Lie group

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In mathematics, a Lie group (named after Sophus Lie) is a real or complex differentiable manifold that is also a group such that the group operations (multiplication and inversion) are differentiable or analytical functions, as the case may be. Lie groups are important in mathematical analysis, physics, and geometry because they serve to describe the symmetry of analytic structures. They were introduced by Sophus Lie in 1870 to study symmetries of differential equations.

While the Euclidean space Rⁿ is a real Lie group (with ordinary addition of vectors as a group operation), more typical examples are some groups of invertible matrices (with matrix multiplication as an operation), for example the SO(3) group of all rotations in 3-dimensional space. See below for a more complete list of examples.

Types of Lie groups

Lie groups are classified with respect to their algebraic properties (simple, semi-simple, solvable, nilpotent, abelian), their connectedness (connected or not connected) and their compactness.

Homomorphisms and isomorphisms

If G and H are Lie groups (both real or complex), then a morphism of Lie groups f : G → H is a group homomorphism that is also a differentiable or analytic function. (It can be shown that it is equivalent to requiring only that it be a continuous function.) The composition of two such homomorphisms is again a homomorphism, and the class of all Lie groups (real or complex), together with these morphisms, forms a category. Two Lie groups are said to be isomorphic if there is a bijective homomorphism between them whose inverse is also a homomorphism.

The Lie algebra associated with a Lie group

To each Lie group, we can associate a Lie algebra that fully captures the local structure of the group. This is done as follows. A vector field in a Lie group G is said to be left invariant if it verifies the following. Define L_g(x) = gx, where g, x are in G. Then the vector field X is left invariant if, for any differentiable or analytic function f: G → F (here F is the field R or C), X(f L_g)=(X f)L_g , for all g in G.

The set of all vector fields in a differential variety is a Lie algebra over Fwhere the product is Lie's corchete. In a group of Lie, invariant vector fields on the left form a sub-algebra, the Lie algebra associated with G, usually denoted by a g Gothic (${displaystyle {mathfrak {g}}}}$). This Lie algebra g is finite-dimensional (has the same dimension as the variety G) which makes it susceptible to classification attempts. Sorting g, one can also get a closer approach to the Lie group G. The theory of representation of Lie's simple groups is the best and most important example.

Each homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras g and h. The association G|- > g is a functor.

Each vector v in g determines a flow line c: R → G whose derivative at every point is given by the left-invariant vector field corresponding to v

${displaystyle c^{prim }(t)=c(t)v}$

and which has the property

${displaystyle c(s+t)=c(s)c(t)}$

for all s and t. The operation on the right hand side is group multiplication on G. The formal similarity of this formula with the one valid for the exponential function justifies the definition

${displaystyle exp(v)=c(1)}$.

This exponential function is an application of the Lie algebra g on the Lie group G. This exponential function is a generalization of the exponential function for real numbers (since R is the Lie algebra of the Lie group of positive real numbers with the usual multiplication), for complex numbers (since that C is the Lie algebra of the Lie group of nonzero complex numbers with the usual multiplication) and for matrices (since M(n, R) with commutator is the Lie algebra of the Lie group GL(n, R) of all invertible matrices).

The exponential gives a diffeomorphism between a neighborhood of 0 in g and a neighborhood of e in G. Because the exponential function is surjective in some neighborhood N of e, it is common to call the elements of Lie algebra infinitesimal generators of the group G. In fact, the subgroup of G generated by N will be the entire group G (assuming G is connected).

The exponential function and the Lie algebra determine the local group structure of each connected Lie group, due to the Campbell-Hausdorff formula: there exists a neighborhood U of the zero element of g, such that for u, v in U we have

${displaystyle exp(u)exp(v)=exp(u+v+1/2[u,v]+1/12[u,v],v]-1/12[[[u,v],u]-dots)}$

where the omitted terms are known and involve Lie brackets of four or more elements. In case u and v commute, this formula reduces to the familiar power law exp(v) exp(u) = exp(u + v).

The global structure of a Lie group is not fully determined, in general, by its Lie algebra; see the table below for examples of various Lie groups sharing the same Lie algebra. We can however say that a connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding property.

If we require the Lie group to be simply connected, then the global structure is determined by its Lie algebra: for every Lie algebra g is finite dimensional over F there is a unique (unless isomorphic) simply connected Lie group G whose Lie algebra is g. On the other hand, each homomorphism between the Lie algebras comes from a unique homomorphism between the corresponding simply connected Lie groups.

List of some real Lie groups and their Lie algebras

group of Lie	description	Comments	Lie algebra	description	dim/R
${displaystyle mathbb {R} ^{n}$	euclid space with addendum	abelian, simply related, not compact	${displaystyle mathbb {R} ^{n}$	Lie's bracket is zero	n
${displaystyle mathbb {R} ^{times}}$	non-nul real numbers with multiplication	abelian, not related, not compact	${displaystyle mathbb {R} }$	Lie's bracket is zero	1
${displaystyle mathbb {R} ^{+}$	positive real numbers with multiplication	abelian, simply related, not compact	${displaystyle mathbb {R} }$	Lie's bracket is zero	1
${displaystyle {mbox{s}}{1}{mathbb {R} /mathbb {Z}} }$	complex numbers of absolute value 1 with multiplication	abelian, related, not simply related, compact	${displaystyle mathbb {R} }$	Lie's bracket is zero	1
${displaystyle mathbb {H} ^{times}}$	nil quaternions with multiplication	related, simply related, not compact	${displaystyle mathbb {H} }$	quaternions, with the Lie corchete given by the switch	4
${displaystyle {mbox{s}}{3},}$	module 1 quaternions with multiplication, 3-sphere	simply related, compact, simple and semi-simple, isomorph to ${displaystyle SU(2);}$ and ${displaystyle {mbox{spin}}(3)}$	${displaystyle mathbb {R} ^{3}$	3- real readers, with the Lie bracket the vector product; isomorph to the quaternions with real part zero, with the Lie bracket given by the switch also isomorph to the ${displaystyle {mathfrak {su}(2)}$ and ${displaystyle {mathfrak {so}(3)}$	3
${displaystyle GL(n,mathbb {R}}}}$	linear general group: real matrices n- by-n invertible	non-related, non-compact	${displaystyle M(n,mathbb {R}}}}$	real matrices n- by-n, with the Lie bracket given by the switcher	${displaystyle n^{2}}$
${displaystyle GL^{+}(n,mathbb {R}}}}$	real matrices n- by-n with positive determinant	non-compact	${displaystyle M(n,mathbb {R}}}}$	real matrices n- by-n, with the Lie bracket given by the switcher	${displaystyle n^{2}}$
${displaystyle SL(n,mathbb {R}}}}$	special linear group: real matrices n- by-n with determinant 1	related, not compact and simple if n▪	${displaystyle {mathfrak {sl}}(n,mathbb {R}}}$	real matrices n- by-n, with trace 0, with the Lie bracket given by the switch	n2-1
${displaystyle O(n,mathbb {R}}}}$	orthogonal group: real matrices n- by-n orthogonal	non-related, compact	${displaystyle {mathfrak {so}(3,mathbb {R}}}}}$	real matrices n- by-n, antisymmetrics, with the Lie bracket given by the switcher; ${displaystyle {mathfrak {so}(3,mathbb {R}}}}}$ isomorph to ${displaystyle {mathfrak {su}(2,mathbb {R}}}}}$ and ${displaystyle mathbb {R} ^{3}$ with the vector product	n(n-1)/2
${displaystyle SO(n,mathbb {R}}}}$	special orthogonal group: real matrices n- by-n orthogonals with determining 1	related, compact, not simply related n1, semisimple, yes n=3 or n ≥5 simple	${displaystyle {mathfrak {so}}(n,mathbb {R}}}$	real matrices n- by-n, antisymmetrics, with the Lie bracket given by the switch	n(n-1)/2
${displaystyle {mbox{spin}}(n,mathbb {R}}}}$	group of thorns	simply related, compact, semisimple, if n=3 or n ≥5 simple	${displaystyle {mathfrak {so}}(n,mathbb {R}}}$	real matrices n- by-n, antisymmetrics, with the Lie bracket given by the switch	n(n-1)/2
${displaystyle {mbox{Sp}}(2n,mathbb {R}}}$	real sympathetic group: real sympathetic matrices	not compact, simple and semisimple	${displaystyle {mathfrak {sp}(2n,mathbb {R}}}$	real matrices that satisfy JA + ATJ = 0 where J is the standard anti-simetric matrix	n(22)n + 1)
${displaystyle {mbox{Sp}}(n,mathbb {R}}}}$	simplectic group: unitary matrices n- by-n Quaternionics	compact, simply related, simple and semisimple if n▪	${displaystyle {mathfrak {sp}}(n)}$	square cuaternionic matrices A satisfying A = −A♪, with the Lie bracket given by the switcher	n(22)n + 1)
${displaystyle U(n),}$	unit group: complex arrays n- by-n unitarian	isomorph to S1 for n=1, not simply related to n.compact. Note: this No. is a group/algebra of Lie complex	${displaystyle {mathfrak {u}}(n)}$	complex matrices n- by-n, that they fulfill A =A♪, with the Lie bracket given by the switcher	n(n-1)/2	n2
${displaystyle SU(n),}$	special unit group: complex arrays n- by-n unit with determinant 1	simply related, compact and n ≥2, simple and semisimple. Note: this No. is a group/algebra of Lie complex	${displaystyle {mathfrak {su}(n)}$	complex matrices ${displaystyle ntimes n}$, that they fulfill A =A♪ with trace 0, with the Lie bracket given by the switch	n2-1

List of some complex Lie groups and their Lie algebras

List of some infinite-dimensional Lie groups

group of Lie	description	Comments	Lie algebra	description	dim/R
${displaystyle mathrm {Diff} ({mathcal {M}}}}}$	Difeomorphisms of ${displaystyle {mathcal {M}}}$	not abeliano utility in general relativity	${displaystyle mathrm {Vec} ({mathcal {M}}}}}$	Vector fields on ${displaystyle {mathcal {M}}}$	${displaystyle aleph _{1}}$
${displaystyle mathrm {SDiff} ({mathcal {M}}}}}$	Difeomorphisms of ${displaystyle {mathcal {M}}}$ that keep the volume	not abeliano usefulness in hydrodynamics	${displaystyle mathrm {SVec} ({mathcal {M}}}}}$	Vector fields on ${displaystyle {mathcal {M}}}$ with null divergence	${displaystyle aleph _{1}}$