Semisimple Lie algebra

A finite-dimensional Lie algebra is semisimple when it has no nonzero solvable ideals.

For a choice of positive roots, the corresponding Borel subalgebra is $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$, the sum of a Cartan subalgebra and all positive root spaces.

A nonabelian Lie algebra is simple when its only ideals are zero and the entire algebra.

The special linear Lie algebra ${\mathfrak{sl}}_n$ consists of the trace-zero $n\times n$ matrices with the commutator bracket.

The two fundamental representations of ${\mathfrak{sl}}_3$ are the defining representation $V({\omega }_1)={\mathbb{C}}^3$ and its dual

For the defining representation $V=V({\omega }_1)$ of ${\mathfrak{sl}}_3$,

The Lie algebra ${\mathfrak{sl}}_2$ has generators $e,h,f$ with $[h,e]=2e$, $[h,f]=-2f$, and $[e,f]=h$.

For every $n\ge 0$ there is one irreducible ${\mathfrak{sl}}_2(\mathbb{C})$-module $V(n)$ of dimension $n+1$, with weights $n,n-2,\dots ,-n$, each of multiplicity one. Every finite-dimensional representation is a direct sum of these modules.

On any finite-dimensional ${\mathfrak{sl}}_2$-module, the lowering operator $f$ is injective from the $m$-weight space to the $(m-2)$-weight space whenever $m>0$. This follows on each irreducible summand from its standard weight string.

The Verma module $M_{\lambda }$ has basis $(f^r{v}_{\lambda }{)}_{r\ge 0}$ andIt is reducible exactly when $\lambda =m\in {\mathbb{Z}}_{\ge 0}$; then its unique proper nonzero submodule is generated by $f^{m+1}{v}_{\lambda }$ and is isomorphic to $M_{-m-2}$.

For a nondegenerate alternating matrix $J$, the symplectic Lie algebra is ${\mathfrak{sp}}_{2n}=\{A:AJ+J{A}^T=0\}$.

In an orthonormal basis $({\epsilon }_i)$, the $C_n$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $i<j$ and $\pm 2{\epsilon }_i$.

The special orthogonal Lie algebra consists of the infinitesimal transformations preserving a nondegenerate symmetric bilinear form.

The Lorentz algebra has rotation generators $J_i$ and boost generators $K_i$ satisfying

Over the complex numbers, $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ generate commuting copies of ${\mathfrak{sl}}_2(\mathbb{C})$, givingFinite-dimensional irreducible representations are labelled $(j_L,j_R)$.

Parity fixes $J_i$, negates $K_i$, and therefore exchanges the two chiral factors. It sends $(j_L,j_R)$ to $(j_R,j_L)$.

The quadratic contraction $M_{\mu \nu }{M}^{\mu \nu }$ is parity even, while $M_{\mu \nu }{\tilde{M}}^{\mu \nu }$ is parity odd. Their linear combinations give the quadratic Casimirs of the two chiral ${\mathfrak{sl}}_2$ factors.

The Lie algebra ${\mathfrak{so}}_5$ has dimension ten and root system $B_2$.

Under the standard block-diagonal subgroup, the vector and adjoint representations branch asUsing ${\mathfrak{so}}_4\cong {\mathfrak{su}}_2\oplus {\mathfrak{su}}_2$, these are $(\frac{1}{2},\frac{1}{2})\oplus (0,0)$ and $(1,0)\oplus (0,1)\oplus (\frac{1}{2},\frac{1}{2})$.

The $B_2$ root system consists of $\pm {\epsilon }_1$, $\pm {\epsilon }_2$, and $\pm {\epsilon }_1\pm {\epsilon }_2$. Its vector representation has weights ${\epsilon }_1,{\epsilon }_2,0,-{\epsilon }_2,-{\epsilon }_1$.

The root systems $B_2$ and $C_2$ are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$.

For simple roots $(1,0)$ and $(-1,1)$, the fundamental weights are ${\omega }_1=(1/2,1/2)$ and ${\omega }_2=(0,1)$. The representation $[1,0]$ is the four-dimensional spin representation with weights $(\pm 1/2,\pm 1/2)$; $[0,1]$ is the five-dimensional vector representation with weights $(\pm 1,0),(0,\pm 1),(0,0)$.

For a complex semisimple Lie algebra, a Cartan subalgebra is a maximal abelian subalgebra consisting of semisimple elements. It is also called a maximal torus in this setting.

For a Cartan subalgebra $\mathfrak{t}$, a semisimple Lie algebra decomposes asThe nonzero functionals $\alpha$ are the roots.

A root system is a finite set of nonzero vectors closed under the reflections they define and satisfying the crystallographic integrality condition when it arises from a semisimple Lie algebra.

A choice of positive roots selects exactly one of $\alpha$ and $-\alpha$ and is closed under addition whenever the sum is a root.

The simple roots are the positive roots that cannot be expressed as sums of two positive roots. Every root is an integer combination of simple roots with coefficients of one sign.

The highest root is the maximal positive root in the root order determined by the simple roots.

The Weyl vector is $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$. It satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for every simple root.

The coroot associated with a nonzero root is ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$.

The root lattice is the integer span of the roots, equivalently of the simple roots.

For weights $\mu ,\lambda$, one writes $\mu ⪯\lambda$ when $\lambda -\mu$ is a nonnegative integer combination of the simple roots.

The weight lattice is the set of vectors $\lambda$ for which $⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer for every root $\alpha$.

An integral weight $\lambda$ is dominant when $⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0$ for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.

The fundamental weights are dual to the simple coroots: $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$.

The fundamental representation associated with a fundamental weight ${\omega }_i$ is the irreducible highest-weight representation $V({\omega }_i)$.

The reflection associated with a root is $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$.

The reflection group of a root system $\Phi$ is the subgroup of the orthogonal group generated by the reflections $s_{\alpha }$ for $\alpha \in \Phi$. It permutes the roots and acts freely and transitively on the fundamental systems.

The Weyl group is generated by the reflections in the roots.

The Coxeter length $\ell (w)$ is the smallest number of simple reflections whose product is $w$. Its parity gives the sign of a Weyl-group element.

If $\lambda$ and $w\lambda$ both lie in the closed dominant chamber, then $w$ is a product of simple reflections whose walls contain $\lambda$. In particular, $w\lambda =\lambda$.

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

In the geometric representation of a Coxeter system,andReplacing “positive” by “negative” reverses either length inequality.

A Dynkin diagram records the angles and relative lengths of the simple roots. A multiple-edge arrow points toward the shorter root.

An extended, or affine, Dynkin diagram adjoins the root ${\alpha }_0=-\theta$, where $\theta$ is the highest root.

The $G_2$ root system has six short and six long roots. For a short simple root ${\alpha }_1$ and a long simple root ${\alpha }_2$, its positive roots are

A fundamental system $\Delta$ is a basis made of roots such that every root is a linear combination of $\Delta$ whose nonzero coefficients all have the same sign. Its members are the simple roots.

The positive system associated with a fundamental system of a root system $\Delta$ consists of the roots whose coordinates in the basis $\Delta$ are nonnegative.

The fundamental chamber is the connected componentof the complement of the reflecting hyperplanes. The Weyl group acts freely and transitively on its chambers.

For a root basis $\Delta$, the closed dominant chamber isEvery Weyl-group orbit meets it in exactly one point.

A highest-weight representation is generated by a weight vector killed by every positive-root subspace.

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations.

If $\lambda ={\sum }_i{m}_i{\omega }_i$ is dominant integral, the tensor product ${⨂}_{i}V({\omega }_i{)}^{\otimes m_i}$ contains $V(\lambda )$ as the irreducible summand generated by the tensor product of highest-weight vectors.

If $\mu$ and $\lambda$ are weights of a finite-dimensional representation, $\mu$ is dominant, and $\mu ⪯\lambda$, thenSuccessively subtract a simple root ${\alpha }_i$ for which $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ and use injectivity of the corresponding ${\mathfrak{sl}}_2$ lowering operator.

A vector $v$ has weight $\mu \in {\mathfrak{t}}^{\ast }$ when $hv=\mu (h)v$ for every $h$ in the Cartan subalgebra.

A weight of a representation is a functional $\mu$ whose weight space is nonzero.

The weight space $V_{\mu }$ consists of all weight vectors of weight $\mu$, together with zero.

The multiplicity of a weight $\mu$ is the dimension of its weight space.

A singular vector is a nonzero weight vector annihilated by the positive nilpotent subalgebra ${\mathfrak{n}}^{+}$.

For a Borel subalgebra $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$ and a weight $\lambda$, the Verma module iswhere ${\mathfrak{n}}^{+}$ kills ${\mathbb{C}}_{\lambda }$ and $h\in \mathfrak{t}$ acts by $\lambda (h)$.

For a module with finite-dimensional weight spaces, its formal character is $\mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{\mu }$ in the completed group algebra of the weight lattice.

For a dominant integral weight $\lambda$, the irreducible character is

The dimension obtained by evaluating the Weyl character formula at the identity is

For the principal specialization, let $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so every simple root takes value two on $h_{\mathrm{pr}}$. The q-character isIt is obtained from the Weyl character formula by substituting $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.

A crystal basis is the combinatorial limit of a basis of a quantum-group representation. Its colored directed graph records the actions of the Kashiwara lowering operators.

On a crystal, ${\tilde{f}}_i$ follows an edge of color $i$ and ${\tilde{e}}_i$ follows that edge backwards.

The tensor product crystal uses the $i$-string data ${\epsilon }_i$ and ${\phi }_i$ to decide on which tensor factor a Kashiwara operator acts. Its connected components are highest-weight crystals and encode the decomposition of the tensor product representation.

The quadratic Casimir element is a central element of $U(\mathfrak{g})$. On a highest-weight module of highest weight $\lambda$, one standard normalization makes it act by $|\lambda +\rho {|}^2-|\rho {|}^2$.

A principal sl2 subalgebra of a semisimple Lie algebra has semisimple generator $h_{\mathrm{pr}}=2{\rho }^{\vee }$. Restriction to it packages a representation's weights into ordinary ${\mathfrak{sl}}_2$ weight strings.