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group_theory:notions:forms_of_a_lie_algebra
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Different Forms of a Lie Algebra

The classification of all simple Lie groups was done with the help of Dynkin diagrams. (For mathematical details see, for example, this thesis).

Each Dynkin diagram corresponds to a complex simple Lie algebra.

(“Groups are usually parameterized with real parameters. This leads to real Lie algebras, i.e. algebras defined on the field of real numbers with real structure constants. In order to cary out the [classification of the simple Lie groups], however, it was necessary to extend the algebra to the complex number field. Source).

There is a famous theorem by Cartan that every complex simple Lie algebra has a compact real form. Expressed differently, this means that every complex simple Lie algebra is the complexification of the Lie algebra of a compact group. The map from the complex simple Lie algebra to the compact real form is usually called Weyl's unitary trick. However, this trick can be applied in various ways and thus it is possible to get various real forms from a complex Lie algebra. The Weyl trick simply means that we multiply some generators with an $i$ and all other generators stay the same. However, this does not necessarily lead to a real Lie algebra. If the structure constants are complex, the Lie algebra is complex. (”a Lie algebra with complex structure constants is complex itself.Source). We can equally start from the compact real form and apply the Weyl trick in all possible ways such that the structure constants remain real:

The problem of finding all possible real algebras concealed in a given complex algebra can be restated as follows: how many ways are there to play the Weyl unitary trick on the compact real form without getting complex structure constants?Source

See also, section 2.4 in http://dias.kb.dk/downloads/dias:67?locale=da.

Weyl's Unitary Trick

The way that the Weyl unitary trick makes complex groups behave like compact ones seems like a magic trick.http://mathoverflow.net/questions/48248/what-do-named-tricks-share

The Weyl unitary trick is a special case of the general procedure of group deformations, where we deform the algebra by multiplying certain generators with an $i$.

With the help of the Weyl trick we can analytically continue non-compact algebras to compact algebras. If we have the Lie algebra $\mathfrak{g}$ of a non-compact semisimple group, we can use to Cartan involution $$\mathfrak{g} = \mathfrak{l} + \mathfrak{p},$$ which is comparable to the singular value decomposition of matrices, to construct a compact algebra $$ g^W = \mathfrak{l} + i \mathfrak{p} = g_\mathbb{C} = \mathbb{C} \otimes g .$$

A famous example is the following complexification of the Lorentz algebra

$$ \mathfrak{lor}_{3,1} = \mathfrak{so}_3 + \mathfrak{p} \rightarrow \mathfrak{so}_3 + i\mathfrak{p} \equiv \mathfrak{lor}_{3,1}^W $$

This is useful, because it allows us to view the representations of the non-compact group $Lor({1,3})$, as representations of the compact group $SO(4) = SU(2) \times SU(2) / \mathbb{Z}_2$.

$SO(4)$ is the real compact form of $SO(4)_\mathbb{C}=SL(2)_\mathbb{C}\times SL(2)_\mathbb{C}/\mathbb{Z}_2$ and $Lor({1,3})$ is a real non-compact form.

Moreover, the simply connected group $SU(2) \times SU(2)$ is the double cover of $SO(4)$ and therefore their Lie algebras are isomorphic. In addition, $\mathfrak{su}(2)$ is a compact real form of $\mathfrak{sl}(2)_\mathbb{C}$. This means that we can study the representations of $\mathfrak{lor}_{3,1}$, by studying the representations of $\mathfrak{sl}(2)_\mathbb{C}$.

Weyl's unitary trick allows us to derive the representations of the Lorentz algebra $\mathfrak{lor}_{3,1}$, by computing the representations of the universal cover of $\mathfrak{so}(4)$, which is $\mathfrak{su}(2) \times \mathfrak{su}(2)$. This is incredibly useful, because all irreducible representations of $\mathfrak{su}(2)$ are known and can be classified easily.

See also: https://arxiv.org/pdf/1606.06966.pdf

Quotes

Weyl’s unitary trick is not a similarity transformationhttps://arxiv.org/pdf/1606.06966.pdf
the essential point is to draw conclusions about certain noncompact groups by making use of a certain compact subgroup (for instance the subgroup pf unitary matrices in the case of matrix groups).http://math.stackexchange.com/questions/174668/weyls-unitarity-trick
It is possible to transform from the diagonal Weyl canonical form (5.14) to Lc by a complex linear transformation called the Weyl unitary trick, a pretentious name for multiplying by i! […] Clearly it can change the character of an algebra and change a compact algebra into a non-compact one and vice versa. If we were to multiply indiscriminantly by i’s, however, we would generate complex structure constants. The problem of finding all possible real algebras concealed in a given complex algebra can be restated as follows: how many ways are there to play the Weyl unitary trick on the compact real form without getting complex structure constants?https://campusvirtual.univalle.edu.co/moodle/pluginfile.php/66702/mod_resource/content/0/Lie-Stetz-U-Oregon.pdf

There exists always a “maximally non-compact Weyl canonical form” of the Lie algebra, which is related to the compact algebra through the Weyl trick.

Original source of the trick.

Why is it called unitary trick?

when G is a complex simple Lie group and K is its maximal compact subgroup, the complex-analytic representations of G correspond in a one-to-way to unitary representations of K. This fact was called the unitarian trick by Hermann Weyl.http://math.ucr.edu/home/baez/qg-fall2008/lie3.pdf

Weyl's unitary trick relates the finite-dimensional representations of different real forms of a given complex Lie algebra. Some of these forms are non-compact. In addition, the trick relates, of course, the complex Lie algebra to its compact real form.

It is in this sense that Weyl's trick allows us to study the representations of a non-compact algebra by studying representations of its compact form.

There is a theorem that ”unitary representations of non-compact groups are infinite-dimensional“ (Ref).

In physics however, we need unitary representations to make sense of the probabilistic interpretation of quantum mechanics (see e.g. page 69 in Schwartz's book). With the help of Weyl's trick, we can

Examples

Source

The Lie algebra $D_2$ has various compact and non-compact forms:

  • If we have $6$ compact generators, we have get the $\mathfrak{so}(4)$ algebra, which is isomorphic to $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$.
  • If we have $3$ compact generators and 3 non-compact generators, we have get the $\mathfrak{so}(3,1)$ algebra, which is isomorphic to $\mathfrak{sl}(2,\mathbb{C}) $.
  • If we have $2$ compact generators and 4 non-compact generators, we have get the $\mathfrak{so}(2,2)$ algebra, which is isomorphic to $\mathfrak{su}(1,1) \oplus \mathfrak{su}(1,1)$.

Source

The $\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ isomorphism can be seen nicely, by looking at the corresponding root diagrams. The root space of $\mathfrak{so}(4)$ contains exactly the roots of $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$ and is therefore semi-simple. See, for example, this chapter.

We get the noncompact form of the classical algebras through the so-called ”Weyl trick“.

The finite dimensional representations of the Lorentz group have long been known. These representations are obtained from the representations of the rotation group in four dimensions by the “unitary trick.http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=5654&context=rtd

We get the Lorentz algebra $\mathfrak{so}(3,1)$ from the $\mathfrak{so}(4)$ through the Weyl trick.

For the Weyl Trick see this chapter and the explicit example here.

group_theory/notions/forms_of_a_lie_algebra.txt · Last modified: 2017/12/06 09:33 (external edit)