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Table of Lie groups
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Lie groups and Lie algebras
Classical groups
Simple Lie groups
Classical
Exceptional
Other Lie groups
Lie algebras
Semisimple Lie algebra
Representation theory
Lie groups in physics
Scientists

This article gives a table of some common Lie groups and their associated Lie algebras.

The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple).

For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics.

Real Lie groups and their algebras

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Column legend

Lie group Description Cpt UC Remarks Lie algebra dim/R
Rn Euclidean space with addition N 0 0 abelian Rn n
R× nonzero real numbers with multiplication N Z2 – abelian R 1
R+ positive real numbers with multiplication N 0 0 abelian R 1
S1 = U(1) the circle group: complex numbers of absolute value 1 with multiplication; Y 0 Z R abelian, isomorphic to SO(2), Spin(2), and R/Z R 1
Aff(1) invertible affine transformations from R to R. N Z2 – solvable, semidirect product of R+ and R× 2
H× non-zero quaternions with multiplication N 0 0 H 4
S3 = Sp(1) quaternions of absolute value 1 with multiplication; topologically a 3-sphere Y 0 0 isomorphic to SU(2) and to Spin(3); double cover of SO(3) Im(H) 3
GL(n,R) general linear group: invertible n×n real matrices N Z2 – M(n,R) n2
GL+(n,R) n×n real matrices with positive determinant N 0 Z  n=2
Z2 n>2 		GL+(1,R) is isomorphic to R+ and is simply connected M(n,R) n2
SL(n,R) special linear group: real matrices with determinant 1 N 0 Z  n=2
Z2 n>2 		SL(1,R) is a single point and therefore compact and simply connected sl(n,R) n2−1
SL(2,R) Orientation-preserving isometries of the Poincaré half-plane, isomorphic to SU(1,1), isomorphic to Sp(2,R). N 0 Z The universal cover has no finite-dimensional faithful representations. sl(2,R) 3
O(n) orthogonal group: real orthogonal matrices Y Z2 – The symmetry group of the sphere (n=3) or hypersphere. so(n) n(n−1)/2
SO(n) special orthogonal group: real orthogonal matrices with determinant 1 Y 0 Z  n=2
Z2 n>2 		Spin(n)
n>2 		SO(1) is a single point and SO(2) is isomorphic to the circle group, SO(3) is the rotation group of the sphere. so(n) n(n−1)/2
SE(n) special euclidean group: group of rigid body motions in n-dimensional space. N 0 se(n) n + n(n−1)/2
Spin(n) spin group: double cover of SO(n) Y 0 n>1 0 n>2 Spin(1) is isomorphic to Z2 and not connected; Spin(2) is isomorphic to the circle group and not simply connected so(n) n(n−1)/2
Sp(2n,R) symplectic group: real symplectic matrices N 0 Z sp(2n,R) n(2n+1)
Sp(n) compact symplectic group: quaternionic n×n unitary matrices Y 0 0 sp(n) n(2n+1)
Mp(2n,R) metaplectic group: double cover of real symplectic group Sp(2n,R) Y 0 Z Mp(2,R) is a Lie group that is not algebraic sp(2n,R) n(2n+1)
U(n) unitary group: complex n×n unitary matrices Y 0 Z R×SU(n) For n=1: isomorphic to S1. Note: this is not a complex Lie group/algebra u(n) n2
SU(n) special unitary group: complex n×n unitary matrices with determinant 1 Y 0 0 Note: this is not a complex Lie group/algebra su(n) n2−1

Real Lie algebras

[edit]
Main article: Classification of low-dimensional real Lie algebras
Lie algebra Description Simple? Semi-simple? Remarks dim/R
R the real numbers, the Lie bracket is zero 1
Rn the Lie bracket is zero n
R3 the Lie bracket is the cross product Yes Yes 3
H quaternions, with Lie bracket the commutator 4
Im(H) quaternions with zero real part, with Lie bracket the commutator; isomorphic to real 3-vectors,

with Lie bracket the cross product; also isomorphic to su(2) and to so(3,R)

Yes Yes 3
M(n,R) n×n matrices, with Lie bracket the commutator n2
sl(n,R) square matrices with trace 0, with Lie bracket the commutator Yes Yes n2−1
so(n) skew-symmetric square real matrices, with Lie bracket the commutator. Yes, except n=4 Yes Exception: so(4) is semi-simple,

but not simple.

n(n−1)/2
sp(2n,R) real matrices that satisfy JA + ATJ = 0 where J is the standard skew-symmetric matrix Yes Yes n(2n+1)
sp(n) square quaternionic matrices A satisfying A = −A∗, with Lie bracket the commutator Yes Yes n(2n+1)
u(n) square complex matrices A satisfying A = −A∗, with Lie bracket the commutator Note: this is not a complex Lie algebra n2
su(n)
n≥2 		square complex matrices A with trace 0 satisfying A = −A∗, with Lie bracket the commutator Yes Yes Note: this is not a complex Lie algebra n2−1

Complex Lie groups and their algebras

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Main article: Complex Lie group

Note that a "complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The dimensions in the table below are dimensions over C. Note that every complex Lie group/algebra can also be viewed as a real Lie group/algebra of twice the dimension.

Lie group Description Cpt UC Remarks Lie algebra dim/C
Cn group operation is addition N 0 0 abelian Cn n
C× nonzero complex numbers with multiplication N 0 Z abelian C 1
GL(n,C) general linear group: invertible n×n complex matrices N 0 Z For n=1: isomorphic to C× M(n,C) n2
SL(n,C) special linear group: complex matrices with determinant

1

N 0 0 for n=1 this is a single point and thus compact. sl(n,C) n2−1
SL(2,C) Special case of SL(n,C) for n=2 N 0 0 Isomorphic to Spin(3,C), isomorphic to Sp(2,C) sl(2,C) 3
PSL(2,C) Projective special linear group N 0 Z2 SL(2,C) Isomorphic to the Möbius group, isomorphic to the restricted Lorentz group SO+(3,1,R), isomorphic to SO(3,C). sl(2,C) 3
O(n,C) orthogonal group: complex orthogonal matrices N Z2 – finite for n=1 so(n,C) n(n−1)/2
SO(n,C) special orthogonal group: complex orthogonal matrices with determinant 1 N 0 Z  n=2
Z2 n>2 		SO(2,C) is abelian and isomorphic to C×; nonabelian for n>2. SO(1,C) is a single point and thus compact and simply connected so(n,C) n(n−1)/2
Sp(2n,C) symplectic group: complex symplectic matrices N 0 0 sp(2n,C) n(2n+1)

Complex Lie algebras

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Main article: Complex Lie algebra

The dimensions given are dimensions over C. Note that every complex Lie algebra can also be viewed as a real Lie algebra of twice the dimension.

Lie algebra Description Simple? Semi-simple? Remarks dim/C
C the complex numbers 1
Cn the Lie bracket is zero n
M(n,C) n×n matrices with Lie bracket the commutator n2
sl(n,C) square matrices with trace 0 with Lie bracket

the commutator

Yes Yes n2−1
sl(2,C) Special case of sl(n,C) with n=2 Yes Yes isomorphic to su(2) C 3
so(n,C) skew-symmetric square complex matrices with Lie bracket

the commutator

Yes, except n=4 Yes Exception: so(4,C) is semi-simple,

but not simple.

n(n−1)/2
sp(2n,C) complex matrices that satisfy JA + ATJ = 0

where J is the standard skew-symmetric matrix

Yes Yes n(2n+1)

The Lie algebra of affine transformations of dimension two, in fact, exist for any field. An instance has already been listed in the first table for real Lie algebras.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Table of Lie groups

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from Grokipedia
A table of Lie groups is a concise tabular summary of the of simple Lie groups, which are connected Lie groups whose Lie algebras are simple and serve as the fundamental building blocks for studying semisimple Lie groups in , , and physics. This , originally developed by in the 1880s and refined by in the 1890s through the analysis of root systems and Cartan subalgebras, reveals that all complex simple Lie algebras—and thus their associated simply connected Lie groups—fall into four infinite classical series and five exceptional cases, uniquely determined up to isomorphism by their Dynkin diagrams. The classical series correspond to matrix groups preserving specific structures: the A_n series (for n ≥ 1) is associated with the SL(n+1, ℂ) and its sl(n+1, ℂ), which has dimension (n+1)^2 - 1 and describes symmetries of vector spaces; the B_n series (n ≥ 2) with the odd SO(2n+1, ℂ) and so(2n+1, ℂ), dimension n(2n+1), preserving quadratic forms in odd dimensions; the C_n series (n ≥ 3) with the Sp(2n, ℂ) and sp(2n, ℂ), dimension n(2n+1), preserving skew-symmetric bilinear forms; and the D_n series (n ≥ 4) with the even SO(2n, ℂ) and so(2n, ℂ), dimension n(2n-1), for even-dimensional quadratic forms. These series cover the vast majority of low-dimensional examples and underpin applications in , such as the rotation group SO(3) from B_1 or D_1 (isomorphic) and the SU(2) from A_1. In addition to these infinite families, there are five exceptional simple Lie groups, which do not fit into the classical patterns but arise from more intricate root systems: G_2 (dimension 14), F_4 (dimension 52), E_6 (dimension 78), E_7 (dimension 133), and E_8 (dimension 248). These exceptional groups, whose Dynkin diagrams feature triple bonds or longer chains, appear in advanced contexts like (e.g., E_8 in heterotic models) and the connections via the correspondence, and their representations are cataloged in resources like the Atlas of Lie Groups and Representations. For real Lie groups, the is more involved, involving multiple real forms for each complex type (e.g., compact, split, or quaternionic forms), often presented in extended tables that include Vogan diagrams and fundamental groups for low-rank cases. Such tables, as compiled in mathematical databases, facilitate computations of representations, character tables, and branching rules, essential for applications in and automorphic forms.

Lie Algebras

Low-Dimensional Lie Algebras

Lie algebras of dimension 1 over either the real numbers R\mathbb{R} or the complex numbers C\mathbb{C} are unique up to isomorphism and abelian, consisting of a 1-dimensional vector space with trivial Lie bracket [X,Y]=0[X, Y] = 0 for all elements X,YX, Y. In dimension 2, the classification is the same over R\mathbb{R} and C\mathbb{C}. There are two isomorphism classes: the abelian Lie algebra, with basis {e1,e2}\{e_1, e_2\} and all brackets zero; and the non-abelian (affine) Lie algebra, with basis {e1,e2}\{e_1, e_2\} and bracket relations [e1,e2]=e1[e_1, e_2] = e_1, [e1,e1]=[e2,e2]=0[e_1, e_1] = [e_2, e_2] = 0. The latter is solvable but not nilpotent. For dimension 3 over R\mathbb{R}, the classification, known as the Bianchi classification, consists of nine types (I through IX), all of which are either solvable or simple. The simple cases are types VIII and IX, corresponding to sl(2,R)\mathfrak{sl}(2, \mathbb{R}) and so(3)\mathfrak{so}(3), respectively. The solvable types include the abelian (type I), the Heisenberg algebra (type II, nilpotent), and others such as type VI (related to the Lorentz algebra in some realizations). The explicit structure is given in the following table, using a basis {e1,e2,e3}\{e_1, e_2, e_3\} for each algebra.
TypeName/DescriptionBracket RelationsProperties
IAbelian[e1,e2]=[e2,e3]=[e3,e1]=0[e_1, e_2] = [e_2, e_3] = [e_3, e_1] = 0Solvable, nilpotent
IIHeisenberg[e1,e2]=0[e_1, e_2] = 0, [e2,e3]=e1[e_2, e_3] = e_1, [e3,e1]=0[e_3, e_1] = 0Solvable, nilpotent
III(Special case of VI with h=−1h = -1)[e1,e2]=0[e_1, e_2] = 0, [e2,e3]=e1+e2[e_2, e_3] = e_1 + e_2, [e3,e1]=−e2[e_3, e_1] = -e_2Solvable
IVSolvable[e1,e2]=0[e_1, e_2] = 0, [e2,e3]=e1−e2[e_2, e_3] = e_1 - e_2, [e3,e1]=e1[e_3, e_1] = e_1Solvable
VAffine-like[e1,e2]=0[e_1, e_2] = 0, [e2,e3]=e2[e_2, e_3] = e_2, [e3,e1]=e1[e_3, e_1] = e_1Solvable
VIh_h (h≤0h \leq 0, h≠−1h \neq -1)Solvable family[e1,e2]=0[e_1, e_2] = 0, [e2,e3]=e1−he2[e_2, e_3] = e_1 - h e_2, [e3,e1]=he1−e2[e_3, e_1] = h e_1 - e_2Solvable; includes Lorentz-like for certain hh
VIIh_h (h≥0h \geq 0)Solvable family[e1,e2]=0[e_1, e_2] = 0, [e2,e3]=e1−he2[e_2, e_3] = e_1 - h e_2, [e3,e1]=he1+e2[e_3, e_1] = h e_1 + e_2Solvable
VIIIsl(2,R)\mathfrak{sl}(2, \mathbb{R})[e1,e2]=−e3[e_1, e_2] = -e_3, [e2,e3]=e1[e_2, e_3] = e_1, [e3,e1]=e2[e_3, e_1] = e_2Simple
IXso(3)\mathfrak{so}(3)[e1,e2]=e3[e_1, e_2] = e_3, [e2,e3]=e1[e_2, e_3] = e_1, [e3,e1]=e2[e_3, e_1] = e_2Simple
Isomorphisms within families occur for specific parameter values, such as h=h′h = h' or h=1/h′h = 1/h' in related forms. Over C\mathbb{C}, the dimension-3 classification is simpler due to the , with no splitting into separate real-parameter families beyond a single parameterized solvable class. Up to , the classes are the abelian algebra (all brackets zero); the Heisenberg algebra (as in real type II); the decomposable solvable algebra with basis {x,y,z}\{x, y, z\} and [x,y]=x[x, y] = x (others zero); the indecomposable solvable algebra with Jordan-block structure, basis {x,y,z}\{x, y, z\} and [x,z]=x+y[x, z] = x + y, [y,z]=y[y, z] = y (others zero); the parameterized solvable family with basis {x,y,z}\{x, y, z\} and [x,z]=x[x, z] = x, [y,z]=αy[y, z] = \alpha y (α∈C∖{0}\alpha \in \mathbb{C} \setminus \{0\}, others zero), where classes differ for distinct α\alpha; and the simple Lie algebra sl(2,C)\mathfrak{sl}(2, \mathbb{C}) (isomorphic to so(3,C)\mathfrak{so}(3, \mathbb{C})), with basis {h,x,y}\{h, x, y\} and [h,x]=2x[h, x] = 2x, [h,y]=−2y[h, y] = -2y, [x,y]=h[x, y] = h. All dimension-3 complex Lie algebras are thus either solvable or simple.

Solvable and Nilpotent Lie Algebras

A Lie algebra g\mathfrak{g} over a field of characteristic zero is called if there exists a positive rr such that the rr-th term of its derived series vanishes, where the derived series is defined recursively by g(0)=g\mathfrak{g}^{(0)} = \mathfrak{g} and g(k+1)=[g(k),g(k)]\mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}] for k≥0k \geq 0. Every finite-dimensional Lie algebra g\mathfrak{g} over an of characteristic zero admits a g=s⋉rad(g)\mathfrak{g} = \mathfrak{s} \ltimes \mathrm{rad}(\mathfrak{g}), where s\mathfrak{s} is a semisimple Levi subalgebra and rad(g)\mathrm{rad}(\mathfrak{g}) is the solvable radical, the maximal solvable ideal of g\mathfrak{g}. A Lie algebra g\mathfrak{g} is nilpotent if its lower central series terminates at the zero ideal, defined by g0=g\mathfrak{g}_0 = \mathfrak{g} and gk+1=[g,gk]\mathfrak{g}_{k+1} = [\mathfrak{g}, \mathfrak{g}_k] for k≥0k \geq 0. Every nilpotent Lie algebra is solvable, but the converse does not hold. Engel's theorem states that a finite-dimensional g\mathfrak{g} over an of characteristic zero is if and only if the adjoint operator adx:g→g\mathrm{ad}_x: \mathfrak{g} \to \mathfrak{g} is for every x∈gx \in \mathfrak{g}. Prominent examples of Lie algebras include the Heisenberg algebras in higher dimensions. The (2n+1)(2n+1)-dimensional Heisenberg algebra over R\mathbb{R} or C\mathbb{C} has basis {p1,…,pn,q1,…,qn,z}\{p_1, \dots, p_n, q_1, \dots, q_n, z\} with nonzero brackets [pi,qj]=δijz[p_i, q_j] = \delta_{ij} z for 1≤i,j≤n1 \leq i,j \leq n, and its lower central series has length 2. Another example is the Lie algebra nm\mathfrak{n}_m of m×mm \times m strictly upper triangular matrices over a field of characteristic zero, which has dimension m(m−1)/2m(m-1)/2 and is with nilpotency index m−1m-1. Solvable Lie algebras arise as Lie algebras of derivations of s. For instance, the Lie algebra of derivations of the polynomial ring kk over a field kk of characteristic zero is two-dimensional, spanned by ∂x\partial_x and x∂xx \partial_x, with bracket [∂x,x∂x]=∂x[\partial_x, x \partial_x] = \partial_x, making it solvable but not . A key criterion for solvability concerns representations: Lie's theorem asserts that if ρ:g→gl(V)\rho: \mathfrak{g} \to \mathfrak{gl}(V) is a finite-dimensional representation of a solvable g\mathfrak{g} on a complex VV, then there exists a basis of VV in which every matrix ρ(x)\rho(x) for x∈gx \in \mathfrak{g} is upper triangular. Over the complex numbers, this implies that every finite-dimensional solvable is triangulable, meaning it admits a faithful representation by upper triangular matrices. Classifications of solvable and nilpotent Lie algebras exist up to 6 over algebraically closed fields of characteristic zero, revealing a variety of structures beyond the low-dimensional cases. Representative examples in dimensions 4 through 6 are summarized below, focusing on ones for brevity, with bracket relations given in adapted bases. In 4, there is a unique filiform algebra up to , characterized by maximal nilpotency index 3. Its bracket table in basis {X0,X1,X2,X3}\{X_0, X_1, X_2, X_3\} is:
X0X_0X1X_1X2X_2X3X_3
X0X_00X2X_2X3X_30
X1X_1−X2-X_2000
X2X_2−X3-X_3000
X3X_30000
In 5, there are nine indecomposable algebras over C\mathbb{C}. Two common ones are the filiform algebra g5,5\mathfrak{g}_{5,5} with basis {x1,x2,x3,x4,x5}\{x_1, x_2, x_3, x_4, x_5\} and brackets [x1,x2]=x3[x_1, x_2] = x_3, [x1,x3]=x4[x_1, x_3] = x_4, [x1,x4]=x5[x_1, x_4] = x_5 (all others zero), and the Heisenberg type g5,1\mathfrak{g}_{5,1} with [x1,x2]=x5[x_1, x_2] = x_5, [x3,x4]=x5[x_3, x_4] = x_5 (all others zero). In dimension 6, there are 30 indecomposable Lie algebras over C\mathbb{C}. Examples include the filiform algebra g6,5\mathfrak{g}_{6,5} (extension of the dimension-5 case by adding [x1,x5]=x6[x_1, x_5] = x_6) and a two-step one L6,10\mathfrak{L}_{6,10} with basis {x1,…,x6}\{x_1, \dots, x_6\} and brackets [x1,x2]=x3[x_1, x_2] = x_3, [x1,x3]=x6[x_1, x_3] = x_6, [x4,x5]=x6[x_4, x_5] = x_6 (all others zero).

Semisimple Lie Algebras

A over the complex numbers C\mathbb{C} is defined as a of simple algebras, where a simple algebra is non-abelian and has no non-trivial ideals. Equivalently, it has a trivial center and a trivial radical (the largest solvable ideal), meaning it is perfect, i.e., equal to its derived algebra [g,g]=g[ \mathfrak{g}, \mathfrak{g} ] = \mathfrak{g}. The Killing form provides a key invariant for semisimple Lie algebras. Defined as the symmetric bilinear form B(X,Y)=Tr⁡(ad⁡X∘ad⁡Y)B(X, Y) = \operatorname{Tr}(\operatorname{ad} X \circ \operatorname{ad} Y) on a finite-dimensional g\mathfrak{g}, where ad⁡\operatorname{ad} denotes the , the Killing form is non-degenerate precisely when g\mathfrak{g} is semisimple. This non-degeneracy follows from Cartan's criterion, which states that a is semisimple if and only if its is non-degenerate. The structure of semisimple Lie algebras is elucidated by Cartan-Weyl theory. A Cartan subalgebra h\mathfrak{h} is a maximal abelian toral subalgebra, i.e., a maximal subspace consisting of semisimple elements that is its own centralizer in g\mathfrak{g}. All Cartan subalgebras are conjugate under the adjoint action of g\mathfrak{g}, and dim⁡h=r\dim \mathfrak{h} = r is the rank of g\mathfrak{g}. The root space decomposition is g=h⊕⨁α∈Φgα\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, where Φ⊂h∗\Phi \subset \mathfrak{h}^* is the root system consisting of non-zero linear functionals α:h→C\alpha: \mathfrak{h} \to \mathbb{C} such that gα={X∈g∣[h,X]=α(h)X}\mathfrak{g}_\alpha = \{ X \in \mathfrak{g} \mid [\mathfrak{h}, X] = \alpha(\mathfrak{h}) X \} is one-dimensional for each α∈Φ\alpha \in \Phi. The roots Φ\Phi form a finite reduced root system, and a basis of simple roots {α1,…,αr}\{ \alpha_1, \dots, \alpha_r \} generates Φ\Phi as Z\mathbb{Z}-combinations with coefficients ≥0\geq 0 or ≤0\leq 0. The Weyl group WW is the finite group generated by reflections sαs_\alpha across hyperplanes orthogonal to roots α∈Φ\alpha \in \Phi, acting on h∗\mathfrak{h}^* and preserving Φ\Phi. The classification of semisimple Lie algebras proceeds via their root systems, encoded by Dynkin diagrams. Each simple root system corresponds to a unique (up to isomorphism) irreducible finite root system, classified by connected Dynkin diagrams of types AnA_n (n≥1n \geq 1), BnB_n (n≥2n \geq 2), CnC_n (n≥3n \geq 3), DnD_n (n≥4n \geq 4), and exceptional types E6,E7,E8,F4,G2E_6, E_7, E_8, F_4, G_2. These diagrams consist of nodes (simple roots) connected by edges indicating the Cartan integers Aij=2(αi,αj)(αi,αi)A_{ij} = 2 \frac{ (\alpha_i, \alpha_j) }{ (\alpha_i, \alpha_i) }, where (⋅,⋅)(\cdot, \cdot) is the inner product induced by the Killing form on h∗\mathfrak{h}^*. Semisimple Lie algebras are direct sums of simple ones, each corresponding to these types. The associated classical simple Lie algebras are sl(n+1,C)\mathfrak{sl}(n+1, \mathbb{C}) for AnA_n, so(2n+1,C)\mathfrak{so}(2n+1, \mathbb{C}) for BnB_n, sp(2n,C)\mathfrak{sp}(2n, \mathbb{C}) for CnC_n, and so(2n,C)\mathfrak{so}(2n, \mathbb{C}) for DnD_n. Explicit root systems illustrate the structure. For the type A1A_1 algebra sl(2,C)\mathfrak{sl}(2, \mathbb{C}), the root system consists of Φ={±α}\Phi = \{ \pm \alpha \}, where α\alpha is the standard root, and dim⁡g=3\dim \mathfrak{g} = 3. For the exceptional type G2G_2, the root system has 12 roots (6 short and 6 long), with rank 2 and dim⁡g=14\dim \mathfrak{g} = 14. In general, the dimension is dim⁡g=r+∣Φ∣\dim \mathfrak{g} = r + |\Phi|, since each root space is one-dimensional. A Chevalley basis provides an integral structure for simple Lie algebras. It consists of basis elements {hi∣i=1,…,r}∪{eα,fα∣α∈Φ+}\{ h_i \mid i=1,\dots,r \} \cup \{ e_\alpha, f_\alpha \mid \alpha \in \Phi^+ \}, where Φ+\Phi^+ are positive , hih_i are coroots in h\mathfrak{h}, and eα,fαe_\alpha, f_\alpha satisfy [eα,fα]=hα[e_\alpha, f_\alpha] = h_\alpha with integer . The are determined by the A=(Aij)A = (A_{ij}), where Aij=⟨αj,hi⟩=2(αi,αj)(αi,αi)A_{ij} = \langle \alpha_j, h_i \rangle = 2 \frac{ (\alpha_i, \alpha_j) }{ (\alpha_i, \alpha_i) }, and the commutation relations follow from Serre relations derived from AA. This basis ensures all cαβγc^\gamma_{\alpha\beta} in [eα,eβ]=cαβγeα+β[e_\alpha, e_\beta] = c^\gamma_{\alpha\beta} e_{\alpha+\beta} are integers. The following table summarizes the simple complex Lie algebras, their ranks rr, dimensions, and Dynkin diagrams (represented textually, with nodes as o---o for single bonds, o=>o for double with arrow indicating length difference, etc.):
TypeRank rrDimensionDynkin Diagram
AnA_n (n≥1n \geq 1)nnn(n+2)n(n+2)o---o---...---o (nn nodes)
BnB_n (n≥2n \geq 2)nn2n2+n2n^2 + no---o---...---o=>o (n−1n-1 single bonds, last double with arrow from long to short)
CnC_n (n≥3n \geq 3)nn2n2+n2n^2 + no---o---...---o<=-o (n−1n-1 single bonds, last double with arrow from short to long)
DnD_n (n≥4n \geq 4)nn2n(n−1)2n(n-1)o---o---...---o
o (last node branches to two)
E6E_6678o---o---o---o---o
|
o (branch at third node)
E7E_77133o---o---o---o---o---o
|
o (branch at third node in the chain)
E8E_88248o---o---o---o---o---o---o
|
o (branch at third node in the chain)
F4F_4452o---o=>o---o (n=2n=2 single-double-single)
G2G_2214o=>o (double with arrow)

Real Lie Groups

Low-Dimensional Real Lie Groups

In one dimension, the connected real Lie groups are the additive group R\mathbb{R} and the circle group S1S^1. Both are abelian, with Lie algebra g=R\mathfrak{g} = \mathbb{R} equipped with the trivial bracket. The group R\mathbb{R} is simply connected and non-compact, and its exponential map exp⁡:R→R\exp: \mathbb{R} \to \mathbb{R} is the identity isomorphism. The group S1S^1 is compact but not simply connected, with fundamental group Z\mathbb{Z}; its universal cover is R\mathbb{R}, and the exponential map exp⁡:R→S1\exp: \mathbb{R} \to S^1 given by t↦e2πitt \mapsto e^{2\pi i t} is a surjective homomorphism with kernel Z\mathbb{Z}. In two dimensions, the connected real Lie groups fall into abelian and non-abelian categories. The abelian ones are R2\mathbb{R}^2 (simply connected, non-compact), R×S1\mathbb{R} \times S^1 (non-compact, not simply connected with universal cover R2\mathbb{R}^2), and the torus T2=S1×S1T^2 = S^1 \times S^1 (compact, not simply connected with universal cover R2\mathbb{R}^2); all share the abelian Lie algebra R2\mathbb{R}^2. The non-abelian case is the solvable group of affine transformations of the line, known as the "ax + b" group, realized as matrices (ab01)\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} with a>0a > 0, b∈Rb \in \mathbb{R}; it is simply connected and non-compact, with Lie algebra spanned by H=(100−1)H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} and X=(0100)X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} satisfying [H,X]=2X[H, X] = 2X. There are no compact connected non-abelian 2D real Lie groups. The exponential map is surjective for the abelian cases but requires careful analysis for the solvable one, reflecting its semi-direct product structure R⋊R+\mathbb{R} \rtimes \mathbb{R}^+. In three dimensions, the connected real Lie groups are classified via the Bianchi types I through IX, corresponding to the real 3D Lie algebras, with groups constructed as simply connected covers or quotients thereof. The solvable and nilpotent examples include: Type I, the abelian R3\mathbb{R}^3 (simply connected, non-compact); Type II, the Heisenberg group of upper-triangular 3x3 matrices with 1s on the diagonal (nilpotent, simply connected, non-compact, with exponential map a bijection); and other solvable types (III–VII), for example, the special Euclidean group of the plane SE(2) = \mathbb{R}^2 \rtimes SO(2) (Bianchi type VII_0), which is non-compact with fundamental group \mathbb{Z}; its universal cover is \mathbb{R}^2 \rtimes \mathbb{R}. The full isometry group has two connected components. The semisimple cases are Type VIII, SL(2,R)SL(2, \mathbb{R}) (simple, non-compact, not simply connected with fundamental group Z\mathbb{Z}, universal cover the metaplectic group); and Type IX, SO(3)SO(3) (compact simple, not simply connected with fundamental group Z/2Z\mathbb{Z}/2\mathbb{Z}). The exponential map exp⁡:sl(2,R)→SL(2,R)\exp: \mathfrak{sl}(2, \mathbb{R}) \to SL(2, \mathbb{R}) is neither surjective nor injective; for example, matrices with trace < -2 are not in the image. In the universal cover (the metaplectic group), the exponential map is surjective onto the cover. The universal cover of SO(3)SO(3) is SU(2)SU(2) (or Spin(3)), a compact simply connected group diffeomorphic to S3S^3, providing a double cover via the adjoint representation. These low-dimensional groups illustrate key topological features: simply connected covers resolve non-trivial fundamental groups (e.g., R\mathbb{R} for S1S^1, SU(2)SU(2) for SO(3)SO(3)), and exponential maps link algebras to groups, being diffeomorphisms for nilpotent cases like the Heisenberg group but only partially covering for semisimple ones like SL(2,R)SL(2, \mathbb{R}). The Bianchi classification ensures all 3D cases are covered, with algebras determining the local structure and topology dictating global connectedness.
GroupDimensionCompact?Lie AlgebraSimply Connected CoverNotes on Exponential Map
R\mathbb{R}1NoAbelian R\mathbb{R}ItselfIsomorphism
S1S^11YesAbelian R\mathbb{R}R\mathbb{R}Surjective, kernel Z\mathbb{Z}
R2\mathbb{R}^22NoAbelian R2\mathbb{R}^2ItselfSurjective
R×S1\mathbb{R} \times S^12NoAbelian R2\mathbb{R}^2R2\mathbb{R}^2Surjective
T2T^22YesAbelian R2\mathbb{R}^2R2\mathbb{R}^2Surjective
ax + b group2NoSolvable non-abelianItselfSurjective
Heisenberg (Bianchi II)3NoNilpotentItselfBijection
SL(2,R)SL(2, \mathbb{R}) (Bianchi VIII)3Nosl(2,R)\mathfrak{sl}(2, \mathbb{R})Metaplectic groupNeither surjective nor injective
SO(3)SO(3) (Bianchi IX)3Yesso(3)\mathfrak{so}(3)SU(2)SU(2)Surjective via cover
SU(2)SU(2)3Yessu(2)\mathfrak{su}(2)ItselfSurjective

Classical Real Lie Groups

The classical real Lie groups arise as the real forms of the complex semisimple Lie groups corresponding to the classical root systems of types A, B, C, and D. These groups are realized as subgroups of the general linear group GL(n, ℝ) preserving specific bilinear or sesquilinear forms, and they include both compact and non-compact variants distinguished by the signature of their Killing form or invariant metrics. The non-compact forms, such as the split real forms, play a central role in the representation theory and geometry of semisimple Lie groups, while the compact forms are the orthogonal and unitary groups in appropriate signatures. The series A groups are exemplified by the SL(n, ℝ), consisting of n × n real matrices with 1, which preserves the standard on ℝⁿ. This group is non-compact with sl(n, ℝ) of traceless real matrices, having dimension n² - 1. Its maximal compact subgroup is SO(n), the special , embedded via orthogonal matrices of 1. For the orthogonal series (types B and D), the indefinite orthogonal groups SO(p, q) preserve a quadratic form of signature (p, q) on ℝ^{p+q}, with p + q = m fixed. The split form SO(n, 1) corresponds to the Lorentz group in Minkowski space, relevant in special relativity, and is non-compact, while SO(n) is the compact form preserving the positive definite Euclidean metric. The Lie algebra so(p, q) has dimension m(m-1)/2, consisting of matrices satisfying Xᵀ J + J X = 0, where J is the diagonal signature matrix. The maximal compact subgroup of SO(p, q) is SO(p) × SO(q). The symplectic series C is represented by Sp(2n, ℝ), the group of 2n × 2n real matrices preserving the standard symplectic form on ℝ^{2n}, which is non-compact with sp(2n, ℝ) of n(2n + 1). Elements satisfy Mᵀ J M = J, where J is the block-diagonal . The maximal compact subgroup is U(n), the acting on ℂⁿ identified with ℝ^{2n}. The real forms for each classical complex type are classified by their Cartan involutions and restricted root systems, with the split form maximizing the rank and the compact form having negative definite Killing form. The following table summarizes the principal real forms for the classical types:
Complex TypeSplit Form (Group/Algebra)Compact Form (Group/Algebra)Other Notable Forms
A_{n-1} (n ≥ 2)SL(n, ℝ) / sl(n, ℝ), dim = n² - 1SU(n) / su(n), dim = n² - 1SU(p, q) / su(p, q) for p + q = n, 1 ≤ p ≤ ⌊n/2⌋
B_n (n ≥ 1)SO(n+1, n) / so(n+1, n), dim = n(2n + 1)SO(2n+1) / so(2n+1), dim = n(2n + 1)—
C_n (n ≥ 1)Sp(2n, ℝ) / sp(2n, ℝ), dim = n(2n + 1)USp(2n) / usp(2n), dim = n(2n + 1)USp(2p, 2q) / usp(2p, 2q) for p + q = n
D_n (n ≥ 4)SO(n, n) / so(n, n), dim = n(2n - 1)SO(2n) / so(2n), dim = n(2n - 1)SO(n+1, n-1) / so(n+1, n-1); SO*(2n) / so*(2n)
These forms are distinguished by the signature of the invariant , with split forms having equal positive and negative eigenvalues in their Cartan decomposition. A key structural property of these non-compact groups is the G = K A N, where K is the maximal compact , A is a maximal abelian in the of the of K (vector part of the Cartan ), and N is the unipotent radical of a minimal parabolic . For SL(n, ℝ), this is SL(n, ℝ) = SO(n) ⋅ (positive diagonal matrices) ⋅ (upper triangular unipotent matrices), providing a and polar coordinates on the group. Similar hold for SO(p, q) and Sp(2n, ℝ), facilitating and unitary representations. As an explicit example, the compact Lie algebra so(3), the real form of type B_1 underlying SO(3), has basis generators corresponding to rotations around the coordinate axes: Lx=(00000−1010),Ly=(001000−100),Lz=(0−10100000).L_x = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \quad L_y = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \quad L_z = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}.
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