, 1) q2 = (0, 1, 0, 1) q3 = (0, 0, 1, 1) Theorem 1. The set of 6 matrices h generates, from the g ...
... the standard Pythagorean quadruples and also triples (when one of the four numbers is zero) belong to this set. We introduce the following generating set of primitive degenerate quadruples: q1 = (1, 0, 0, 1) q2 = (0, 1, 0, 1) q3 = (0, 0, 1, 1) Theorem 1. The set of 6 matrices h generates, from the g ...
Invariant Laplacians
... [1.0.2] Remark: These Lie algebras will prove to be R-vectorspaces with a R-bilinear operation, x × y → [x, y], which is why they are called algebras. However, this binary operation is different from more typical ring or algebra multiplications: it is not associative. [1.0.3] Example: The condition ...
... [1.0.2] Remark: These Lie algebras will prove to be R-vectorspaces with a R-bilinear operation, x × y → [x, y], which is why they are called algebras. However, this binary operation is different from more typical ring or algebra multiplications: it is not associative. [1.0.3] Example: The condition ...
Topology of Lie Groups Lecture 1
... Proof: By spectral theorem and the remerk above, we may as well assume that H = diag (λ1 , . . . , λn ). Now, exp H ∈ G implies that for all integers k, ekλ1 , . . . , ekλn satisfy a set of polynomials. By simple application of Vander Monde, this implies that etλ1 , . . . , etλn satisfy the same set ...
... Proof: By spectral theorem and the remerk above, we may as well assume that H = diag (λ1 , . . . , λn ). Now, exp H ∈ G implies that for all integers k, ekλ1 , . . . , ekλn satisfy a set of polynomials. By simple application of Vander Monde, this implies that etλ1 , . . . , etλn satisfy the same set ...
Lecture 10 - Harvard Math Department
... The following result provides an intrinsic characterization of von Neumann algebras: Theorem 1. Let A be a C ∗ -algebra. Suppose there exists a Banach space E and a Banach space isomorphism A ' E ∨ . Then there exists a von Neumann algebra B and an isomorphism of C ∗ -algebras A → B (in other words, ...
... The following result provides an intrinsic characterization of von Neumann algebras: Theorem 1. Let A be a C ∗ -algebra. Suppose there exists a Banach space E and a Banach space isomorphism A ' E ∨ . Then there exists a von Neumann algebra B and an isomorphism of C ∗ -algebras A → B (in other words, ...
Homework assignment 2 p 21 Exercise 2. Let Solution: Solution: Let
... The bar denotes complex conjugation, i.e. a + bi = a − bi. Show that V is a vector space over the field of real numbers. Give an example of a function in V that is not real-valued. First, check that if functions f, g satisfy equation (1), then f + g and λf for a real λ also satisfy it. This is becau ...
... The bar denotes complex conjugation, i.e. a + bi = a − bi. Show that V is a vector space over the field of real numbers. Give an example of a function in V that is not real-valued. First, check that if functions f, g satisfy equation (1), then f + g and λf for a real λ also satisfy it. This is becau ...
AFFINE LIE ALGEBRAS, THE SYMMETRIC GROUPS, AND
... are increasing on every row and column. The degree nλ of the representation ρλ is the number of such paths and the set of standard tableaux may be taken as a basis. The defining property of this representation is that it is simultaneously induced from the trivial representation of the subgroup of Sd ...
... are increasing on every row and column. The degree nλ of the representation ρλ is the number of such paths and the set of standard tableaux may be taken as a basis. The defining property of this representation is that it is simultaneously induced from the trivial representation of the subgroup of Sd ...
Alice Guionnet`s Review Session Exercise
... An−1 the following enequality holds λi+1 ≤ λ1 (An−1 ) ≤ λi (An ) for all 1 ≤ i ≤ n − 1 6. Show that for any two Hermitian n × n matrices A and B the following enequality holds n X ...
... An−1 the following enequality holds λi+1 ≤ λ1 (An−1 ) ≤ λi (An ) for all 1 ≤ i ≤ n − 1 6. Show that for any two Hermitian n × n matrices A and B the following enequality holds n X ...
Homework #2
... 6. Let L : V 7→ V be a linear operator such that L2 = L (here L2 = L ◦ L). Such a mapping is said to be a projection operator. Show that: (a) ker(L) ∩ Ran(L) = {0} (b) if v ∈ V is given, then there are x ∈ ker(L), y ∈ Ran(L) such that v = x + y (c) V = ker(L) ⊕ Ran(L). 7. The first four Legendre pol ...
... 6. Let L : V 7→ V be a linear operator such that L2 = L (here L2 = L ◦ L). Such a mapping is said to be a projection operator. Show that: (a) ker(L) ∩ Ran(L) = {0} (b) if v ∈ V is given, then there are x ∈ ker(L), y ∈ Ran(L) such that v = x + y (c) V = ker(L) ⊕ Ran(L). 7. The first four Legendre pol ...