Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.
Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group formula_1). Lie groups are widely used in many parts of modern mathematics and physics.
Lie groups were first found by studying matrix subgroups formula_2 contained in formula_3 or formula_4, the groups of formula_5 invertible matrices over formula_6 or formula_7. 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.
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 R3, 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
means that "μ" is a smooth mapping of the product manifold into "G". The two requirements can be combined to the single requirement that the mapping
be a smooth mapping of the product manifold into "G".
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 
with formula_17 a "fixed" irrational number, is a subgroup of the torus formula_18 that is not a Lie group when given the subspace topology. If we take any small neighborhood formula_19 of a point formula_20 in formula_21, for example, the portion of formula_21 in formula_19 is disconnected. The group formula_21 winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of formula_18.
The group formula_21 can, however, be given a different topology, in which the distance between two points formula_27 is defined as the length of the shortest path "in the group " formula_21 joining formula_29 to formula_30. In this topology, formula_21 is identified homeomorphically with the real line by identifying each element with the number formula_32 in the definition of formula_21. With this topology, formula_21 is just the group of real numbers under addition and is therefore a Lie group.
The group formula_21 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 formula_36 denote the group of formula_5 invertible matrices with entries in formula_7. Any closed subgroup of formula_36 is a Lie group; 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 and Rossmann. 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.
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: formula_59), and similarly, using an alternate metric completion of formula_60, 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. First, we define an immersely linear Lie group to be a subgroup "G" of the general linear group formula_36 such that
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:
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 formula_6 (with the group operation being addition) and the circle group formula_70 of complex numbers with absolute value one (with the group operation being multiplication). The formula_70 group is often denoted as formula_72, the group of formula_73 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 formula_74 (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples."
Constructions.
There are several standard ways to form new Lie groups from old ones:
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:
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 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 it is not even obvious that the Lie algebra is independent of the representation we use. To get around these problems we give 
the general definition of the Lie algebra of a Lie group (in 4 steps):
This Lie algebra formula_93 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
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.
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 formula_95 be a Lie group homomorphism and let formula_96 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 formula_96 is a map between the corresponding Lie algebras:
One can show that formula_96 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.
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, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.
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. (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.
A Lie group formula_2 is said to be simply connected if every loop in formula_2 can be shrunk continuously to a point in formula_2. This notion is important because of the following result that has simple connectedness as a hypothesis:
Lie's third theorem says that every finite-dimensional 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 formula_113, and the compact symplectic group Sp(n).
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.
The exponential map from the Lie algebra formula_114 of the general linear group formula_115 to formula_115 is defined by the matrix exponential, given by the usual power series:
for matrices formula_87. If formula_2 is a closed subgroup of formula_115, then the exponential map takes the Lie algebra of formula_2 into formula_2; thus, we have an exponential map for all matrix groups. Every element of formula_2 that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.
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 formula_87 in the Lie algebra formula_93 of formula_2 (i.e., the tangent space to formula_2 at the identity), one proves that there is a unique one-parameter subgroup formula_128 such that formula_129. Saying that formula_130 is a one-parameter subgroup means simply that formula_130 is a smooth map into formula_2 and that 
for all formula_134 and formula_135. The operation on the right hand side is the group multiplication in formula_2. The formal similarity of this formula with the one valid for the exponential function justifies the definition
This is called the exponential map, and it maps the Lie algebra formula_93 into the Lie group formula_2. It provides a diffeomorphism between a neighborhood of 0 in formula_93 and a neighborhood of formula_141 in formula_2. This exponential map is a generalization of the exponential function for real numbers (because formula_6 is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because formula_7 is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because formula_145 with the regular commutator is the Lie algebra of the Lie group formula_146 of all invertible matrices).
Because the exponential map is surjective on some neighbourhood formula_147 of formula_141, it is common to call elements of the Lie algebra infinitesimal generators of the group formula_2. The subgroup of formula_2 generated by formula_147 is the identity component of formula_2.
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 formula_19 of the zero element of formula_93, such that for formula_155 we have
where the omitted terms are known and involve Lie brackets of four or more elements. In case formula_87 and formula_158 commute, this formula reduces to the familiar exponential law formula_159
The exponential map relates Lie group homomorphisms. That is, if formula_160 is a Lie group homomorphism and formula_161 the induced map on the corresponding Lie algebras, then for all formula_162 we have
In other words, the following diagram commutes,
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 "C"∞ Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.
Lie subgroup.
A Lie subgroup formula_21 of a Lie group formula_2 is a Lie group that is a subset of formula_2 and such that the inclusion map from formula_21 to formula_2 is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of formula_2 admits a unique smooth structure which makes it an embedded Lie subgroup of formula_2—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.
Examples of non-closed subgroups are plentiful; for example take formula_2 to be a torus of dimension 2 or greater, and let formula_21 be a one-parameter subgroup of "irrational slope", i.e. one that winds around in "G". Then there is a Lie group homomorphism formula_173 with formula_174. The closure of formula_21 will be a sub-torus in formula_2.
The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group formula_2 and the subalgebras of the Lie algebra of formula_2. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of formula_2 which determines which subalgebras correspond to closed subgroups.
Representations.
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, formula_180. Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator formula_181 commutes with the action of SO(3) on the wave function formula_182. (One important example of such a system is the Hydrogen atom.) This assumption does not necessarily mean that the solutions formula_182 are rotationally invariant functions. Rather, it means that the "space" of solutions to formula_180 is invariant under rotations (for each fixed value of formula_185). This space, therefore, constitutes a representation of SO(3). These representations have been 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. 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.
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. 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" A"n", B"n", C"n" and D"n", 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. 
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 
so that we have a sequence of normal subgroups
Then
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.
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: