# About the Lie algebra of the angular momentum Poisson bracket structure [duplicate]

The components of the classical angular momentum $$L_i$$, satisfy the Poisson bracket relation $$\{L_i,L_j\}=\epsilon_{ijk}L_k,\tag{1}$$ and forms a Lie algebra (i.e, anti-symmetric, obeys the Jacobi identity etc).

In the theory of Lie groups, the commutator $$[\hat{L}_i,\hat{L}_j]=i\epsilon_{ijk}\hat{L}_k\tag{2}$$ is definitely called a $$\mathfrak{so}(3)$$ Lie algebra, because the $$\hat{L}_i$$'s generate $${\rm SO}(3)$$ group elements $$\left(\exp\left(-i\theta_i\hat{L}_i\right)\right)$$.

I wonder whether the PB Lie algebra $$(1)$$ is also called a $$\mathfrak{so}(3)$$ Lie algebra. And if so, why? What is the relation of $$(1)$$ to the $${\rm SO}(3)$$ group, if any?

• Yes, it is the $\mathfrak{so}(3)$ Lie algebra. physics.stackexchange.com/q/532270 and links therein might help – Nihar Karve Dec 29 '20 at 12:59
• In fact, the linked question (with its most popular answer) appears as a near duplicate. – Cosmas Zachos Dec 29 '20 at 16:31
• Note the linear operators $\{L_i, \bullet \}\equiv \partial L_i /\partial q^j \partial \bullet /\partial p^j - \partial L_i /\partial p^j \partial \bullet /\partial q^j$ furnish a faithful realization of the elements $\hat L_i$ of (2). Verify this by acting on an arbitrary phase-space function f(x,p). – Cosmas Zachos Dec 29 '20 at 16:46
• – Cosmas Zachos Dec 29 '20 at 18:59
• I have voted to reopen because the duplicate does not address the issue of the finite transformation $\in SO(3)$, which seems core to this question. – ZeroTheHero Dec 31 '20 at 15:44