Semisimple Lie algebra

History[edit]

The semisimple Lie algebras over the complex numbers were first classified by Wilhelm Killing (1888–90), though his proof lacked rigor. His proof was made rigorous by Élie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year-old Eugene Dynkin in 1947. Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials and can be found in any standard reference, such as (Humphreys 1972).

As noted in #Structure, semisimple Lie algebras over ${\displaystyle \mathbb {C} }$ (or more generally an algebraically closed field of characteristic zero) are classified by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams. Examples of semisimple Lie algebras, the classical Lie algebras, with notation coming from their Dynkin diagrams, are:


The restriction ${\displaystyle n>1}$ in the ${\displaystyle D_{n}}$ family is needed because ${\displaystyle {\mathfrak {so}}_{2}}$ is one-dimensional and commutative and therefore not semisimple.


These Lie algebras are numbered so that n is the rank. Almost all of these semisimple Lie algebras are actually simple and the members of these families are almost all distinct, except for some collisions in small rank. For example ${\displaystyle {\mathfrak {so}}_{4}\cong {\mathfrak {so}}_{3}\oplus {\mathfrak {so}}_{3}}$ and ${\displaystyle {\mathfrak {sp}}_{2}\cong {\mathfrak {so}}_{5}}$. These four families, together with five exceptions (E₆, E₇, E₈, F₄, and G₂), are in fact the only simple Lie algebras over the complex numbers.

Let ${\displaystyle {\mathfrak {g}}}$ be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in #Structure, ${\textstyle {\mathfrak {g}}={\mathfrak {h}}\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }}$ where ${\displaystyle \Phi }$ is the root system. Choose the simple roots in ${\displaystyle \Phi }$; a root ${\displaystyle \alpha }$ of ${\displaystyle \Phi }$ is then called positive and is denoted by ${\displaystyle \alpha >0}$ if it is a linear combination of the simple roots with non-negative integer coefficients. Let ${\textstyle {\mathfrak {b}}={\mathfrak {h}}\oplus \bigoplus _{\alpha >0}{\mathfrak {g}}_{\alpha }}$, which is a maximal solvable subalgebra of ${\displaystyle {\mathfrak {g}}}$, the Borel subalgebra.


Let V be a (possibly-infinite-dimensional) simple ${\displaystyle {\mathfrak {g}}}$-module. If V happens to admit a ${\displaystyle {\mathfrak {b}}}$-weight vector ${\displaystyle v_{0}}$,^[13] then it is unique up to scaling and is called the highest weight vector of V. It is also an ${\displaystyle {\mathfrak {h}}}$-weight vector and the ${\displaystyle {\mathfrak {h}}}$-weight of ${\displaystyle v_{0}}$, a linear functional of ${\displaystyle {\mathfrak {h}}}$, is called the highest weight of V. The basic yet nontrivial facts^[14] then are (1) to each linear functional ${\displaystyle \mu \in {\mathfrak {h}}^{*}}$, there exists a simple ${\displaystyle {\mathfrak {g}}}$-module ${\displaystyle V^{\mu }}$ having ${\displaystyle \mu }$ as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between ${\displaystyle {\mathfrak {h}}^{*}}$ and the set of the equivalence classes of simple ${\displaystyle {\mathfrak {g}}}$-modules admitting a Borel-weight vector.


For applications, one is often interested in a finite-dimensional simple ${\displaystyle {\mathfrak {g}}}$-module (a finite-dimensional irreducible representation). This is especially the case when ${\displaystyle {\mathfrak {g}}}$ is the Lie algebra of a Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the positive Weyl chamber ${\displaystyle C\subset {\mathfrak {h}}^{*}}$, we mean the convex cone ${\displaystyle C=\{\mu \in {\mathfrak {h}}^{*}|\mu (h_{\alpha })\geq 0,\alpha \in \Phi >0\}}$ where ${\displaystyle h_{\alpha }\in [{\mathfrak {g}}_{\alpha },{\mathfrak {g}}_{-\alpha }]}$ is a unique vector such that ${\displaystyle \alpha (h_{\alpha })=2}$. The criterion then reads:^[15]


A linear functional ${\displaystyle \mu }$ satisfying the above equivalent condition is called a dominant integral weight. Hence, in summary, there exists a bijection between the dominant integral weights and the equivalence classes of finite-dimensional simple ${\displaystyle {\mathfrak {g}}}$-modules, the result known as the theorem of the highest weight. The character of a finite-dimensional simple module in turns is computed by the Weyl character formula.


The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional module of a semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$ is completely reducible; i.e., it is a direct sum of simple ${\displaystyle {\mathfrak {g}}}$-modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra.