![[math.RT] 30 Jun 2006 A generalized Cartan](http://s1.studyres.com/store/data/016692548_1-536349640838ef57792bbd8394ad9b17-300x300.png)
Trivial remarks about tori.
... is just the bi-K-invariant functions from G to C with compact support, and with multiplication given by convolution. ...
... is just the bi-K-invariant functions from G to C with compact support, and with multiplication given by convolution. ...
The Etingof-Kazhdan construction of Lie bialgebra deformations.
... functorial isomorphism JV W : F (V ) ⊗ F (W ) → F (V ⊗ W ) such that F (ΦV W U )JV ⊗W,U ◦ (JV W ⊗ 1) = JV,W ⊗U ◦ (1 ⊗ JW U ), and JV 1 = J1V = 1. In order to do so we consider a different realisation of the functor which, while convenient for us now, is necessary to generalise to the infinite dimens ...
... functorial isomorphism JV W : F (V ) ⊗ F (W ) → F (V ⊗ W ) such that F (ΦV W U )JV ⊗W,U ◦ (JV W ⊗ 1) = JV,W ⊗U ◦ (1 ⊗ JW U ), and JV 1 = J1V = 1. In order to do so we consider a different realisation of the functor which, while convenient for us now, is necessary to generalise to the infinite dimens ...
LIE-ADMISSIBLE ALGEBRAS AND THE VIRASORO
... functionals µ, λ : W → F such that cei = ei c = c ◦ ei = ei ◦ c = µ(ei )ei + λ(ei )c. If u = ej , v = c and w = ei in (9), then it follows that µ(ei − ej )[(i − j )ei+ j + φ(ei , ej )c] = 0 and hence µ(ei ) = µ(ej ) for i 6= j . Since [ei , c ◦ c] = 0 by (7) for all i ∈ Z, cc = c ◦ c = βc for some β ...
... functionals µ, λ : W → F such that cei = ei c = c ◦ ei = ei ◦ c = µ(ei )ei + λ(ei )c. If u = ej , v = c and w = ei in (9), then it follows that µ(ei − ej )[(i − j )ei+ j + φ(ei , ej )c] = 0 and hence µ(ei ) = µ(ej ) for i 6= j . Since [ei , c ◦ c] = 0 by (7) for all i ∈ Z, cc = c ◦ c = βc for some β ...
5. The algebra of complex numbers We use complex
... Once you have a single root, say r, for a polynomial p(x), you can divide through by (x − r) and get a polynomial of smaller degree as quotient, which then also has a complex root, and so on. The result is that a polynomial p(x) = axn + · · · of degree n factors completely into linear factors over t ...
... Once you have a single root, say r, for a polynomial p(x), you can divide through by (x − r) and get a polynomial of smaller degree as quotient, which then also has a complex root, and so on. The result is that a polynomial p(x) = axn + · · · of degree n factors completely into linear factors over t ...
p-Groups - Brandeis
... This action has a fixed point, namely x = 1. [x is a fixed point if g · x = x for all g in the group. When the action is conjugation this is the same as saying that x commutes with every element of the group, i.e., x ∈ Z(P ) ∩ N .] However, we know that the number of fixed points is congruent modulo ...
... This action has a fixed point, namely x = 1. [x is a fixed point if g · x = x for all g in the group. When the action is conjugation this is the same as saying that x commutes with every element of the group, i.e., x ∈ Z(P ) ∩ N .] However, we know that the number of fixed points is congruent modulo ...
- Journal of Algebra and Related Topics
... functions. The next lemma characterizes AB and D in terms of the elements of S WAP and S LMC , respectively. Lemma 3.1. Let S be a semitopological semigroup. (i) A function f ∈ WAP(S) is in AB(S) if and only if µν(f ) = νµ(f ) and Tµν f = Tνµ f for all µ, ν ∈ S WAP [4, Lemma 3.1 ]. (ii) A function f ...
... functions. The next lemma characterizes AB and D in terms of the elements of S WAP and S LMC , respectively. Lemma 3.1. Let S be a semitopological semigroup. (i) A function f ∈ WAP(S) is in AB(S) if and only if µν(f ) = νµ(f ) and Tµν f = Tνµ f for all µ, ν ∈ S WAP [4, Lemma 3.1 ]. (ii) A function f ...
AN APPLICATION OF A FUNCTIONAL INEQUALITY TO QUASI-INVARIANCE IN INFINITE DIMENSIONS
... We start by describing an abstract setting of how finite-dimensional approximations can be used to prove such a quasi-invariance. In [9] this method was applied to projective and inductive limits of finite-dimensional Lie groups acting on themselves by left or right multiplication. In that setting a ...
... We start by describing an abstract setting of how finite-dimensional approximations can be used to prove such a quasi-invariance. In [9] this method was applied to projective and inductive limits of finite-dimensional Lie groups acting on themselves by left or right multiplication. In that setting a ...
On the maximal number of facets of 0/1 polytopes
... condition supp θ~ ≥ n/r is satisfied, and hence, if β < c1 (γ)/r then ~x θ, well-defined and unique. We will estimate the measure of M r . Lemma 3.2 There exists r > 0 such that: if n ≥ 3 then ...
... condition supp θ~ ≥ n/r is satisfied, and hence, if β < c1 (γ)/r then ~x θ, well-defined and unique. We will estimate the measure of M r . Lemma 3.2 There exists r > 0 such that: if n ≥ 3 then ...
(Urysohn`s Lemma for locally compact Hausdorff spaces).
... (b) Show that if f, g ∈ C0 (X), then f g ∈ C0 (X), where f g denotes the point-wise multiplication. Furthermore, show that if f ∈ Cc (X) and g ∈ C0 (X), then f g ∈ Cc (X). (c) Prove that Cc (X) is dense in C0 (X) with the following outline. (i) For f ∈ C0 (X), we find for all ε > 0, fε ∈ Cc (X) such ...
... (b) Show that if f, g ∈ C0 (X), then f g ∈ C0 (X), where f g denotes the point-wise multiplication. Furthermore, show that if f ∈ Cc (X) and g ∈ C0 (X), then f g ∈ Cc (X). (c) Prove that Cc (X) is dense in C0 (X) with the following outline. (i) For f ∈ C0 (X), we find for all ε > 0, fε ∈ Cc (X) such ...
Change log for Magma V2.20-6 - Magma Computational Algebra
... elements has been fixed. This manifested in poor performance in some algorithms. Corrections have been made to the real precision used in computing the LLL (basis) of an order. Insufficient precision lead to a result which was far from LLL in some (unusual) cases, and this in turn caused a crash in ...
... elements has been fixed. This manifested in poor performance in some algorithms. Corrections have been made to the real precision used in computing the LLL (basis) of an order. Insufficient precision lead to a result which was far from LLL in some (unusual) cases, and this in turn caused a crash in ...
Chapter 8 The Log-Euclidean Framework Applied to
... Section 6.9, Proposition 17, and also Theorem 6). Let S(n) denote the set of all real matrices whose eigenvalues, λ + iµ, lie in the horizontal strip determined by the condition −π < µ < π. Then, we have the following theorem: Theorem 8.1.1 The restriction of the exponential map to S(n) is a diffeom ...
... Section 6.9, Proposition 17, and also Theorem 6). Let S(n) denote the set of all real matrices whose eigenvalues, λ + iµ, lie in the horizontal strip determined by the condition −π < µ < π. Then, we have the following theorem: Theorem 8.1.1 The restriction of the exponential map to S(n) is a diffeom ...
A counterexample to discrete spectral synthesis
... conclusion, let us point out that the pathology of E has its source in the fact that the multiplier algebra for E does not contain most functions z of the form z(m) = 03B6m for fixed complex 03B6 of modulus 1. Or what is the same, 0 is not translation-invariant. For suppose, that the commutative sem ...
... conclusion, let us point out that the pathology of E has its source in the fact that the multiplier algebra for E does not contain most functions z of the form z(m) = 03B6m for fixed complex 03B6 of modulus 1. Or what is the same, 0 is not translation-invariant. For suppose, that the commutative sem ...
Eigenvalues, eigenvectors, and eigenspaces of linear operators
... values the eigenvalues λ can be? In the finite diAn example transformation that has 0 as an mensional case, it comes down to finding the roots eigenvalue is a projection, like (x, y, z) 7→ (x, y, 0) of a particular polynomial, called the characteristic that maps space to the xy-plane. For this proje ...
... values the eigenvalues λ can be? In the finite diAn example transformation that has 0 as an mensional case, it comes down to finding the roots eigenvalue is a projection, like (x, y, z) 7→ (x, y, 0) of a particular polynomial, called the characteristic that maps space to the xy-plane. For this proje ...
univariate case
... Clearly, if p(x) is nonnegative, then its degree must be an even number. The set of nonnegative polynomials has very interesting properties. Perhaps the most appealing one for our purposes is the following: Theorem 12. Consider the set Pn of nonnegative univariate polynomials of degree less than or ...
... Clearly, if p(x) is nonnegative, then its degree must be an even number. The set of nonnegative polynomials has very interesting properties. Perhaps the most appealing one for our purposes is the following: Theorem 12. Consider the set Pn of nonnegative univariate polynomials of degree less than or ...
When is a group homomorphism a covering homomorphism?
... that ϕ is surjective and that the kernel of ϕ is discrete. In general, these conditions are not sufficient to assure that ϕ is a covering homomorphism. As an example, let α be a real irrational number and let G be the subgroup of the torus S 1 × S 1 whose elements are those of the form (eit , eiαt ) ...
... that ϕ is surjective and that the kernel of ϕ is discrete. In general, these conditions are not sufficient to assure that ϕ is a covering homomorphism. As an example, let α be a real irrational number and let G be the subgroup of the torus S 1 × S 1 whose elements are those of the form (eit , eiαt ) ...