This shows that the space of left invariant vector fields (vector fields satisfying Lg * Xh Xgh for every h in g, where Lg * denotes the differential of Lg ) on a lie group is a lie algebra under the lie bracket of vector. Any tangent vector at the identity of a lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v g Lg *. This identifies the tangent space teg at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a lie algebra, called the lie algebra of g, usually denoted by a fraktur. Thus the lie bracket on gdisplaystyle mathfrak g is given explicitly by v, w v,. This lie algebra gdisplaystyle mathfrak g is finite-dimensional and it has the same dimension as the manifold. The lie algebra of G determines g up to "local isomorphism where two lie groups are called locally isomorphic if they look the same near the identity element. Problems about lie groups are often solved by first solving the corresponding problem for the lie algebras, and the result for groups then usually follows easily.
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Before giving the abstract definition we give a few examples: The lie algebra of global the vector space r n is just R n with the lie bracket given by a,. (In general the lie bracket of a connected lie group is always 0 if and only if the lie group is abelian.) The lie algebra of the general linear group GL( writing n, c ) of invertible matrices is the vector space M( n, c ). If g is a closed subgroup of GL( n, c ) then the lie algebra of G can be thought of informally as the matrices m of M( n, r ) such that 1 ε m is in g, where ε is an infinitesimal positive number. For example, the orthogonal group O( n, r ) consists of matrices A with aa t 1, so the lie algebra consists of the matrices m with (1 ε m 1 ε m )T 1, which is equivalent to m m T 0 because. The preceding description can be made more rigorous as follows. The lie algebra of a closed subgroup g of GL( n, c may be computed as lie(G)XM(n;C)exp(tX)G for all t in R,displaystyle operatorname lie (G)Xin M(n;mathbb C )operatorname exp (tX)in Gtext for all ttext in mathbb mathbb r, 5 6 where exp( tX ) is defined using the matrix. It can then be shown that the lie algebra of g is a real vector space that is closed under the bracket operation, x,yxyyxdisplaystyle x, yxy-yx. 7 The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a lie group as a group of matrices, but not all lie groups can be represented in this. 8 to get around these problems we give the general definition of the lie algebra of a lie group (in 4 steps vector fields on any smooth manifold M can be thought of as derivations x of the ring of smooth functions on the manifold. If g is any group acting smoothly on the manifold m, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the lie bracket and therefore also forms a lie algebra. We apply this construction to the case when the manifold m is the underlying space of a lie group g, with g acting on g m by left translations Lg ( h ) .
In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure (compatible with its other structures). Related notions edit some examples of groups that are not lie groups (except in the trivial sense that any group can be viewed as a 0-dimensional lie group, with the discrete topology are: Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector. These are not lie groups as they are not finite-dimensional manifolds. Some totally disconnected groups, such as the galois group of an infinite extension of fields, or the additive group of the p -adic numbers. These are not lie groups because their underlying spaces are not real manifolds. (Some of these groups are " summary p -adic lie groups".) In general, only topological groups having similar local properties to r n for some positive integer n can be lie groups (of course they must also have a differentiable structure). Basic concepts edit The lie algebra associated with a lie group edit main article: lie grouplie algebra correspondence to every lie group we can associate a lie algebra whose underlying vector space is the tangent space of the lie group at the identity element and. Informally we can think of elements of the lie algebra as elements of the group that are " infinitesimally close" to the identity, and the lie bracket of the lie algebra is related to the commutator of two such infinitesimal elements.
Along with the a-b-c-d thesis series of simple lie best groups, the exceptional groups complete the list of simple lie groups. The symplectic group Sp(2 n, r ) consists of all 2 n 2 n matrices preserving a symplectic form on R. It is a connected lie group of dimension 2 n. Constructions edit There are several standard ways to form new lie groups from old ones: The product of two lie groups is a lie group. Any topologically closed subgroup of a lie group is a lie group. This is known as the Closed subgroup theorem or Cartan's theorem. The"ent of a lie group by a closed normal subgroup is a lie group. The universal cover of a connected lie group is a lie group. For example, the group r is the universal cover of the circle group.
There are (up to isomorphism) only two lie algebras of dimension two. The associated simply connected lie groups are R2displaystyle mathbb R 2 (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples." Additional examples edit The group SU(2) is the group of 22displaystyle 2times. Topologically, su(2) is the 3-sphere S3displaystyle S3 ; as a group, it may be identified with the group of unit quaternions. The heisenberg group is a connected nilpotent lie group of dimension 3, playing a key role in quantum mechanics. The lorentz group is a 6-dimensional lie group of linear isometries of the minkowski space. The poincaré group is a 10-dimensional lie group of affine isometries of the minkowski space. The exceptional lie groups of types G 2, f 4, e 6, e 7, e 8 have dimensions 14, 52, 78, 133, and 248.
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Related concepts edit a complex lie group is defined in the same way using complex manifolds rather than real ones (example: SL(2, c and similarly, using an alternate metric completion of q, one can define administrative a p -adic lie group over the p -adic numbers. Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, montgomery and Zippin showed that if g is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a lie group (see. If the underlying manifold is allowed to be infinite-dimensional (for example, a hilbert manifold then one arrives at the notion of an infinite-dimensional lie group. It is possible to define analogues of many lie groups over finite fields, and these give most of the examples of finite simple groups. The language of category theory provides a concise definition for lie groups: a lie group is a group object in the category of smooth manifolds.
This is important, because it allows generalization of the notion of a lie group to lie supergroups. More examples of lie groups edit see also: Table essay of lie groups and List of simple lie groups lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups and these give most of the more common examples of lie groups. Dimensions one and two edit The only connected lie groups with dimension one are the real line Rdisplaystyle mathbb r (with the group operation being addition) and the group S1displaystyle S1 of complex numbers with absolute value one (with the group operation being multiplication). The S1displaystyle S1 group is often denoted as U(1)displaystyle U(1), the group of 11displaystyle 1times 1 unitary matrices. In two dimensions, if we restrict attention to simply connected groups, then they are classified by their lie algebras.
The group Hdisplaystyle h winds repeatedly around the torus and forms a dense subgroup of T2displaystyle mathbb. The group Hdisplaystyle h can, however, be given a different topology, in which the distance between two points h1,h2Hdisplaystyle h_1,h_2in h is defined as the length of the shortest path in the group H joining h1displaystyle h_1 to h2displaystyle h_2. In this topology, hdisplaystyle h is identified homeomorphically with the real line by identifying each element with the number θdisplaystyle theta in the definition of Hdisplaystyle. With this topology, hdisplaystyle h is just the group of real numbers under addition and is therefore a lie group. The group Hdisplaystyle h is an example of a "lie subgroup" of a lie group that is not closed.
See the discussion below of lie subgroups in the section on basic concepts. Matrix lie groups edit let GL( n ; C ) denote the group of n n invertible matrices with entries. Any closed subgroup of GL( n, c ) is a lie group; 2 lie groups of this sort are called matrix lie groups. Since most of the interesting examples of lie groups can be realized as matrix lie groups, some textbooks restrict attention to this class, including those of Hall 3 and Rossmann. 4 Restricting attention to matrix lie groups simplifies the definition of the lie algebra and the exponential map. The following are standard examples of matrix lie groups. The special linear groups over r and c, sl( n, r ) and SL( n, c consisting of n n matrices with determinant one and entries in r or c the unitary groups and special unitary groups, U( n ) and SU( n consisting.
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Thus, the shredder group consists of matrices of the form A(ab01 a 0,bR. Displaystyle a 0 bin mathbb. Counterexample edit a portion of the group Hdisplaystyle h inside T2displaystyle mathbb. Small neighborhoods of the element hHdisplaystyle hin h are disconnected in the subset topology on Hdisplaystyle h we now present an example of a group with an uncountable number of elements that is not a lie group under a certain topology. The group given by hleftleft. Left(beginmatrixe2pi itheta 00 e2pi iatheta endmatrixright)righttheta in mathbb R rightsubset mathbb T 2leftleft. Left(beginmatrixe2pi itheta 00 e2pi iphi endmatrixright)righttheta, phi in mathbb R right, with aprqdisplaystyle ain mathbb P mathbb R setminus mathbb q a fixed irrational number, is a subgroup of the torus T2displaystyle mathbb T 2 that is not a lie group when given the subspace. 1 If we take any small neighborhood Udisplaystyle u of a point hdisplaystyle h in Hdisplaystyle h, for example, the portion of Hdisplaystyle h in Udisplaystyle u is disconnected.
First examples edit operatorname gl (2,mathbf r aad-bcneq 0right. This is a four-dimensional noncompact real lie group; it is an open subset of R4displaystyle mathbb. This group is disconnected ; it has two connected components corresponding to the ireland positive and negative values of the determinant. The rotation matrices form a subgroup of GL(2, r denoted by so(2, r ). It is a lie group in its own right: specifically, a one-dimensional compact connected lie group which is diffeomorphic to the circle. Using the rotation angle φdisplaystyle varphi as a parameter, this group can be parametrized as follows: operatorname so (2,mathbf R )leftbeginpmatrixcos varphi -sin varphi sin varphi cos varphi endpmatrix:varphi in mathbf R /2pi mathbf Z right. Addition of the angles corresponds to multiplication of the elements of SO(2, r and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps. The affine group in one dimension is a two-dimensional matrix lie group, consisting of 22displaystyle 2times 2 real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being.
rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of lie groups are especially important, and are studied in representation theory. In the 1940s1950s, Ellis Kolchin, armand Borel, and Claude Chevalley realised that many foundational results concerning lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of lie groups over adele rings ; p -adic lie groups play an important role, via their connections with Galois representations in number theory. Definitions and examples edit a real lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ:gggμ(x,y)xydisplaystyle mu :Gtimes Gto Gquad mu (x,y)xy means that μ is a smooth mapping of the product manifold g g into. These two requirements can be combined to the single requirement that the mapping (x,y)x1ydisplaystyle (x,y)mapsto x-1y be a smooth mapping of the product manifold into.
Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric night properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the euclidean space r 3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective. This idea later led to the notion of a g-structure, where g is a lie group of "local" symmetries of a manifold. Lie groups (and their associated lie algebras ) play a major role in modern physics, with the lie group typically playing the role of a symmetry of a physical system. Here, the representations of the lie group (or of its lie algebra ) are especially important. Representation theory is used extensively in particle physics.
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Not to be confused with, group of lie type. In mathematics, a, lie group (pronounced /li/ "lee is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Norwegian mathematician. Sophus lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations, in much the same way as permutation entry groups are used in, galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of lie's principal motivations. Contents, overview edit, lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its "infinitesimal group" and which has since become known as its lie algebra.