I thought that groups with the same Lie Algebra are automatically equivalent, but there appear to be some exceptions to this?
What sort of exceptions are there and why?
2025-01-12 23:36:41.1736725001
How are groups with the same Lie Algebra inequivalent?
862 Views Asked by MadScientist https://math.techqa.club/user/madscientist/detail At
1
There are 1 best solutions below
Related Questions in GROUP-THEORY
- Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads
- Proper and discontinuous action of a group
- Category Theory compared with Meta-Grammars (or Hyper-Grammars) in Programming Languages
- Prove a subgroup is normal
- Is a finite group $G$ determined by the sequence $p(G,k)$ of probabilities that $G$ is generated by $k$ random elements?
- Conjugacy classes for rotations of $D_{2n}$
- Understanding the concept
- To prove a statement about finite groups of even order.
- Normal subgroup of prime order in the center
- Showing that the groups (Q,+) and (Q⁺,*) are not isomorphic
Related Questions in LIE-GROUPS
- Invariant measure of Lorentz Group
- Vectors fixed under compact subgroup
- Peter-Weyl theorem versions
- ODEs are invariant under the given Lie groups?
- Right translations of left-invariant forms are left-invariant
- How can I show that the following pairs of permutations are in the same conjugacy class in S5:
- How are groups with the same Lie Algebra inequivalent?
- Generators of a semi simple lie algebra must be traceless
- a method to find the infinitesimal generator of an ODE
- Proving that $U(n)\subset SO(2n)$
Related Questions in LIE-ALGEBRAS
- How do I make sense of terms $X^j\partial_j(Y^i)$ in the Lie bracket of vector fields?
- When derivations are exactly homomorphisms?
- Ideals in Lie algebras
- Lie bracket on $\Gamma(TM\oplus (M\times \mathfrak{g}))$?
- 2-dimensional derived subalgebra of 3-dimensional Lie algebra is abelian
- How are groups with the same Lie Algebra inequivalent?
- Generators of a semi simple lie algebra must be traceless
- From Generators of Lie Groups to Representations
- How does a Lie algebra act on a tensor product of L-modules?
- Representation of a Kac-Moody algebra
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
The 3-sphere (unit quaternions) and SO(3) have the same Lie algebra. Why? Because one is a double cover of the other, and the Lie algebra can be defined purely locally (via vector fields, for instance).
So I guess a general answer is "covering groups are a general case where same algebra doesn't (necessarily) mean same group."
(Sometimes you get a covering group that IS isomorphic, as in the covering $\theta \mapsto 2\theta$ on $S^1$.)