What is Lie groups: Definition and 101 Discussions

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



{\displaystyle {\text{SO}}(3)}
). Lie groups are widely used in many parts of modern mathematics and physics.
Lie groups were first found by studying matrix subgroups


{\displaystyle G}
contained in






{\displaystyle {\text{GL}}_{n}(\mathbb {R} )}






{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}
, the groups of


{\displaystyle n\times n}
invertible matrices over


{\displaystyle \mathbb {R} }


{\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.

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  1. cianfa72

    I 3-parameter group of symmetries on the plane

    Consider ##\mathbb R^2## as the Euclidean plane. Since it is maximally symmetric it has a 3-parameter group of Killing vector fields (KVFs). Pick orthogonal cartesian coordinates centered at point P. Then the 3 KVFs are given by: $$K_1=\partial_x, K_2=\partial_y, K_3=-y\partial_x + x...
  2. cianfa72

    I ##SU(2, \mathbb C)## parametrization using Euler angles

    Hi, I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements \begin{pmatrix} e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\ ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}...
  3. cianfa72

    I ##SL(2,\mathbb R)## Lie group as manifold

    Hi, consider the set of the following parametrized matrices $$ \begin{bmatrix} 1+a & b \\ c & \frac {1 + bc} {1 + a} \\ \end{bmatrix} $$ They are member of the group ##SL(2,\mathbb R)## (indeed their determinant is 1). The group itself is homemorphic to a quadric in ##\mathbb R^4##. I believe...
  4. K

    I Conjugation vs Change of Basis

    For transformations, A and B are similar if A = S-1BS where S is the change of basis matrix. For Lie groups, the adjoint representation Adg(b) = gbg-1, describes a group action on itself. The expressions have similar form except for the order of the inverses. Is there there any connection...
  5. P

    A Differential structure of the group of automorphism of a Lie group

    I am working on this I am having trouble with b and c: b) Suppose ##(f_n)_{n=1}^{\infty}## is a sequence in ##Aut(G)##, such that ##(T_e(f_n))_{n=1}^{\infty} \to \psi## converges in ##Aut(\mathfrak g)## I want to show that ## f := \lim_{n\to \infty} f_n## exists as an continuous...
  6. T

    A Classification of reductive groups via root datum

    I have a couple of questions about classification of reductive groups over algebraically closed field (up to isomorphism) by so called root datum. In the linked discussion is continued that Obviously, a root datum ##(X^*, \Phi, X_*, \phi^{\vee})## contains full information ("building plan")...
  7. A

    About representations of Lie groups

    I am not quite sure of how this works, i.e. of what exactly I need to do with the hint. Any explanantion would be helpful!
  8. A

    A About calculating a fundamental group

    What is the way to compute ##\pi_1(PGL_2(R))##? Is it related to defining an action of ##PGL_2(R)## on ##S^3##? it would be helpful if you can provide me with relevant information regarding this
  9. A

    A About computing the tangent space at 1 of certain lie groups

    Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...
  10. J

    Model with SU(2) gauge symmetry and SO(3) global symmetry

    1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
  11. K

    I Transition Functions and Lie Groups

    I understand that on Riemannian manifolds, the transition functions that glue charts together are coordinate transformations (Jacobian matrices). However, I am not quite sure how transition functions work in the context of Lie groups and Fiber bundles. Do we consider the manifolds to be flat...
  12. S

    I Matrix construction for spinors

    I'm reading the book QFT by Ryder, in the section where ##\rm{SU(2)}## is discussed. First, he considered the group of ##2 \times 2## unitary matrices ##U## with unit determinant such that it has the form, $$U =\begin{bmatrix} a & b \\ -b^* & a^* \end{bmatrix}, \qquad \xi = \begin{bmatrix}...
  13. LucaC

    A Invariance of ##SO(3)## Lie group when expressed via Euler angles

    I am trying to understand the properties of the ##SO(3)## Lie Group but when expressed via Euler angles instead of rotation matrix or quaternions. I am building an Invariant Extended Kalman Filter (IEKF), which exploits the invariance property of ##SO(3)## dynamics ##\mathbf{\dot{R}} =...
  14. S

    Algebra Book on Lie algebra & Lie groups for advanced math undergrad

    Posting for my son (who does not have an account here): He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...
  15. L

    A Unitary representations of Lie group from Lie algebra

    In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...
  16. Jason Bennett

    (Physicist version of) Taylor expansions

    3) Taylor expansion question in the context of Lie algebra elements: Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g( \alpha)...
  17. Jason Bennett

    Lorentz algebra elements in an operator representation

    1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...
  18. S

    Algebra Lie Groups and Lie Algebras by Robert Gilmore

    Anyone reading Lie Groups and Lie Algebras and Some of Their Applications by Robert Gilmore , might be interested in a series of YouTube videos by "XylyXylyX" that follows the book. The first lecture is:
  19. E

    I Derivative of the Ad map on a Lie group

    Hi, let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##. In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a...
  20. G

    A Is the Exponential Map Always Surjective from Lie Algebras to Lie Groups?

    Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...
  21. W

    I Proving Commutator Identity for Baker-Campbell-Hausdorff Formula

    I'm having a little trouble proving the following identity that is used in the derivation of the Baker-Campbell-Hausdorff Formula: $$[e^{tT},S] = -t[S,T]e^{tT}$$ It is assumed that [S,T] commutes with S and T, these being linear operators. I tried opening both sides and comparing terms to no...
  22. L

    A Why does the Lie group ##SO(N)## have ##n=\frac{N(N-1)}{2}## real parameters?

    When we have a Lie group, we want to obtain number of real parameters. In case of orthogonal matrices we have equation R^{\text{T}}R=I, that could be written in form \sum_i R_{i,j}R_{i,k}=\delta_{j,k}. For this real algebra ##SO(N)## there is ##n=\frac{N(N-1)}{2}## real parameters. Why this is...
  23. W

    I Parametrization manifold of SL(2,R)

    I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a...
  24. W

    I Exponential map for Lie groups

    I’ve read about the exponential map that for Lie groups the exponential map is actually the exponential function. But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric. But usually nobody gives you a metric with a Lie group. So how can I get...
  25. N

    I Computation of the left invariant vector field for SO(3)

    I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward. I have been looking at these notes: https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...
  26. tomdodd4598

    I Do the SU(n) generators represent any observables?

    Hey there, I've recently been trying to get my head around Yang-Mills gauge theory and was just wandering: do the Pauli matrices for su(2), Gell-Mann matrices for su(3), etc. represent any important observable quantities? After all, they are Hermitian operators and act on the doublets and...
  27. Luck0

    A Diagonalization of adjoint representation of a Lie Group

    So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
  28. Luck0

    I Question about Haar measures on lie groups

    I'm not sure if this question belongs to here, but here it goes Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is...
  29. M

    I Lie groups left invariant vector fields

    hello every one . can someone please find the left invariant vector fields or the generator of SO(2) using Dr. Frederic P. Schuller method ( push-forward,composition of maps and other stuff) Dr Frederic found the left invariant vector fields of SL(2,C) and then translated them to the identity...
  30. W

    What is SU*(N)? Definition and Explanation

    I've run across a Lie group notation that I am unfamiliar with and having trouble googling (since google won't seem to search on * characters literally). Does anyone know the definition of the "star groups" notated e.g. SU*(N), SO*(N) ?? The paper I am reading states for example that SO(5,1)...
  31. O

    Generators of Lie Groups and Angular Velocity

    I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy) I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix. (I understand how I obtain this equation... that is not the issue.) Now I am making the leap to learning about...
  32. Rasalhague

    I Simply-connected, complex, simple Lie groups

    I've been looking at John Baez's lecture notes "Lie Theory Through Examples". In the first chapter, he says Dynkin diagrams classify various types of object, including "simply-connected, complex, simple Lie groups." He discusses the An case in detail. But what are the simply-connected, complex...
  33. Xico Sim

    A Matrix Lie groups and its Lie Algebra

    Hi! I'm studying Lie Algebras and Lie Groups. I'm using Brian Hall's book, which focuses on matrix lie groups for a start, and I'm loving it. However, I'm really having a hard time connecting what he does with what physicists do (which I never really understood)... Here goes one of my questions...
  34. O

    Praise HANK YOU All - 2 Years of Learning Differential Forms & Exterior Algebra

    No question this time. Just a simple THANK YOU For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups. My math background was very deficient: I am a 55 year old retired (a good life) professor of...
  35. mnb96

    A Derivative of smooth paths in Lie groups

    Hello, Given a Lie group G and a smooth path γ:[-ε,ε]→G centered at g∈G (i.e., γ(0)=g), and assuming I have a chart Φ:G→U⊂ℝn, how do I define the derivative \frac{d\gamma}{dt}\mid_{t=0} ? I already know that many books define the derivative of matrix Lie groups in terms of an "infinitesimal...
  36. mnb96

    Definition of chart for Lie groups

    Hello, I'm reading a book on Lie group theory, and before giving the definition of a Lie group G, the author defines the concept of chart as a pair (U(g), f) where: i) U(g) is a neighborhood of g∈G ii) f : U(g)→f(U(g))⊂ℝn is an invertible map such that f(U(g)) is an open subset of ℝn. My...
  37. mnb96

    How Is the Abelianization of a Lie Group Defined?

    Hi, the abelianization of a group G is given by the quotient G/[G,G], where [G,G] is the commutator subgroup of G. When dealing with finite groups, the commutator subgroup is given by the (normal) subgroup generated by all the commutators of G. If we consider instead the case of G being a Lie...
  38. Q

    Proving Correspondence between SO(3)/SO(2) and S^2

    Homework Statement Take the subgroup isomorphic to SO(2) in the group SO(3) to be the group of matrices of the form \begin{pmatrix} g & & 0 \\ & & 0 \\ 0 & 0 & 1 \end{pmatrix}, g\in{}SO(2). Show that there is a one-to-one correspondence between the coset space of SO(3) by this subgroup and...
  39. P

    Controllability of non-linear systems via Lie Brackets

    In http://www.me.berkeley.edu/ME237/6_cont_obs.pdf , page 65, the controllability matrix is defined as: $$C=[g_1, g_m,\dots,[g_i,g_j],[ad_{g_i}^k,g_j],\dots,[f,g_i],\dots,[ad_f^k,g_i],\dots]$$ where the systems is in general given by $$\dot{x}=f(x)+\sum_i^m{g_i(x)\mu_i}$$ Lets say you have a...
  40. JonnyMaddox

    Lie Groups, Lie Algebra and Vectorfields

    Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let a and g be elements of a Lie group G, the left translation L_{a}: G \rightarrow G of g by a are defined by : L_{a}g=ag which induces a map L_{a*}...
  41. K

    What are the algebra prerequisites for Lie groups?

    I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract...
  42. H

    Homomorphism between Lie groups

    Hello I've been reading some Group theory texts and would like to clarify something. Let's say we have two Lie groups A and B, with corresponding Lie algebras a and b. Does the fact that a and b share the same Lie Bracket structure, as in if we can find a map M:a->b which obeys...
  43. P

    Galois Theory, Differential Equations, and Lie Groups?

    I apologize for the informal and un-rigourous question. I have heard, in passing, that doing Galois Theory over Lie Groups instead of discrete groups is connected to solutions of differential equations instead of algebraic equations. First of all, is this correct? If so, what is this...
  44. B

    Why do several Lie groups have the same Lie algebra when acting on a space?

    Why is it that several lie groups can have the same Lie algebra? could it have to do with the space where they act transitively? Could two different Lie groups acting transitively on the same space have the same Lie algebra?
  45. B

    Homogeneous spacetime - Lie groups

    All Bianchi type spacetimes have metrics that admits a 3-dimensional killing algebra. They are in general not isotropic. Bianchi type IX have a killing algebra that is isomorphic to SO(3), i.e. the rotation group. But what does it mean? If the fourdimensional spacetime is invariant under the...
  46. Mandelbroth

    Formulating a Method of Steepest Ascent on Lie Groups

    Suppose we have a compact Lie group ##G##, and a differentiable function ##f:G_0\to\mathbb{R}## from the identity component of ##G## to the real numbers. I'm looking to maximize the value of this function. Being something of a neophyte at optimization, especially of this kind, I decided to...
  47. G

    Why Is U(n) Considered Connected When O(n) Is Not?

    I surely am missing something about the notion of connectedness, and I clarify this by means of an example: O(n), the orthogonal group, has two subsets with detO=1 and detO=-1. Now, the maximally connected component of O(n) is SO(n), which is the subgroup with detO=1 including the Identity...
  48. L

    Differentiation Problem on Lie Groups

    Suppose θ is a differential 1 form defined on a manifold and with values in the Lie algebra of a Lie group,G. On MxG define the 1 form, ad(g)θ ,where θ is extended by letting it be zero on the tangent space to G How do you compute the exterior derivative, dad(g)θ ? BTW: For matrix...
  49. S

    Quantum Gravity: Lie Groups vs. Banach Algebras & Spectral Theory

    Quantum Gravity: "Lie Groups" vs. "Banach Algebras & Spectral Theory" I'm interested in researching quantum gravity & non-commutative geometry. I am planning to take one math course outside of my physics classes this Fall to help, but can't decide between two: "Lie groups" or "Banach algebras &...