Defining representation of a classical Lie algebra

Definition

The defining representation of a classical matrix Lie algebra $\mathfrak g\subseteq\mathfrak{gl}(V)$ is the natural action on $V$ :

\rho:\mathfrak g\hookrightarrow\mathfrak{gl}(V), \qquad \rho(X)v=Xv.

For the complex simple classical families, this gives

\begin{array}{c|c|c} \text{type}&\mathfrak g&V\\ \hline A_{n-1}&\mathfrak{sl}_n(\mathbb C)&\mathbb C^n\\ B_n&\mathfrak{so}_{2n+1}(\mathbb C)&\mathbb C^{2n+1}\\ C_n&\mathfrak{sp}_{2n}(\mathbb C)&\mathbb C^{2n}\\ D_n&\mathfrak{so}_{2n}(\mathbb C)&\mathbb C^{2n}. \end{array}

The same phrase is used for $\mathfrak{gl}_n$ and for real forms acting on their natural real or complex vector spaces.

Preserved tensors

The matrix realization can be characterized by tensors preserved infinitesimally. The algebra $\mathfrak{sl}(V)$ consists of endomorphisms preserving a nonzero volume form to first order. The algebra $\mathfrak{so}(V,q)$ preserves a nondegenerate symmetric bilinear form $q$ , while $\mathfrak{sp}(V,\omega)$ preserves a nondegenerate alternating form $\omega$ :

q(Xv,w)+q(v,Xw)=0, \qquad \omega(Xv,w)+\omega(v,Xw)=0.

Highest weights and constructions

With the standard numbering of simple roots, the defining representation has highest weight $\omega_1$ in each classical family. It is therefore a fundamental representation. Tensor, symmetric, and exterior powers of the defining module produce many other representations.

For $\mathfrak{sl}_n$ , every fundamental representation is an exterior power $\Lambda^k\mathbb C^n$ . For orthogonal algebras, the defining module is also called the vector representation to distinguish it from spin representations, which live naturally on the spin group.

Group versus Lie algebra

The defining representations of $SL_n$ , $SO_n$ , and $Sp_{2n}$ differentiate to the displayed Lie-algebra representations. For the simply connected group $\operatorname{Spin}(n)$ , however, the vector representation is the composite

\operatorname{Spin}(n)\longrightarrow SO(n)\longrightarrow GL(\mathbb R^n)

and has kernel $\{\pm1\}$ . Thus it is faithful as a representation of $\mathfrak{so}_n$ but not as a representation of $\operatorname{Spin}(n)$ .

“Standard representation” can mean a different preferred module in other contexts, especially for induced representations of reductive groups. The adjective “defining” should be tied to an explicit matrix realization $\mathfrak g\subseteq\mathfrak{gl}(V)$ .

References

William Fulton and Joe Harris, Representation Theory: A First Course, Springer, 1991, §§15–20. Publisher record.
James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§8 and 13. Publisher record.