Type A root system - Knowlpedia

Definition

For $n\geq1$ , the root system of type $A_n$ is

A_n=\{e_i-e_j:1\leq i\neq j\leq n+1\}

in the $n$ -dimensional Euclidean space

V=\left\{(x_1,\ldots,x_{n+1})\in\mathbb R^{n+1}:\sum_i x_i=0\right\}.

It is an irreducible simply laced root system of rank $n$ .

Simple roots and diagram

A standard base of simple roots is

\alpha_i=e_i-e_{i+1},\qquad 1\leq i\leq n.

Its Dynkin diagram is a chain of $n$ vertices. The system has $n(n+1)$ roots, and its Weyl group is the symmetric group $S_{n+1}$ , acting by permuting coordinates.

Lie-algebra realization

The roots of $\mathfrak{sl}_{n+1}(\mathbb C)$ relative to its diagonal trace-zero Cartan subalgebra are the functionals $e_i-e_j$ . Hence type $A_n$ corresponds under the classification of complex simple Lie algebras to

\mathfrak{sl}_{n+1}(\mathbb C),

whose dimension is $(n+1)^2-1=n(n+2)$ . See also the special linear Lie algebra.

Low-rank cases

The system $A_1$ consists of two opposite roots. The accidental diagram isomorphism $A_3\cong D_3$ is why the irreducible type $D_n$ family is conventionally indexed from $n=4$ .

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§11–12. Publisher record.
Nicolas Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Springer, 2002, Chapter VI, §4. Publisher record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §§2, 5, 7–8. arXiv record.