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Matrix Lie groups Let {\displaystyle \operatorname {GL} (n,\mathbb {C} )}\operatorname {GL}(n,{\mathbb {C}}) denote the group of {\displaystyle n\times n}n\times n invertible matrices with entries in {\displaystyle \mathbb {C} }\mathbb {C} . Any closed subgroup of {\displaystyle \operatorname {GL} (n,\mathbb {C} )}\operatorname {GL}(n,{\mathbb {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 {\displaystyle \mathbb {R} }\mathbb {R} and {\displaystyle \mathbb {C} }\mathbb {C} , {\displaystyle \operatorname {SL} (n,\mathbb {R} )}{\displaystyle \operatorname {SL} (n,\mathbb {R} )} and {\displaystyle \operatorname {SL} (n,\mathbb {C} )}{\displaystyle \operatorname {SL} (n,\mathbb {C} )}, consisting of {\displaystyle n\times n}n\times n matrices with determinant one and entries in {\displaystyle \mathbb {R} }\mathbb {R} or {\displaystyle \mathbb {C} }\mathbb {C} The unitary groups and special unitary groups, {\displaystyle {\text{U}}(n)}{\displaystyle {\text{U}}(n)} and {\displaystyle {\text{SU}}(n)}{\displaystyle {\text{SU}}(n)}, consisting of {\displaystyle n\times n}n\times n complex matrices satisfying {\displaystyle U^{*}=U^{-1}}{\displaystyle U^{*}=U^{-1}} (and also {\displaystyle \det(U)=1}{\displaystyle \det(U)=1} in the case of {\displaystyle {\text{SU}}(n)}{\displaystyle {\text{SU}}(n)}) The orthogonal groups and special orthogonal groups, {\displaystyle {\text{O}}(n)}{\displaystyle {\text{O}}(n)} and {\displaystyle {\text{SO}}(n)}{\displaystyle {\text{SO}}(n)}, consisting of {\displaystyle n\times n}n\times n real matrices satisfying {\displaystyle R^{\mathrm {T} }=R^{-1}}{\displaystyle R^{\mathrm {T} }=R^{-1}} (and also {\displaystyle \det(R)=1}\det(R)=1 in the case of {\displaystyle {\text{SO}}(n)}{\displaystyle {\text{SO}}(n)}) All of the preceding examples fall under the heading of the classical groups.



Related concepts A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: {\displaystyle \operatorname {SL} (2,\mathbb {C} )}\operatorname {SL} (2,\mathbb {C} )), and similarly, using an alternate metric completion of {\displaystyle \mathbb {Q} }\mathbb {Q} , one can define a p-adic Lie group over the p-adic numbers, a topological group in which each point has a p-adic neighborhood. 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 also Hilbert–Smith conjecture). 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. Topological definition A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds.[5] First, we define an immersely linear Lie group to be a subgroup G of the general linear group {\displaystyle \operatorname {GL} (n,\mathbb {C} )}\operatorname {GL}(n,{\mathbb {C}}) such that for some neighborhood V of the identity element e in G, the topology on V is the subspace topology of {\displaystyle \operatorname {GL} (n,\mathbb {C} )}\operatorname {GL}(n,{\mathbb {C}}) and V is closed in {\displaystyle \operatorname {GL} (n,\mathbb {C} )}\operatorname {GL}(n,{\mathbb {C}}). G has at most countably many connected components. (For example, a closed subgroup of {\displaystyle \operatorname {GL} (n,\mathbb {C} )}\operatorname {GL}(n,{\mathbb {C}}); that is, a matrix Lie group satisfies the above conditions.) Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows: Given a Lie group G in the usual manifold sense, the Lie group–Lie algebra correspondence (or a version of Lie's third theorem) constructs an immersed Lie subgroup {\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )}{\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )} such that {\displaystyle G,G'}{\displaystyle G,G'} share the same Lie algebra; thus, they are locally isomorphic. Hence, G satisfies the above topological definition. Conversely, let G be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group {\displaystyle G'}G' that is locally isomorphic to G. Then, by a version of the closed subgroup theorem, {\displaystyle G'}G' is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; [6] so the group operations are real-analytic and G itself is a real-analytic manifold. The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group. More examples of Lie groups See also: Table 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 The only connected Lie groups with dimension one are the real line {\displaystyle \mathbb {R} }\mathbb {R} (with the group operation being addition) and the circle group {\displaystyle S^{1}}S^{1} of complex numbers with absolute value one (with the group operation being multiplication). The {\displaystyle S^{1}}S^{1} group is often denoted as {\displaystyle U(1)}U(1), the group of {\displaystyle 1\times 1}1\times 1 unitary matrices. In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are {\displaystyle \mathbb {R} ^{2}}\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 The group SU(2) is the group of {\displaystyle 2\times 2}2\times 2 unitary matrices with determinant {\displaystyle 1}1. Topologically, {\displaystyle {\text{SU}}(2)}{\displaystyle {\text{SU}}(2)} is the {\displaystyle 3}3-sphere {\displaystyle S^{3}}S^{3}; as a group, it may be identified with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension {\displaystyle 3}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 G2, F4, E6, E7, E8 have dimensions 14, 52, 78, 133, and 248. Along with the A-B-C-D series of simple Lie groups, the exceptional groups complete the list of simple Lie groups. The symplectic group {\displaystyle {\text{Sp}}(2n,\mathbb {R} )}{\displaystyle {\text{Sp}}(2n,\mathbb {R} )} consists of all {\displaystyle 2n\times 2n}{\displaystyle 2n\times 2n} matrices preserving a symplectic form on {\displaystyle \mathbb {R} ^{2n}}{\mathbb {R}}^{{2n}}. It is a connected Lie group of dimension {\displaystyle 2n^{2}+n}{\displaystyle 2n^{2}+n}. Constructions 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 quotient 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 {\displaystyle \mathbb {R} }\mathbb {R} is the universal cover of the circle group {\displaystyle S^{1}}S^{1}. 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 Some examples of groups that are not Lie groups (except in the trivial sense that any group having at most countably many elements 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 space, or the space of smooth functions from a manifold {\displaystyle X}X to a Lie group {\displaystyle G}G, {\displaystyle C^{\infty }(X,G)}{\displaystyle C^{\infty }(X,G)}. 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 "p-adic Lie groups".) In general, only topological groups having similar local properties to Rn for some positive integer n can be Lie groups (of course they must also have a differentiable structure). Basic concepts The Lie algebra associated with a Lie group Main article: Lie group–Lie 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 which completely captures the local structure of the group. 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. Before giving the abstract definition we give a few examples: The Lie algebra of the vector space Rn is just Rn with the Lie bracket given by [A, B] = 0. (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(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by [A, B] = AB - BA. 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 + em is in G, where e is an infinitesimal positive number with e2 = 0 (of course, no such real number e exists). For example, the orthogonal group O(n, R) consists of matrices A with AAT = 1, so the Lie algebra consists of the matrices m with (1 + em)(1 + em)T = 1, which is equivalent to m + mT = 0 because e2 = 0. 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 {\displaystyle \operatorname {Lie} (G)=\{X\in M(n;\mathbb {C} )|\operatorname {exp} (tX)\in G{\text{ for all }}t{\text{ in }}\mathbb {\mathbb {R} } \},}{\displaystyle \operatorname {Lie} (G)=\{X\in M(n;\mathbb {C} )|\operatorname {exp} (tX)\in G{\text{ for all }}t{\text{ in }}\mathbb {\mathbb {R} } \},}[7][3] where exp(tX) is defined using the matrix exponential. It can then be shown that the Lie algebra of G is a real vector space that is closed under the bracket operation, {\displaystyle [X,Y]=XY-YX}{\displaystyle [X,Y]=XY-YX}.[8] 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 way, and even it is not obvious that the Lie algebra is independent of the representation we use.[9] 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, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY - YX, because the Lie bracket of any two derivations is a derivation. 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) = gh. 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 fields. 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*v. 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 {\displaystyle {\mathfrak {g}}.}{\mathfrak {g}}. Thus the Lie bracket on {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} is given explicitly by [v, w] = [v^, w^]e. This Lie algebra {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} is finite-dimensional and it has the same dimension as the manifold G. 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. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras. We could also define a Lie algebra structure on Te using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as -1 on the tangent space Te. The Lie algebra structure on Te can also be described as follows: the commutator operation (x, y) ? xyx-1y-1 on G × G sends (e, e) to e, so its derivative yields a bilinear operation on TeG. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields. Homomorphisms and isomorphisms If G and H are Lie groups, then a Lie group homomorphism f : G ? H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic.[10] The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let {\displaystyle \phi \colon G\to H}\phi \colon G\to H be a Lie group homomorphism and let {\displaystyle \phi _{*}}\phi _{*} be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements then {\displaystyle \phi _{*}}\phi _{*} is a map between the corresponding Lie algebras: {\displaystyle \phi _{*}\colon {\mathfrak {g}}\to {\mathfrak {h}}}\phi _{*}\colon {\mathfrak {g}}\to {\mathfrak {h}} One can show that {\displaystyle \phi _{*}}\phi _{*} is actually a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Lie group versus Lie algebra isomorphisms Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.[11] On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups SU(2) and SO(3). These two groups have isomorphic Lie algebras,[12] but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.[13] On the other hand, if we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic.[14] (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups. Simply connected Lie groups See also: Lie group–Lie algebra correspondence and Fundamental group § Lie groups A Lie group {\displaystyle G}G is said to be simply connected if every loop in {\displaystyle G}G can be shrunk continuously to a point in {\displaystyle G}G. This notion is important because of the following result that has simple connectedness as a hypothesis: Theorem[15]: Suppose {\displaystyle G}G and {\displaystyle H}H are Lie groups with Lie algebras {\displaystyle {\mathfrak {g}}}{\mathfrak {g}} and {\displaystyle {\mathfrak {h}}}{\mathfrak {h}} and that {\displaystyle f:{\mathfrak {g}}\rightarrow {\mathfrak {h}}}{\displaystyle f:{\mathfrak {g}}\rightarrow {\mathfrak {h}}} is a Lie algebra homomorphism. If {\displaystyle G}G is simply connected, then there is a unique Lie group homomorphism {\displaystyle \phi :G\rightarrow H}{\displaystyle \phi :G\rightarrow H} such that {\displaystyle \phi _{*}=f}{\displaystyle \phi _{*}=f}, where {\displaystyle \phi _{*}}{\displaystyle \phi _{*}} is the differential of {\displaystyle \phi }\phi at the identity. Lie's third theorem says that every finite-dimensional


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