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Movie Title Year Distributor Notes Rev Formats 40 of the World's Most Beautiful She-Males 2000 Leisure Time Entertainment American She Males 1 2005 Avica Entertainment DO Attack of the Fifty Foot Tranny 1999 Vivid Top DRO Busty Transsexual Beauty Queens 2 2007 Avalon Enterprises MastOnly DRO Drag How To 1 2001 Bizarre Video Drag How To 2 2001 Bizarre Video O Drag Queen Boulevard 1999 130 C Street Corporation DO Hard To The Touch 2006 Smash Pictures 1 DRO Horny Little Devils 1998 Devil's Film O Hot Chicks with Hung Dicks 1 2010 Blue Coyote Pictures DRO Kimberly Devine - Transexual Teaser 2003 JB Video D Nasty She-Male Sex 2000 Leisure Time Entertainment Planet Giselle 1 2004 Spring Break Bottom Portrait Of A TS 1997 Bizarre Video DRO Prick Chicks 8 2003 Underground Video D Queen 1999 Devil's Film 1 RO Real Transsexuals 1998 Metro Top 1 DO Shakin' It With She Males 8 1998 Vivid She Male Husbands 1999 Boy Chick Productions Facial Bottom DO She Male Interns 1998 Boy Chick Productions O She Male Jet Set 5 2003 Altered State 1 DO She Male Sex Pistols at the Rock Hard Cafe 2 2003 Androgeny MastOnly DO She Male Sex Pistols Unleashed 1 2007 Avalon Enterprises She Male Splendor 1998 He Girl Video DO
She Male Strokers 3 2004 Exquisite MastOnly DRO She Male Swat Team 2 1998 Bizarre Video She Males Enslaved 2 2002 Bizarre Video She's Got Balls 2002 Caballero Home Video Top DRO She-Male Cherry Busters 2 2002 Leoram Inc Top O She-Male Nation 1 2000 Bizarre Video She-Male Sorority Secrets 1999 Leoram Inc Top She-Males In Heat 2001 Leisure Time Entertainment Shemale Sorority Secrets 2002 Leoram Inc Straight Guys Who Like Tranny Cock 5 2005 Devil's Film DRO Summer Girls ... And Some Are Not 1 2001 Robert Hill Releasing Top DRO Summer Girls ... And Some Are Not 5 2003 Robert Hill Releasing DRO T Girl Fantasies 2 2002 Don Goo Bottom 1 DRO That 70's Transsexual Show 2001 Devil's Film DR Their Feminine Side 1999 Executive (Leisure Time Entertainment) Trannie Hides The Cigar 1999 He Girl Video DO Trannie Trivia 2000 Boy Chick Productions O Tranny Transformation 2 2003 Bizarre Video DO Trans Gender Bender 3 2002 Midnight Video DO Transparent 1998 All Worlds Top DRO Transsexual Adventures 1998 Bizarre Video Transsexual Beauty Queens - Dildo Mania 2009 Androgeny DRO Transsexual Beauty Queens 12 2001 Avalon Enterprises Top O



Transsexual Beauty Queens Unleashed 1 2005 Androgeny O Transsexual Beauty Queens: Divine Divas 3 2012 Androgeny DO Transsexual Beauty Queens: Hung Honey's 4 2008 Androgeny O Transsexual Beauty Queens: Hung Honey's 9 2010 Androgeny DRO Transsexual Dream Girls 4 2002 Odyssey Top DO Transsexual Dream Girls 8 2002 Odyssey Top DO Transsexual Dynasty 3 1998 Bizarre Video Transsexual Dynasty 4 1998 Bizarre Video DO Transsexual Extreme 2 2005 Bizarre Video DRO Transsexual High School Reunion 1998 Bizarre Video NonSex Transsexual Party Monsters 2000 130 C Street Corporation Top Transsexual Prostitutes 4: Smokin in the Girls Room 1998 Devil's Film DR Transsexual Prostitutes 7: Best Little Whorehouse in California 1998 Devil's Film DRO Transsexual Slumber Party 1998 Bizarre Video Transsexual Submission 2 2000 Bizarre Video DO Transsexual Submission 3 2000 Bizarre Video TS Dating Game 1998 Bizarre Video TS Sex School 1998 Bizarre Video NonSex DO Ultimate She-Male 4: Kimberley 2006 Leoram Inc Top O You Got She-Mail 2011 Channel 1 Releasing real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group. An example of a simply connected group is the special unitary group SU(2), which as a manifold is the 3-sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n), the spin group (double cover of rotation group) Spin(n) for {\displaystyle n\geq 3}n\geq 3, and the compact symplectic group Sp(n).[16] Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups. The exponential map Main article: Exponential map (Lie theory) See also: derivative of the exponential map and normal coordinates The exponential map from the Lie algebra {\displaystyle M(n;\mathbb {C} )}{\displaystyle M(n;\mathbb {C} )} of the general linear group {\displaystyle GL(n;\mathbb {C} )}{\displaystyle GL(n;\mathbb {C} )} to {\displaystyle GL(n;\mathbb {C} )}{\displaystyle GL(n;\mathbb {C} )} is defined by the matrix exponential, given by the usual power series: {\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots }{\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots } for matrices {\displaystyle X}X. If {\displaystyle G}G is a closed subgroup of {\displaystyle GL(n;\mathbb {C} )}{\displaystyle GL(n;\mathbb {C} )}, then the exponential map takes the Lie algebra of {\displaystyle G}G into {\displaystyle G}G; thus, we have an exponential map for all matrix groups. Every element of {\displaystyle G}G that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.[17] The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows. For each vector {\displaystyle X}X in the Lie algebra {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} of {\displaystyle G}G (i.e., the tangent space to {\displaystyle G}G at the identity), one proves that there is a unique one-parameter subgroup {\displaystyle c:\mathbb {R} \rightarrow G}{\displaystyle c:\mathbb {R} \rightarrow G} such that {\displaystyle c'(0)=X}{\displaystyle c'(0)=X}. Saying that {\displaystyle c}c is a one-parameter subgroup means simply that {\displaystyle c}c is a smooth map into {\displaystyle G}G and that {\displaystyle c(s+t)=c(s)c(t)\ }c(s+t)=c(s)c(t)\ for all {\displaystyle s}s and {\displaystyle t}t. The operation on the right hand side is the group multiplication in {\displaystyle G}G. The formal similarity of this formula with the one valid for the exponential function justifies the definition {\displaystyle \exp(X)=c(1).\ }{\displaystyle \exp(X)=c(1).\ } This is called the exponential map, and it maps the Lie algebra {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} into the Lie group {\displaystyle G}G It provides a diffeomorphism between a neighborhood of 0 in {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} and a neighborhood of {\displaystyle e}e in {\displaystyle G}G. This exponential map is a generalization of the exponential function for real numbers (because {\displaystyle \mathbb {R} }\mathbb {R} is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because {\displaystyle \mathbb {C} }\mathbb {C} is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because {\displaystyle M(n,\mathbb {R} )}{\displaystyle M(n,\mathbb {R} )} with the regular commutator is the Lie algebra of the Lie group {\displaystyle GL(n,\mathbb {R} )}{\displaystyle GL(n,\mathbb {R} )} of all invertible matrices). Because the exponential map is surjective on some neighbourhood {\displaystyle N}N of {\displaystyle e}e, it is common to call elements of the Lie algebra infinitesimal generators of the group {\displaystyle G}G. The subgroup of {\displaystyle G}G generated by {\displaystyle N}N is the identity component of {\displaystyle G}G. The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood {\displaystyle U}U of the zero element of {\displaystyle {\mathfrak {g}}}{\mathfrak {g}}, such that for {\displaystyle X,Y\in U}{\displaystyle X,Y\in U} we have {\displaystyle \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac {1}{12}}[\,[X,Y],Y]-{\tfrac {1}{12}}[\,[X,Y],X]-\cdots \right),}{\displaystyle \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac {1}{12}}[\,[X,Y],Y]-{\tfrac {1}{12}}[\,[X,Y],X]-\cdots \right),} where the omitted terms are known and involve Lie brackets of four or more elements. In case {\displaystyle X}X and {\displaystyle Y}Y commute, this formula reduces to the familiar exponential law {\displaystyle \exp(X)\exp(Y)=\exp(X+Y)}{\displaystyle \exp(X)\exp(Y)=\exp(X+Y)} The exponential map relates Lie group homomorphisms. That is, if {\displaystyle \phi :G\to H}\phi :G\to H is a Lie group homomorphism and {\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}}\phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}} the induced map on the corresponding Lie algebras, then for all {\displaystyle x\in {\mathfrak {g}}}x\in {\mathfrak {g}} we have {\displaystyle \phi (\exp(x))=\exp(\phi _{*}(x)).\,}\phi (\exp(x))=\exp(\phi _{*}(x)).\, In other words, the following diagram commutes,[Note 1] ExponentialMap-01.png (In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.) The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C8 Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1. Lie subgroup A Lie subgroup {\displaystyle H}H of a Lie group {\displaystyle G}G is a Lie group that is a subset of {\displaystyle G}G and such that the inclusion map from {\displaystyle H}H to {\displaystyle G}G is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of {\displaystyle G}G admits a unique smooth structure which makes it an embedded Lie subgroup of {\displaystyle G}G—i.e. a Lie subgroup such that the inclusion map is a smooth embedding. Examples of non-closed subgroups are plentiful; for example take {\displaystyle G}G to be a torus of dimension 2 or greater, and let {\displaystyle H}H be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism {\displaystyle \varphi :\mathbb {R} \to G}{\displaystyle \varphi :\mathbb {R} \to G} with {\displaystyle \mathrm {im} (\varphi )=H}{\displaystyle \mathrm {im} (\varphi )=H}. The closure of {\displaystyle H}H will be a sub-torus in {\displaystyle G}G. The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group {\displaystyle G}G and the subalgebras of the Lie algebra of {\displaystyle G}G.[18] Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of {\displaystyle G}G which determines which subalgebras correspond to closed subgroups. Representations Main article: Representation of a Lie group See also: Compact group § Representation theory of a connected compact Lie group, and Lie algebra representation One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, {\displaystyle {\hat {H}}\psi =E\psi }{\displaystyle {\hat {H}}\psi =E\psi }. Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator {\displaystyle {\hat {H}}}{\hat {H}} commutes with the action of SO(3) on the wave function {\displaystyle \psi }\psi . (One important example of such a system is the Hydrogen atom.) This assumption does not necessarily mean that the solutions {\displaystyle \psi }\psi are rotationally invariant functions. Rather, it means that the space of solutions to {\displaystyle {\hat {H}}\psi =E\psi }{\displaystyle {\hat {H}}\psi =E\psi } is invariant under rotations (for each fixed value of {\displaystyle E}E). This space, therefore, constitutes a representation of SO(3). These representations have been classified and the classification leads to a substantial simplification of the problem, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation. The case of a connected compact Lie group K (including the just-mentioned case of SO(3)) is particularly tractable.[19] In that case, every finite-dimensional representation of K decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by Hermann Weyl. The classification is in terms of the "highest weight" of the representation. The classification is closely related to the classification of representations of a semisimple Lie algebra. One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the representations of the group SL(2,R) and the representations of the Poincaré group. Early history According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), Sophus Lie himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation (ibid). Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years (ibid, p. 2). Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe (ibid, p. 76). In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformationsgruppen, published in 1888, 1890, and 1893. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.[20] Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p. 43). Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions. Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject. Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen (The composition of continuous finite transformation groups) (Hawkins, p. 100). The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights. In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris. Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups.[21] The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley. The concept of a Lie group, and possibilities of classification Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point. The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these. Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness. A first key result is the Levi decomposition, which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup. Connected compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S1 and simple compact Lie groups (which correspond to connected Dynkin diagrams). Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Solvable groups are too messy to classify except in a few small dimensions. Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions. Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL(2, R) is simple according to the second definition but not according to the first. They have all been classified (for either definition). Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.[22] They are central extensions of products of simple Lie groups. The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write Gcon for the connected component of the identity Gsol for the largest connected normal solvable subgroup Gnil for the largest connected normal nilpotent subgroup so that we have a sequence of normal subgroups 1 ? Gnil ? Gsol ? Gcon ? G. Then G/Gcon is discrete Gcon/Gsol is a central extension of a product of simple connected Lie groups. Gsol/Gnil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S1. Gnil/1 is nilpotent, and therefore its ascending central series has all quotients abelian. This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension. The diffeomorphism group of a Lie group acts transitively on the Lie group Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity) Infinite-dimensional Lie groups Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed to Euclidean space in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold. The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined. Some of the examples that have been studied include: The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivation of this fact) is the symmetry algebra of two-dimensional conformal field theory. Diffeomorphism groups of compact manifolds of larger dimension are regular Fréchet Lie groups; very little about their structure is known. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity. The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras. There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on.[23] One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem. In M-theory, for example, a 10 dimensional SU(N) gauge theory becomes an 11 dimensional theory when N becomes infinite


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