Complex Lie algebra sl2(C)

The complex Lie algebra $\mathfrak{sl}_2(\mathbb C)$ is

\mathfrak{sl}_2(\mathbb C) =\{X\in M_2(\mathbb C):\operatorname{tr}X=0\}, \qquad [X,Y]=XY-YX.

It is a three-dimensional simple Lie algebra of rank $1$ , with root system and Dynkin type $A_1$ . Its defining representation is the two-dimensional module $\mathbb C^2$ , and its adjoint representation has dimension $3$ .

Standard basis and roots

The matrices

E=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad F=\begin{pmatrix}0&0\\1&0\end{pmatrix},\qquad H=\begin{pmatrix}1&0\\0&-1\end{pmatrix}

satisfy

[H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=H.

Thus $\mathfrak h=\mathbb C H$ is a Cartan subalgebra, and

\mathfrak{sl}_2(\mathbb C)=\mathbb C F\oplus\mathbb C H\oplus\mathbb C E

is its root-space decomposition. The two roots take values $-2$ and $2$ on $H$ ; rescaling this coordinate does not change the abstract root system $A_1$ .

Finite-dimensional representations

For every integer $m\geq 0$ , the symmetric power

\operatorname{Sym}^m(\mathbb C^2)

is the irreducible highest-weight module of highest weight $m$ and dimension $m+1$ . Every finite-dimensional irreducible $\mathfrak{sl}_2(\mathbb C)$ -module is obtained this way. In particular, $\operatorname{Sym}^2(\mathbb C^2)$ is the three-dimensional adjoint module.

The $E,F,H$ relations occur inside every complex semisimple Lie algebra: the root spaces for a root $\alpha$ and $-\alpha$ , together with the coroot, generate an $\mathfrak{sl}_2$ -subalgebra. This is why rank-one calculations control root strings and much of highest-weight representation theory.

Groups and real forms

The connected simply connected complex Lie group with this Lie algebra is $SL(2,\mathbb C)$ ; its adjoint quotient is $PSL(2,\mathbb C)=SL(2,\mathbb C)/\{\pm I\}$ . These complex groups should not be confused with real forms having the same complexification. The compact real form is $\mathfrak{su}(2)$ , while the split real form is $\mathfrak{sl}_2(\mathbb R)$ . As a real Lie algebra, $\mathfrak{sl}_2(\mathbb C)$ has real dimension $6$ and is isomorphic to $\mathfrak{so}(1,3)$ .

Role in the E7 three-generation construction

In the three-generation construction, three distinguished $\mathfrak{sl}_2$ -subalgebras occur inside a generation-symmetry $\mathfrak{sl}_3$ . The centralizer of each one in $\mathfrak e_7$ is a copy of $\mathfrak{so}_{12}(\mathbb C)$ . The relevant branching is

\mathfrak e_7 \cong (\mathbf 3,\mathbf 1) \oplus(\mathbf 1,\mathbf{66}) \oplus(\mathbf 2,\mathbf{32}),

where $\mathbf 3$ and $\mathbf{66}$ are the adjoint modules of $\mathfrak{sl}_2$ and $\mathfrak{so}_{12}$ , respectively, and $\mathbf{32}$ is a half-spin module.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, Sections 7 and 11. Publisher record.
William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, Sections 11.1--11.2. Publisher record.
John C. Baez, Three Generations in E7, 2026. arXiv:2608.06271.