Complex Lie algebra vs Real Lie algebra in Physics

Question

A Lie algebra is a vector space $\mathfrak{g}$ over some field $F$ together with a binary operation $$\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$$ called the Lie bracket satisfying the following axioms: Bilinearity, Alternativity, Jacobi identity, Anticommutativity.

(Correct me if I am wrong)* If the Lie algebra over the field $F$ is a complex number, we have a complex Lie algebra. If the Lie algebra over the field $F$ is a real number, we have a real Lie algebra.

Given a complex Lie algebra $\mathfrak g$, a real Lie algebra $\mathfrak{g}_0$ is said to be a real form of $\mathfrak g$ if the complexification $$\mathfrak{g}_0 \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathfrak{g}$$ is isomorphic to $\mathfrak{g}$.

**My question is that:

1. Where do we encounter Real Lie algebra in Physics? (this should be abundant.) Do we have examples of both compact and noncompact real Lie group?

1. Where do we encounter Complex Lie algebra in Physics? (this should be rare? or abundant?) Do we have examples of both compact and noncompact complex Lie group?

This post imported from StackExchange Physics at 2020-12-04 11:35 (UTC), posted by SE-user annie marie heart

Comments

If the Lie algebra over the field $F$ is a complex number... An algebra is not a number.

This post imported from StackExchange Physics at 2020-12-04 11:35 (UTC), posted by SE-user G. Smith

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I meant the field

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OK, but that’s not what you wrote.

This post imported from StackExchange Physics at 2020-12-04 11:35 (UTC), posted by SE-user G. Smith

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Wikipedia: $\mathfrak{su}(n)$ is a real Lie algebra. (Despite the fact that one of the Pauli matrices involves $i$.) $SU(n)$ is not a complex Lie group.

This post imported from StackExchange Physics at 2020-12-04 11:35 (UTC), posted by SE-user G. Smith

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field

Answer 1

One interesting example: For the one-dimensional quantum simple harmonic oscillator, the operators $1$, $\hat{x}$ and $\hat{p}$ generate a real Lie algebra (the Heisenberg algebra). However, it is often useful to work instead in terms of the raising and lowering operators, so that our generators are $1, \hat{a},\hat{a}^\dagger$. Since $\hat{a} = \hat{x} + i\hat{p}$, the algebra they generate is the complexification of the Heisenberg algebra.

There is a useful sense in which the Lie algebra of any Lie group is "naturally" real, since it's a tangent space of a manifold. For example $GL_n(\mathbb{C})$, considered as an abstract Lie group, is a $2n^2$-dimensional manifold, so the Lie algebra is "really" a real vector space with $2n^2$ basis vectors. In physics we usually think of it as a complex vector space with $n^2$ basis vectors, but we don't have to. Every Lie algebra I can think of in physics comes from some Lie group, so complex Lie algebras come only from either complexification (as in the raising/lowering operator case) or from treating a $2n$-dimensional real Lie algebra as an $n$-dimensional complex Lie algebra. (I don't have a name for the latter process, or much understanding of it - suggestions welcome).

This post imported from StackExchange Physics at 2020-12-04 11:35 (UTC), posted by SE-user Daniel