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Split Screen 2 2019 Juicy Entertainment DO Teeny Black Chicks Trying White Dicks 12 2017 Pulse Distribution 1 DRO Want To Be A Slut 2019 leilanileixxx.com Every analytically integral element is integral in the Lie algebra sense, but not the other way around.[13] (This phenomenon reflects that, in general, not every representation of the Lie algebra {\displaystyle {\mathfrak {k}}}\mathfrak{k} comes from a representation of the group K.) On the other hand, if K is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.[14] The Weyl character formula Main article: Weyl character formula If {\displaystyle \Pi :K\rightarrow \operatorname {GL} (V)}{\displaystyle \Pi :K\rightarrow \operatorname {GL} (V)} is representation of K, we define the character of {\displaystyle \Pi }\Pi to be the function {\displaystyle \mathrm {X} :K\rightarrow \mathbb {C} }{\displaystyle \mathrm {X} :K\rightarrow \mathbb {C} } given by {\displaystyle \mathrm {X} (x)=\operatorname {trace} (\Pi (x)),\quad x\in K}{\displaystyle \mathrm {X} (x)=\operatorname {trace} (\Pi (x)),\quad x\in K}. This function is easily seen to be a class function, i.e., {\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)}{\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)} for all {\displaystyle x}x and {\displaystyle y}y in K. Thus, {\displaystyle \mathrm {X} }\mathrm{X} is determined by its restriction to T.

The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the Peter–Weyl theorem, is that the characters form an orthonormal basis for the set of square-integrable class functions in K. A second key result is the Weyl character formula, which gives an explicit formula for the character—or, rather, the restriction of the character to T—in terms of the highest weight of the representation. In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established after the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of K, the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in a totally different way from the usual Lie algebra construction using Verma modules. In Weyl's approach, the construction is based on the Peter–Weyl theorem and an analytic proof of the Weyl character formula.[15] Ultimately, the irreducible representations of K are realized inside the space of continuous functions on K. The SU(2) case We now consider the case of the compact group SU(2). The representations are often considered from the Lie algebra point of view, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form {\displaystyle {\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}}{\displaystyle {\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}}. According to the example discussed above in the section on representations of T, the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers {\displaystyle m}m. The general theory then tells us that for each {\displaystyle m}m, there is a unique irreducible representation of SU(2) with highest weight {\displaystyle m}m. Much information about the representation corresponding to a given {\displaystyle m}m is encoded in its character. Now, the Weyl character formula says, in this case, that the character is given by {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\sin((m+1)\theta )}{\sin(\theta )}}.}{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\sin((m+1)\theta )}{\sin(\theta )}}.} We can also write the character as sum of exponentials as follows: {\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta }+e^{-im\theta }.}{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta }+e^{-im\theta }.} (If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.) From this last expression and the standard formula for the character in terms of the weights of the representation, we can read off that the weights of the representation are {\displaystyle m,m-2,\ldots ,-(m-2),-m}{\displaystyle m,m-2,\ldots ,-(m-2),-m}, each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are {\displaystyle m+1}m+1 weights, each with multiplicity 1, the dimension of the representation is {\displaystyle m+1}m+1. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation. An outline of the proof We now outline the proof of the theorem of the highest weight, following the original argument of Hermann Weyl. We continue to let {\displaystyle K}K be a connected compact Lie group and {\displaystyle T}T a fixed maximal torus in {\displaystyle K}K. We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.[16] The tools for the proof are the following: The torus theorem. The Weyl integral formula. The Peter–Weyl theorem for class functions, which states that the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions on {\displaystyle K}K. With these tools in hand, we proceed with the proof. The first major step in the argument is to prove the Weyl character formula. The formula states that if {\displaystyle \Pi }\Pi is an irreducible representation with highest weight {\displaystyle \lambda }\lambda , then the character {\displaystyle \mathrm {X} }\mathrm{X} of {\displaystyle \Pi }\Pi satisfies: {\displaystyle \mathrm {X} (e^{H})={\frac {\sum _{w\in W}\det(w)e^{i\langle w\cdot (\lambda +\rho ),H\rangle }}{\sum _{w\in W}\det(w)e^{i\langle w\cdot \rho ,H\rangle }}}}{\displaystyle \mathrm {X} (e^{H})={\frac {\sum _{w\in W}\det(w)e^{i\langle w\cdot (\lambda +\rho ),H\rangle }}{\sum _{w\in W}\det(w)e^{i\langle w\cdot \rho ,H\rangle }}}} for all {\displaystyle H}H in the Lie algebra of {\displaystyle T}T. Here {\displaystyle \rho }\rho is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of {\displaystyle i}i in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the {\displaystyle L^{2}}L^{2} norm of the character is 1. Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1. Next, we let {\displaystyle \Phi _{\lambda }}\Phi_\lambda denote the function on the right-hand side of the character formula. We show that even if {\displaystyle \lambda }\lambda is not known to be the highest weight of a representation, {\displaystyle \Phi _{\lambda }}\Phi_\lambda is a well-defined, Weyl-invariant function on {\displaystyle T}T, which therefore extends to a class function on {\displaystyle K}K. Then using the Weyl integral formula, one can show that as {\displaystyle \lambda }\lambda ranges over the set of dominant, analytically integral elements, the functions {\displaystyle \Phi _{\lambda }}\Phi_\lambda form an orthonormal family of class functions. We emphasize that we do not currently know that every such {\displaystyle \lambda }\lambda is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions {\displaystyle \Phi _{\lambda }}\Phi_\lambda, and these functions are orthonormal. Now comes the conclusion. The set of all {\displaystyle \Phi _{\lambda }}\Phi_\lambda—with {\displaystyle \lambda }\lambda ranging over the dominant, analytically integral elements—forms an orthonormal set in the space of square integrable class functions. But by the Weyl character formula, the characters of the irreducible representations form a subset of the {\displaystyle \Phi _{\lambda }}\Phi_\lambda's. And by the Peter–Weyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions. If there were some {\displaystyle \lambda }\lambda that is not the highest weight of a representation, then the corresponding {\displaystyle \Phi _{\lambda }}\Phi_\lambda would not be the character of a representation. Thus, the characters would be a proper subset of the set of {\displaystyle \Phi _{\lambda }}\Phi_\lambda's. But then we have an impossible situation: an orthonormal basis (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of {\displaystyle \Phi _{\lambda }}\Phi_\lambda's). Thus, every {\displaystyle \lambda }\lambda must actually be the highest weight of a representation. Duality The topic of recovering a compact group from its representation theory is the subject of the Tannaka–Krein duality, now often recast in terms of Tannakian category theory. From compact to non-compact groups The influence of the compact group theory on non-compact groups was formulated by Weyl in his unitarian trick. Inside a general semisimple Lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by Harish-Chandra, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory

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