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| The '''Cartan decomposition''' is a decomposition of a [[Semisimple Lie algebra|semisimple]] [[Lie group]] or [[Lie algebra]], which plays an important role in their structure theory and [[representation theory]]. It generalizes the [[polar decomposition]] or [[singular value decomposition]] of matrices. Its history can be traced to the 1880s work of [[Élie Cartan]] and [[Wilhelm Killing]]. [http://books.google.com/books?id=udj-1UuaOiIC&pg=PA46&dq=history+cartan+decomposition&hl=en&sa=X&ei=aa-wUuCDEMGmkQfNqoHABg&ved=0CDQQ6AEwAQ#v=onepage&q=history%20cartan%20decomposition&f=false]
| | Anxiousness problems encompass a group of circumstances that are characterised by feelings of maximum anxiousness coupled with bodily, psychological, and behavioral responses. Of all of the disorders categorized underneath anxiety disorders, panic assault is among the most common along with obsessive compulsive dysfunction (OCD).<br><br><br><br>In addition to the above, I started utilizing a Progesterone (20 mg measured dose) cream, which helped with my menopausal symptoms. I was examined and found to be producing too much estrogen, was having mental fog, evening sweats, temper swings and difficulty sleeping. This helped and I can buy it on-line or from a health food retailer - all pure type. Nourish your physique instead of attempting to fix it. Get enough sleep in fact. For these with issues with sleeping (together with me) peppermint works for that too. Hops is a powerful sleep assist as nicely. Drink a hops tea earlier than mattress. Panic Disorder is a broad-brush label at present applied to a variety ofconditions having clearly distinct etiologies, prognoses, and appropriatetreatment strategies. Click right here to start out the check<br><br>Diet additionally has a direct effect on stress and anxiety levels. It's observed that girls who eat quick spicy meals endure with larger ranges of tension and hyper rigidity. Pregnant women who eat lot of recent vegetables and fruit are much less liable to get affected with such issues. Consuming good amount of proteins also helps in dealing with stress ranges and nervousness assaults. Panic dysfunction is recognized extra typically in ladies than in males. Although the age of onset varies considerably, it's mostly skilled for the primary time between late adolescence and the mid-30's. Up to one-half of those identified with panic disorder even have agoraphobia. Whereas panic dysfunction can actually be critical, it is not instantly organ-threatening. Remedy Therapy<br><br>Consuming alcohol lose weight fast is commonly regarded as a approach to chill out and settle down. But consuming an excessive amount of alcohol not solely increases lactic acid levels in the blood , it additionally causes fluctuations in blood sugar ranges. This leads to elevated nervousness , irritability, and disturbed sleep patterns. It is best to drink sparsely and In the event you drink alcohol, do so carefully and never related to any habit. Nervousness disorders are dangerous and needs to be treated oncediscovered. There are several ways whereby a person can seek treatment foranxiety problems, be it by medical or non-medical means. Herbs are foundto be efficient in suppressing anxiousness and convey no aspect-results. Am I willing to place up with unpleasant unwanted side effects in return for nervousness reduction?<br><br>One scorching subject of research is the connection between anxiousness issues and genetics, because anxiety problems clearly run in households. At Women to Girls, we think genes are a think about some anxiousness disorders, but typically not the most important factor. Most of the time, anxious ladies grew up in anxious households. Anxiety is usually a realized habits that may be unlearned — even when it's severe. We'll return to this topic after we explore generalized anxiousness dysfunction, which affects many extra ladies than the severe nervousness problems. Generalized anxiousness dysfunction (GAD)<br><br>When you have almost any concerns concerning where in addition to how to employ simple diet burn fat, you possibly can contact us on the web site. |
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| == Cartan involutions on Lie algebras ==
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| Let <math>\mathfrak{g}</math> be a real [[Semisimple Lie algebra|semisimple]] [[Lie algebra]] and let <math>B(\cdot,\cdot)</math> be its [[Killing form]]. An involution on <math>\mathfrak{g}</math> is a Lie algebra [[automorphism]] <math>\theta</math> of <math>\mathfrak{g}</math> whose square is equal to the identity. Such an involution is called a '''Cartan involution''' on <math>\mathfrak{g}</math> if <math>B_\theta(X,Y) := -B(X,\theta Y)</math> is a [[positive definite bilinear form]].
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| Two involutions <math>\theta_1</math> and <math>\theta_2</math> are considered equivalent if they differ only by an [[inner automorphism]].
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| Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.
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| === Examples ===
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| {{^|NOTE: Blank lines between items helped source readability, but screwed up list formatting}}
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| * A Cartan involution on <math>\mathfrak{sl}_n(\mathbb{R})</math> is defined by <math>\theta(X)=-X^T</math>, where <math>X^T</math> denotes the transpose matrix of <math>X</math>.
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| * The identity map on <math>\mathfrak{g}</math> is an involution, of course. It is the unique Cartan involution of <math>\mathfrak{g}</math> if and only if the Killing form of <math>\mathfrak{g}</math> is negative definite. Equivalently, <math>\mathfrak{g}</math> is the Lie algebra of a compact semisimple Lie group.
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| * Let <math>\mathfrak{g}</math> be the complexification of a real semisimple Lie algebra <math>\mathfrak{g}_0</math>, then complex conjugation on <math>\mathfrak{g}</math> is an involution on <math>\mathfrak{g}</math>. This is the Cartan involution on <math>\mathfrak{g}</math> if and only if <math>\mathfrak{g}_0</math> is the Lie algebra of a compact Lie group.
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| * The following maps are involutions of the Lie algebra <math>\mathfrak{su}(n)</math> of the special unitary group [[SU(n)]]:
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| ** the identity involution <math>\theta_0(X) = X</math>, which is the unique Cartan involution in this case;
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| ** <math>\theta_1 (X) = - X^T</math> which on <math>\mathfrak{su}(n)</math> is also the complex conjugation;
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| ** if <math>n = p+q</math> is odd, <math>\theta_2 (X) = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix} X \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math>. These are all equivalent, but not equivalent to the identity involution (because the matrix <math>\begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}</math> does not belong to <math>\mathfrak{su}(n)</math>.)
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| ** if <math>n = 2m</math> is even, we also have <math>\theta_3 (X) = \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix} X^T \begin{pmatrix} 0 & I_m \\ -I_m & 0 \end{pmatrix}.</math>
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| == Cartan pairs ==
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| Let <math>\theta</math> be an involution on a Lie algebra <math>\mathfrak{g}</math>. Since <math>\theta^2=1</math>, the linear map <math>\theta</math> has the two eigenvalues <math>\pm1</math>. Let <math>\mathfrak{k}</math> and <math>\mathfrak{p}</math> be the corresponding eigenspaces, then <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math>. Since <math>\theta</math> is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that
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| : <math>[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}</math>, <math>[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}</math>, and <math>[\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}</math>.
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| Thus <math>\mathfrak{k}</math> is a Lie subalgebra, while any subalgebra of <math>\mathfrak{p}</math> is commutative.
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| Conversely, a decomposition <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math> with these extra properties determines an involution <math>\theta</math> on <math>\mathfrak{g}</math> that is <math>+1</math> on <math>\mathfrak{k}</math> and <math>-1</math> on <math>\mathfrak{p}</math>.
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| Such a pair <math>(\mathfrak{k}, \mathfrak{p})</math> is also called a '''Cartan pair''' of <math>\mathfrak{g}</math>.
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| The decomposition <math>\mathfrak{g} = \mathfrak{k}+\mathfrak{p}</math> associated to a Cartan involution is called a '''Cartan decomposition''' of <math>\mathfrak{g}</math>. The special feature of a Cartan decomposition is that the Killing form is negative definite on <math>\mathfrak{k}</math> and positive definite on <math>\mathfrak{p}</math>. Furthermore, <math>\mathfrak{k}</math> and <math>\mathfrak{p}</math> are orthogonal complements of each other with respect to the Killing form on <math>\mathfrak{g}</math>.
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| == Cartan decomposition on the Lie group level ==
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| Let <math>G</math> be a [[Semisimple Lie group|semisimple]] [[Lie group]] and <math>\mathfrak{g}</math> its [[Lie algebra]]. Let <math>\theta</math> be a Cartan involution on <math>\mathfrak{g}</math> and let <math>(\mathfrak{k},\mathfrak{p})</math> be the resulting Cartan pair. Let <math>K</math> be the [[analytic subgroup]] of <math>G</math> with Lie algebra <math>\mathfrak{k}</math>. Then:
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| * There is a Lie group automorphism <math>\Theta</math> with differential <math>\theta</math> that satisfies <math>\Theta^2=1</math>.
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| * The subgroup of elements fixed by <math>\Theta</math> is <math>K</math>; in particular, <math>K</math> is a closed subgroup.
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| * The mapping <math>K\times\mathfrak{p} \rightarrow G</math> given by <math>(k,X) \mapsto k\cdot \mathrm{exp}(X)</math> is a diffeomorphism.
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| * The subgroup <math>K</math> contains the center <math>Z</math> of <math>G</math>, and <math>K</math> is compact modulo center, that is, <math>K/Z</math> is compact.
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| * The subgroup <math>K</math> is the maximal subgroup of <math>G</math> that contains the center and is compact modulo center.
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| The automorphism <math>\Theta</math> is also called '''global Cartan involution''', and the diffeomorphism <math>K\times\mathfrak{p} \rightarrow G</math> is called '''global Cartan decomposition'''.
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| For the general linear group, we get <math> X \mapsto (X^{-1})^T </math> as the Cartan involution.
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| A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras <math>\mathfrak{a}</math> in <math>\mathfrak{p}</math> are unique up to conjugation by ''K''. Moreover
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| :<math>\displaystyle{\mathfrak{p}= \bigcup_{k\in K} \mathrm{Ad}\, k \cdot \mathfrak{a}.}</math>
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| In the compact and noncompact case this Lie algebraic result implies the decomposition
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| :<math>\displaystyle{G=KAK,}</math>
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| where ''A'' = exp <math>\mathfrak{a}</math>. Geometrically the image of the subgroup ''A'' in ''G'' / ''K'' ia a [[totally geodesic]] submanifold.
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| == Relation to polar decomposition ==
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| Consider <math>\mathfrak{gl}_n(\mathbb{R})</math> with the Cartan involution <math>\theta(X)=-X^T</math>. Then <math>\mathfrak{k}=\mathfrak{so}_n(\mathbb{R})</math> is the real Lie algebra of skew-symmetric matrices, so that <math>K=\mathrm{SO}(n)</math>, while <math>\mathfrak{p}</math> is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from <math>\mathfrak{p}</math> onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the [[polar decomposition]] of a matrix. Notice that the polar decomposition of an invertible matrix is unique.
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| == See also ==
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| * [[Lie group decompositions]]
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| == References ==
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| * {{citation|first=Sigurdur|last= Helgason|title=Differential geometry, Lie groups, and symmetric spaces|year=1978|publisher=Academic Press|isbn= 0-8218-2848-7}}
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| *[[A. W. Knapp]], ''Lie groups beyond an introduction'', ISBN 0-8176-4259-5, Birkhäuser.
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| [[Category:Lie groups]]
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| [[Category:Lie algebras]]
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Anxiousness problems encompass a group of circumstances that are characterised by feelings of maximum anxiousness coupled with bodily, psychological, and behavioral responses. Of all of the disorders categorized underneath anxiety disorders, panic assault is among the most common along with obsessive compulsive dysfunction (OCD).
In addition to the above, I started utilizing a Progesterone (20 mg measured dose) cream, which helped with my menopausal symptoms. I was examined and found to be producing too much estrogen, was having mental fog, evening sweats, temper swings and difficulty sleeping. This helped and I can buy it on-line or from a health food retailer - all pure type. Nourish your physique instead of attempting to fix it. Get enough sleep in fact. For these with issues with sleeping (together with me) peppermint works for that too. Hops is a powerful sleep assist as nicely. Drink a hops tea earlier than mattress. Panic Disorder is a broad-brush label at present applied to a variety ofconditions having clearly distinct etiologies, prognoses, and appropriatetreatment strategies. Click right here to start out the check
Diet additionally has a direct effect on stress and anxiety levels. It's observed that girls who eat quick spicy meals endure with larger ranges of tension and hyper rigidity. Pregnant women who eat lot of recent vegetables and fruit are much less liable to get affected with such issues. Consuming good amount of proteins also helps in dealing with stress ranges and nervousness assaults. Panic dysfunction is recognized extra typically in ladies than in males. Although the age of onset varies considerably, it's mostly skilled for the primary time between late adolescence and the mid-30's. Up to one-half of those identified with panic disorder even have agoraphobia. Whereas panic dysfunction can actually be critical, it is not instantly organ-threatening. Remedy Therapy
Consuming alcohol lose weight fast is commonly regarded as a approach to chill out and settle down. But consuming an excessive amount of alcohol not solely increases lactic acid levels in the blood , it additionally causes fluctuations in blood sugar ranges. This leads to elevated nervousness , irritability, and disturbed sleep patterns. It is best to drink sparsely and In the event you drink alcohol, do so carefully and never related to any habit. Nervousness disorders are dangerous and needs to be treated oncediscovered. There are several ways whereby a person can seek treatment foranxiety problems, be it by medical or non-medical means. Herbs are foundto be efficient in suppressing anxiousness and convey no aspect-results. Am I willing to place up with unpleasant unwanted side effects in return for nervousness reduction?
One scorching subject of research is the connection between anxiousness issues and genetics, because anxiety problems clearly run in households. At Women to Girls, we think genes are a think about some anxiousness disorders, but typically not the most important factor. Most of the time, anxious ladies grew up in anxious households. Anxiety is usually a realized habits that may be unlearned — even when it's severe. We'll return to this topic after we explore generalized anxiousness dysfunction, which affects many extra ladies than the severe nervousness problems. Generalized anxiousness dysfunction (GAD)
When you have almost any concerns concerning where in addition to how to employ simple diet burn fat, you possibly can contact us on the web site.