In mathematics, a Lie group (pronounced Lee) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For any rotation of the circle, there exists the same symmetry, and concatenation of such rotations makes them into the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics and physics. Lie groups were first found by studying matrix subgroups G {\displaystyle G} contained in GL n ( R ) {\displaystyle {\text{GL}}_{n}(\mathbb {R} )} or GL n ( C ) {\displaystyle {\text{GL}}_{n}(\mathbb {C} )} , the groups of n × n {\displaystyle n\times n} invertible matrices over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } . These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.
History Sophus Lie considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Thomas 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. 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. 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. 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. 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. 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; 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). 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. The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.
Overview
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. 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 properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R 3 {\displaystyle \mathbb {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 group. 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. Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of 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 1940s–1950s, 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 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 : G × G → G μ ( x , y ) = x y {\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy}
means that μ {\displaystyle \mu } is a smooth mapping of the product manifold G × G {\displaystyle G\times G} into G {\displaystyle G} . The two requirements can be combined to the single requirement that the mapping
( x , y ) ∈ G × G ↦ x − 1 y {\displaystyle (x,y)\in G\times G\mapsto x^{-1}y}
be a smooth mapping of the product manifold into G {\displaystyle G} .
First examples The 2 × 2 {\displaystyle 2\times 2} real invertible matrices form a group under multiplication, called general linear group of degree 2 and denoted by GL ( 2 , R ) {\displaystyle \operatorname {GL} (2,\mathbb {R} )} or by GL 2 ( R ) {\displaystyle \operatorname {GL} _{2}(\mathbb {R} )} : GL ( 2 , R ) = { A = ( a b c d ) : det A = a d − b c ≠ 0 } . {\displaystyle \operatorname {GL} (2,\mathbb {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\det A=ad-bc\neq 0\right\}.} This is a four-dimensional noncompact real Lie group; it is an open subset of R 4 {\displaystyle \mathbb {R} ^{4}} . This group is disconnected; it has two connected components containing matrices with positive and negative determinants, respectively. The rotation matrices form a subgroup of GL ( 2 , R ) {\displaystyle \operatorname {GL} (2,\mathbb {R} )} , denoted by SO ( 2 , R ) {\displaystyle \operatorname {SO} (2,\mathbb {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 φ {\displaystyle \varphi } as a parameter, this group can be parametrized as follows: SO ( 2 , R ) = { ( cos φ − sin φ sin φ cos φ ) : φ ∈ R / 2 π Z } . {\displaystyle \operatorname {SO} (2,\mathbb {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\varphi \in \mathbb {R} \ /\ 2\pi \mathbb {Z} \right\}.} Addition of the angles corresponds to multiplication of the elements of SO ( 2 , R ) {\displaystyle \operatorname {SO} (2,\mathbb {R} )} , and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps. The affine group of one dimension is a two-dimensional matrix Lie group, consisting of 2 × 2 {\displaystyle 2\times 2} real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1 {\displaystyle 1} . Thus, the group consists of matrices of the form A = ( a b 0 1 ) , a > 0 , b ∈ R . {\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .}
Non-example
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
H = { ( e 2 π i θ 0 0 e 2 π i a θ ) : θ ∈ R } ⊂ T 2 = { ( e 2 π i θ 0 0 e 2 π i ϕ ) : θ , ϕ ∈ R } , {\displaystyle H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right):\,\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right):\,\theta ,\phi \in \mathbb {R} \right\},}
with a ∈ R ∖ Q {\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} } a fixed irrational number, is a subgroup of the torus T 2 {\displaystyle \mathbb {T} ^{2}} that is not a Lie group when given the subspace topology. If we take any small neighborhood U {\displaystyle U} of a point h {\displaystyle h} in H {\displaystyle H} , for example, the portion of H {\displaystyle H} in U {\displaystyle U} is disconnected. The group H {\displaystyle H} winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of T 2 {\displaystyle \mathbb {T} ^{2}} .
The group H {\displaystyle H} can, however, be given a different topology, in which the distance between two points h 1 , h 2 ∈ H {\displaystyle h_{1},h_{2}\in H} is defined as the length of the shortest path in the group H {\displaystyle H} joining h 1 {\displaystyle h_{1}} to h 2 {\displaystyle h_{2}} . In this topology, H {\displaystyle H} is identified homeomorphically with the real line by identifying each element with the number θ {\displaystyle \theta } in the definition of H {\displaystyle H} . With this topology, H {\displaystyle H} is just the group of real numbers under addition and is therefore a Lie group. The group H {\displaystyle 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 Let GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} denote the group of n × n {\displaystyle n\times n} invertible matrices with entries in C {\displaystyle \mathbb {C} } . Any closed subgroup of GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} is a Lie group; Lie groups of this kind 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, Rossmann, and Stillwell. 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 {\displaystyle \mathbb {R} } and C {\displaystyle \mathbb {C} } , SL ( n , R ) {\displaystyle \operatorname {SL} (n,\mathbb {R} )} and SL ( n , C ) {\displaystyle \operatorname {SL} (n,\mathbb {C} )} , consisting of n × n {\displaystyle n\times n} matrices with determinant 1 {\displaystyle 1} and entries in R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} }
The unitary groups and special unitary groups, U ( n , C ) {\displaystyle \operatorname {U} (n,\mathbb {C} )} and SU ( n , C ) {\displaystyle \operatorname {SU} (n,\mathbb {C} )} , consisting of n × n {\displaystyle n\times n} complex matrices satisfying U ∗ = U − 1 {\displaystyle U^{*}=U^{-1}} (and also det ( U ) = 1 {\displaystyle \det(U)=1} in the case of SU ( n , C ) {\displaystyle \operatorname {SU} (n,\mathbb {C} )} ), where U ∗ {\displaystyle U^{*}} is the conjugate transpose of U {\displaystyle U}
The orthogonal groups and special orthogonal groups, O ( n , R ) {\displaystyle \operatorname {O} (n,\mathbb {R} )} and SO ( n , R ) {\displaystyle \operatorname {SO} (n,\mathbb {R} )} , consisting of n × n {\displaystyle n\times n} real matrices satisfying R T = R − 1 {\displaystyle R^{\mathrm {T} }=R^{-1}} (and also det ( R ) = 1 {\displaystyle \det(R)=1} in the case of SO ( n , R ) {\displaystyle \operatorname {SO} (n,\mathbb {R} )} ), where R T {\displaystyle R^{\mathrm {T} }} is the transpose of R {\displaystyle R}
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: SL ( 2 , C ) {\displaystyle \operatorname {SL} (2,\mathbb {C} )} ), and holomorphic maps. Similarly, using an alternate metric completion of Q {\displaystyle \mathbb {Q} } , one can define a p-adic Lie group over the p-adic numbers, a topological group which is also an analytic p-adic manifold, such that the group operations are analytic. In particular, 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. This categorical point of view leads also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement.
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 nor topological manifolds. Precisely, a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to a matrix Lie group, a closed subgroup of GL ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} 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 a closed Lie subgroup G ′ ⊂ GL ( n , C ) {\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )} such that G , G ′ {\displaystyle G,G'} share the same Lie algebra; thus, they are locally isomorphic. Hence, G {\displaystyle G} satisfies the above topological definition. Conversely, let G {\displaystyle G} be a topological group that is a Lie group in the above topological sense and choose a matrix Lie group G ′ {\displaystyle G'} that is locally isomorphic to G {\displaystyle G} around the respective identities. Then, by a version of the closed subgroup theorem, G ′ {\displaystyle 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; so the group operations are real-analytic and G {\displaystyle 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
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 R {\displaystyle \mathbb {R} } (with the group operation being addition) and the circle group S 1 {\displaystyle S^{1}} of complex numbers with absolute value one (with the group operation being multiplication). The S 1 {\displaystyle S^{1}} group is often denoted as U ( 1 ) {\displaystyle \operatorname {U} (1)} , the group of 1 × 1 {\displaystyle 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 R 2 {\displaystyle \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 2 × 2 {\displaystyle 2\times 2} unitary matrices with determinant 1 {\displaystyle 1} . Topologically, SU ( 2 ) {\displaystyle {\text{SU}}(2)} is the 3 {\displaystyle 3} -sphere S 3 {\displaystyle 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 3 {\displaystyle 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 Sp ( 2 n , R ) {\displaystyle {\text{Sp}}(2n,\mathbb {R} )} consists of all 2 n × 2 n {\displaystyle 2n\times 2n} matrices preserving the symplectic form Ω = ( 0 I n − I n 0 ) {\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}}} on R 2 n {\displaystyle \mathbb {R} ^{2n}} . It is a connected Lie group of dimension 2 n 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 R {\displaystyle \mathbb {R} } is the universal cover of the circle group S 1 {\displaystyle 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 {\displaystyle 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 X {\displaystyle X} to a Lie group G {\displaystyle G} , C ∞ ( 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 {\displaystyle p} -adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are " p {\displaystyle p} -adic Lie groups".) In general, only topological groups having similar local properties to R n {\displaystyle \mathbb {R} ^{n}} for some positive integer n {\displaystyle n} can be Lie groups (of course they must also have a differentiable structure).
Basic concepts
The Lie algebra associated with a Lie group
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,