Kristen Kane Un Official Fan Site Tribute

Kristen Kane : This Is An Un Official Fan Site Tribute
Kristen Kane Kristin Hunter, Kristin Kane
Porn Queen Actress Superstar

Kristen Kane

Movie Title Year Distributor Notes Rev Formats 72 Cheerleader Orgy 2000 Vivid LezOnly O All Pissed Off 3 2000 Wildlife NonSex Pee 1 DO Beach Bunnies with Big Brown Eyes 11 2001 Seymore Butts Anal Facial A2M 1 O Bottom Dweller 6: Sex After Death 2000 Elegant Angel 6 DO Bottom Dweller Orgies 2002 Elegant Angel 1 DRO Coed Cocksuckers 19 2000 Zane Entertainment Group 1 Different Strokes 6: Rodney's Birthday Blast 2000 Odyssey Facial CumSwap 1 DRO Ecstasy 5 2001 Wildlife 2 DRO Filthy First Timers 15 2000 Elegant Angel Facial 1 O Handjobs 5 2000 Wildlife HJOnly DRO I Swallow 10 2000 Odyssey Facial 1 DRO Lust In The Hood 2000 Extreme Associates Facial 1 Monster Facials 1 2002 Rodnievision Facial 2 DRO More Dirty Debutantes 147 2000 4-Play Video Anal Facial 1 DR Perverted Stories 31 2001 JM Productions Facial DP 2 DO Screw My Wife Please 14 (And Make Her Cream) 2000 Wildlife Anal Facial 1 DRO Shut Up and Blow Me 22 2000 All Good Video BJOnly Facial 1 DO Symphony In Bondage 2000 California Star NonSex University Coeds 26 2000 Dane Productions Anal Facial Bald 1
Rank one example There is only one root system of rank 1, consisting of two nonzero vectors {\displaystyle \{\alpha ,-\alpha \}}\{\alpha ,-\alpha \}. This root system is called {\displaystyle A_{1}}A_{1}. Rank two examples In rank 2 there are four possibilities, corresponding to {\displaystyle \sigma _{\alpha }(\beta )=\beta +n\alpha }\sigma _{\alpha }(\beta )=\beta +n\alpha , where {\displaystyle n=0,1,2,3}n=0,1,2,3.[8] The figure at right shows these possibilities, but with some redundancies: {\displaystyle A_{1}\times A_{1}}{\displaystyle A_{1}\times A_{1}} is isomorphic to {\displaystyle D_{2}}D_{2} and {\displaystyle B_{2}}B_{2} is isomorphic to {\displaystyle C_{2}}C_{2}. Note that a root system is not determined by the lattice that it generates: {\displaystyle A_{1}\times A_{1}}A_{1}\times A_{1} and {\displaystyle B_{2}}B_{2} both generate a square lattice while {\displaystyle A_{2}}A_{2} and {\displaystyle G_{2}}G_{2} generate a hexagonal lattice, only two of the five possible types of lattices in two dimensions. Whenever F is a root system in E, and S is a subspace of E spanned by ? = F n S, then ? is a root system in S. Thus, the exhaustive list of four root systems of rank 2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank. In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees. Root systems arising from semisimple Lie algebras See also: Semisimple Lie algebra § Cartan subalgebras and root systems, and Root system of a semi-simple Lie algebra If {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} is a complex semisimple Lie algebra and {\displaystyle {\mathfrak {h}}}{\mathfrak {h}} is a Cartan subalgebra, we can construct a root system as follows. We say that {\displaystyle \alpha \in {\mathfrak {h}}^{*}}{\displaystyle \alpha \in {\mathfrak {h}}^{*}} is a root of {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} relative to {\displaystyle {\mathfrak {h}}}{\mathfrak {h}} if {\displaystyle \alpha \neq 0}\alpha \neq 0 and there exists some {\displaystyle X\neq 0\in {\mathfrak {g}}}{\displaystyle X\neq 0\in {\mathfrak {g}}} such that

{\displaystyle [H,X]=\alpha (H)X}{\displaystyle [H,X]=\alpha (H)X} for all {\displaystyle H\in {\mathfrak {h}}}{\displaystyle H\in {\mathfrak {h}}}. One can show[9] that there is an inner product for which the set of roots forms a root system. The root system of {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} is a fundamental tool for analyzing the structure of {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} and classifying its representations. (See the section below on Root systems and Lie theory.) History The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem[10]).[11] He used them in his attempt to classify all simple Lie algebras over the field of complex numbers. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.[12] Killing investigated the structure of a Lie algebra {\displaystyle L}L, by considering what is now called a Cartan subalgebra {\displaystyle {\mathfrak {h}}}{\mathfrak {h}}. Then he studied the roots of the characteristic polynomial {\displaystyle \det(\operatorname {ad} _{L}x-t)}{\displaystyle \det(\operatorname {ad} _{L}x-t)}, where {\displaystyle x\in {\mathfrak {h}}}x\in {\mathfrak {h}}. Here a root is considered as a function of {\displaystyle {\mathfrak {h}}}{\mathfrak {h}}, or indeed as an element of the dual vector space {\displaystyle {\mathfrak {h}}^{*}}{\mathfrak {h}}^{*}. This set of roots form a root system inside {\displaystyle {\mathfrak {h}}^{*}}{\mathfrak {h}}^{*}, as defined above, where the inner product is the Killing form.[13] Elementary consequences of the root system axioms The integrality condition for {\displaystyle \langle \beta ,\alpha \rangle }{\displaystyle \langle \beta ,\alpha \rangle } is fulfilled only for ß on one of the vertical lines, while the integrality condition for {\displaystyle \langle \alpha ,\beta \rangle }{\displaystyle \langle \alpha ,\beta \rangle } is fulfilled only for ß on one of the red circles. Any ß perpendicular to a (on the Y axis) trivially fulfills both with 0, but does not define an irreducible root system. Modulo reflection, for a given a there are only 5 nontrivial possibilities for ß, and 3 possible angles between a and ß in a set of simple roots. Subscript letters correspond to the series of root systems for which the given ß can serve as the first root and a as the second root (or in F4 as the middle 2 roots). The cosine of the angle between two roots is constrained to be one-half of the square root of a positive integer. This is because {\displaystyle \langle \beta ,\alpha \rangle }\langle \beta ,\alpha \rangle and {\displaystyle \langle \alpha ,\beta \rangle }\langle \alpha ,\beta \rangle are both integers, by assumption, and {\displaystyle \langle \beta ,\alpha \rangle \langle \alpha ,\beta \rangle =2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\cdot 2{\frac {(\alpha ,\beta )}{(\beta ,\beta )}}=4{\frac {(\alpha ,\beta )^{2}}{\vert \alpha \vert ^{2}\vert \beta \vert ^{2}}}=4\cos ^{2}(\theta )=(2\cos(\theta ))^{2}\in \mathbb {Z} .}\langle \beta ,\alpha \rangle \langle \alpha ,\beta \rangle =2{\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}\cdot 2{\frac {(\alpha ,\beta )}{(\beta ,\beta )}}=4{\frac {(\alpha ,\beta )^{2}}{\vert \alpha \vert ^{2}\vert \beta \vert ^{2}}}=4\cos ^{2}(\theta )=(2\cos(\theta ))^{2}\in \mathbb {Z} . Since {\displaystyle 2\cos(\theta )\in [-2,2]}2\cos(\theta )\in [-2,2], the only possible values for {\displaystyle \cos(\theta )}\cos(\theta ) are {\displaystyle 0,\pm {\tfrac {1}{2}},\pm {\tfrac {\sqrt {2}}{2}},\pm {\tfrac {\sqrt {3}}{2}}}{\displaystyle 0,\pm {\tfrac {1}{2}},\pm {\tfrac {\sqrt {2}}{2}},\pm {\tfrac {\sqrt {3}}{2}}} and {\displaystyle \pm {\tfrac {\sqrt {4}}{2}}=\pm 1}{\displaystyle \pm {\tfrac {\sqrt {4}}{2}}=\pm 1}, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of a other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2a or -2a, are out. The diagram at right shows that an angle of 60° or 120° corresponds to roots of equal length, while an angle of 45° or 135° corresponds to a length ratio of {\displaystyle {\sqrt {2}}}{\sqrt {2}} and an angle of 30° or 150° corresponds to a length ratio of {\displaystyle {\sqrt {3}}}{\sqrt {3}}. In summary, here are the only possibilities for each pair of roots.[14] Angle of 90 degrees; in that case, the length ratio is unrestricted. Angle of 60 or 120 degrees, with a length ratio of 1. Angle of 45 or 135 degrees, with a length ratio of {\displaystyle {\sqrt {2}}}{\sqrt {2}}. Angle of 30 or 150 degrees, with a length ratio of {\displaystyle {\sqrt {3}}}{\sqrt 3}. Positive roots and simple roots The labeled roots are a set of positive roots for the {\displaystyle G_{2}}G_{2} root system, with {\displaystyle \alpha _{1}}\alpha _{1} and {\displaystyle \alpha _{2}}\alpha _{2} being the simple roots Given a root system {\displaystyle \Phi }\Phi we can always choose (in many ways) a set of positive roots. This is a subset {\displaystyle \Phi ^{+}}\Phi ^{+} of {\displaystyle \Phi }\Phi such that For each root {\displaystyle \alpha \in \Phi }\alpha \in \Phi exactly one of the roots {\displaystyle \alpha }\alpha , –{\displaystyle \alpha }\alpha is contained in {\displaystyle \Phi ^{+}}\Phi ^{+}. For any two distinct {\displaystyle \alpha ,\beta \in \Phi ^{+}}\alpha ,\beta \in \Phi ^{+} such that {\displaystyle \alpha +\beta }\alpha +\beta is a root, {\displaystyle \alpha +\beta \in \Phi ^{+}}\alpha +\beta \in \Phi ^{+}. If a set of positive roots {\displaystyle \Phi ^{+}}\Phi ^{+} is chosen, elements of {\displaystyle -\Phi ^{+}}-\Phi ^{+} are called negative roots. A set of positive roots may be constructed by choosing a hyperplane {\displaystyle V}V not containing any root and setting {\displaystyle \Phi ^{+}}\Phi ^{+} to be all the roots lying on a fixed side of {\displaystyle V}V. Furthermore, every set of positive roots arises in this way.[15] An element of {\displaystyle \Phi ^{+}}\Phi ^{+} is called a simple root if it cannot be written as the sum of two elements of {\displaystyle \Phi ^{+}}\Phi ^{+}. (The set of simple roots is also referred to as a base for {\displaystyle \Phi }\Phi .) The set {\displaystyle \Delta }\Delta of simple roots is a basis of {\displaystyle E}E with the following additional special properties:[16] Every root {\displaystyle \alpha \in \Phi }\alpha \in \Phi is a linear combination of elements of {\displaystyle \Delta }\Delta with integer coefficients. For each {\displaystyle \alpha \in \Phi }\alpha \in \Phi , the coefficients in the previous point are either all non-negative or all non-positive. For each root system {\displaystyle \Phi }\Phi there are many different choices of the set of positive roots—or, equivalently, of the simple roots—but any two sets of positive roots differ by the action of the Weyl group.[17] Dual root system, coroots, and integral elements See also: Langlands dual group The dual root system If F is a root system in E, the coroot a? of a root a is defined by {\displaystyle \alpha ^{\vee }={2 \over (\alpha ,\alpha )}\,\alpha .}\alpha ^{\vee }={2 \over (\alpha ,\alpha )}\,\alpha . The set of coroots also forms a root system F? in E, called the dual root system (or sometimes inverse root system). By definition, a? ? = a, so that F is the dual root system of F?. The lattice in E spanned by F? is called the coroot lattice. Both F and F? have the same Weyl group W and, for s in W, {\displaystyle (s\alpha )^{\vee }=s(\alpha ^{\vee }).}(s\alpha )^{\vee }=s(\alpha ^{\vee }). If ? is a set of simple roots for F, then ?? is a set of simple roots for F?.[18] In the classification described below, the root systems of type {\displaystyle A_{n}}A_{n} and {\displaystyle D_{n}}D_{n} along with the exceptional root systems {\displaystyle E_{6},E_{7},E_{8},F_{4},G_{2}}{\displaystyle E_{6},E_{7},E_{8},F_{4},G_{2}} are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the {\displaystyle B_{n}}B_{n} and {\displaystyle C_{n}}C_{n} root systems are dual to one another, but not isomorphic (except when {\displaystyle n=2}n=2). Integral elements See also: Weight (representation theory) § Weights in the representation theory of semisimple Lie algebras A vector {\displaystyle \lambda }\lambda in E is called integral[19] if its inner product with each coroot is an integer: {\displaystyle 2{\frac {(\lambda ,\alpha )}{(\alpha ,\alpha )}}\in \mathbb {Z} ,\quad \alpha \in \Phi .}{\displaystyle 2{\frac {(\lambda ,\alpha )}{(\alpha ,\alpha )}}\in \mathbb {Z} ,\quad \alpha \in \Phi .} Since the set of {\displaystyle \alpha ^{\vee }}{\displaystyle \alpha ^{\vee }} with {\displaystyle \alpha \in \Delta }{\displaystyle \alpha \in \Delta } forms a base for the dual root system, to verify that {\displaystyle \lambda }\lambda is integral, it suffices to check the above condition for {\displaystyle \alpha \in \Delta }{\displaystyle \alpha \in \Delta }. The set of integral elements is called the weight lattice associated to the given root system. This term comes from the representation theory of semisimple Lie algebras, where the integral elements form the possible weights of finite-dimensional representations. The definition of a root system guarantees that the roots themselves are integral elements. Thus, every integer linear combination of roots is also integral. In most cases, however, there will be integral elements that are not integer combinations of roots. That is to say, in general the weight lattice does not coincide with the root lattice. Classification of root systems by Dynkin diagrams See also: Dynkin diagram Pictures of all the connected Dynkin diagrams A root system is irreducible if it can not be partitioned into the union of two proper subsets {\displaystyle \Phi =\Phi _{1}\cup \Phi _{2}}\Phi =\Phi _{1}\cup \Phi _{2}, such that {\displaystyle (\alpha ,\beta )=0}(\alpha ,\beta )=0 for all {\displaystyle \alpha \in \Phi _{1}}\alpha \in \Phi _{1} and {\displaystyle \beta \in \Phi _{2}}\beta \in \Phi _{2} . Irreducible root systems correspond to certain graphs, the Dynkin diagrams named after Eugene Dynkin. The classification of these graphs is a simple matter of combinatorics, and induces a classification of irreducible root systems. Constructing the Dynkin diagram Given a root system, select a set ? of simple roots as in the preceding section. The vertices of the associated Dynkin diagram correspond to the roots in ?. Edges are drawn between vectors as follows, according to the angles. (Note that the angle between simple roots is always at least 90 degrees.) No edge if the vectors are orthogonal, An undirected single edge if they make an angle of 120 degrees, A directed double edge if they make an angle of 135 degrees, and A directed triple edge if they make an angle of 150 degrees. The term "directed edge" means that double and triple edges are marked with an arrow pointing toward the shorter vector. (Thinking of the arrow as a "greater than" sign makes it clear which way the arrow is supposed to point.) Note that by the elementary properties of roots noted above, the rules for creating the Dynkin diagram can also be described as follows. No edge if the roots are orthogonal; for nonorthogonal roots, a single, double, or triple edge according to whether the length ratio of the longer to shorter is 1, {\displaystyle {\sqrt {2}}}{\sqrt {2}}, {\displaystyle {\sqrt {3}}}{\sqrt 3}. In the case of the {\displaystyle G_{2}}G_{2} root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of {\displaystyle {\sqrt {3}}}{\sqrt 3}). Thus, the Dynkin diagram has two vertices joined by a triple edge, with an arrow pointing from the vertex associated to the longer root to the other vertex. (In this case, the arrow is a bit redundant, since the diagram is equivalent whichever way the arrow goes.) Classifying root systems Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices.[20] Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one can match up roots, starting with the roots in the base, and show that the systems are in fact the same.[21] Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. A root systems is irreducible if and only if its Dynkin diagrams is connected.[22] The possible connected diagrams are as indicated in the figure. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system). If {\displaystyle \Phi }\Phi is a root system, the Dynkin diagram for the dual root system {\displaystyle \Phi ^{\vee }}\Phi ^{\vee } is obtained from the Dynkin diagram of {\displaystyle \Phi }\Phi by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that {\displaystyle B_{n}}B_{n} and {\displaystyle C_{n}}C_{n} are dual to each other. Weyl chambers and the Weyl group See also: Coxeter group § Affine Coxeter groups The shaded region is the fundamental Weyl chamber for the base {\displaystyle \{\alpha _{1},\alpha _{2}\}}{\displaystyle \{\alpha _{1},\alpha _{2}\}} If {\displaystyle \Phi \subset E}{\displaystyle \Phi \subset E} is a root system, we may consider the hyperplane perpendicular to each root {\displaystyle \alpha }\alpha . Recall that {\displaystyle \sigma _{\alpha }}{\displaystyle \sigma _{\alpha }} denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of {\displaystyle E}E generated by all the {\displaystyle \sigma _{\alpha }}{\displaystyle \sigma _{\alpha }}'s. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set ? of simple roots, we may define the fundamental Weyl chamber associated to ? as the set of points {\displaystyle v\in E}{\displaystyle v\in E} such that {\displaystyle (\alpha ,v)>0}{\displaystyle (\alpha ,v)>0} for all {\displaystyle \alpha \in \Delta }{\displaystyle \alpha \in \Delta }. Since the reflections {\displaystyle \sigma _{\alpha },\,\alpha \in \Phi }{\displaystyle \sigma _{\alpha },\,\alpha \in \Phi } preserve {\displaystyle \Phi }\Phi , they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers. The figure illustrates the case of the {\displaystyle A_{2}}A_{2} root system. The "hyperplanes" (in this case, one dimensional) orthogonal

nude bikini pics clinton photos chelsea pictures desnuda fotos naked laura porn free porno fan and linda video site lisa kelly playboy topless lolo joan xxx official sex traci ferrari lords eva photo the nue tube pic videos sexy smith ana leah welch lovelace you remini club loren giacomo karen elizabeth carangi fake julia trinity ava kate fenech dana pozzi images gallery edwige moana victoria kristel joanna pornstar foto sylvia rachel pamela principal clips movies lauren shania valerie fabian collins nia rio del robin rhodes hart jane stevens measurements susan taylor jenny sanchez moore lane antonelli lancaume nancy roselyn emily hartley boobs brooke angie kim web demi bonet carrie allen grant hot esther deborah with braga jones fansite yates freeones
lee heather tina inger severance christina louise lopez gina wallpaper nacked ann film nackt fisher carey corinne shue ass vancamp clery model shannon elisabeth panties biografia angelina sofia erin monroe dazza charlene janet doris vanessa anna belinda reguera diane paula fucking scene peeples sonia shauna autopsy monica sharon patricia alicia plato bardot
melissa movie picture cynthia nicole maria star nina julie mary gemser naomi williams torrent nuda barbara twain anderson gia nudes fakes larue pussy actress upskirt san raquel jennifer tits mariah meg sandra big michelle roberts marie lumley tewes clip salma vergara jada cristal day shields cassidy sandrelli penthouse dickinson goldie nud angel brigitte drew fucked amanda shemale olivia website milano ellen ellison vidcaps hayek stone download carmen bessie swimsuit vera zeta locklear shirley anal gray cindy marilyn connie kayla sucking streep cock jensen john tiffani stockings hawn for weaver rue barrymore catherine bellucci rebecca bondage feet applegate jolie sigourney wilkinson nipples juliet revealing teresa magazine kennedy ashley what bio biography agutter wood her jordan hill com jessica pornos blowjob
lesbian nued grace hardcore regera palmer asia theresa leeuw heaton juhi alyssa pinkett rene actriz black vicky jamie ryan gillian massey short shirtless scenes maggie dreyfus lynne mpegs melua george thiessen jean june crawford alex natalie bullock playmate berry andrews maren kleevage quennessen pix hair shelley tiffany gunn galleries from russo dhue lebrock leigh fuck stefania tilton laurie russell vids bessie swimsuit vera zeta shirley locklear anal gray cindy marilyn connie kayla sucking streep cock jensen john tiffani stockings hawn for weaver rue catherine barrymore bellucci rebecca bondage feet applegate jolie george thiessen jean june crawford alex sigourney wilkinson nipples juliet revealing teresa magazine kennedy ashley what bio biography agutter jordan wood her hill com jessica pornos blowjob lesbian nued grace
hardcore regera palmer asia theresa leeuw heaton juhi alyssa pinkett rene actriz black vicky rutherford lohan winslet spungen shawnee swanson newton hannah leslie silverstone did frann wallpapers kidman louis kristy valeria lang fiorentino deanna rita hillary katie granny girls megan tori paris arquette amber sue escort chawla dorothy jessie anthony courtney shot sites kay meryl judy candice desnudo wallace gertz show teen savannah busty schneider glass thong spears young erika aniston stiles capshaw loni imagenes von myspace jena daryl girl hotmail nicola savoy
garr bonnie sexe play adriana donna angelique love actor mitchell unger sellecca adult hairstyles malone teri hayworth lynn harry kara rodriguez films welles peliculas kaprisky uschi blakely halle lindsay miranda jami jamie ryan gillian massey short scenes shirtless maggie dreyfus lynne mpegs melua natalie bullock playmate berry andrews maren kleevage quennessen pix hair shelley tiffany gunn