Semisimple Lie algebra

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

A semisimple Lie algebra over a field of characteristic zero is split if it has a Cartan subalgebra whose adjoint action is simultaneously diagonalizable over that field. Its root-space decomposition then needs no extension of scalars. The sl2R Lie algebra is split, with the real diagonal Cartan subalgebra. The real special unitary Lie algebra $\mathfrak{su}(2)$ is not: its adjoint operators are skew-adjoint for the positive definite inner product obtained by negating its Killing form. Thus a split adjoint operator would have to be zero.

A compact real form of a complex semisimple Lie algebra $\mathfrak{g}$ is a real Lie subalgebra $\mathfrak{u}$ whose complexification of a Lie algebra is $\mathfrak{g}$ and whose Killing form is negative definite. Its simply connected Lie group is compact. The positive internal inner product used in unitary gauge theory is therefore $-\kappa$. For ${\mathfrak{sl}}_2(\mathbb{C})$ the SU(2) Lie algebra is a compact real form, whereas ${\mathfrak{sl}}_2(\mathbb{R})$ is a different real form with indefinite Killing form.

A Lie algebra $L$ is perfect when $[L,L]=L$. Every semisimple Lie algebra over a field of characteristic zero is perfect.

Perfectness is a property of a Lie algebra, defined by equality with its commutator algebra; semisimplicity is a sufficient condition in characteristic zero.

Every finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.

For a finite-dimensional continuous complex representation $R$ of a compact group, normalize Haar measure to mass one and average any positive-definite Hermitian inner product:The resulting inner product is positive-definite and invariant. If $W$ is an invariant subspace, then $W^{\perp }$ is invariant as well, since $⟨R(g)v,w{⟩}_K=⟨v,R(g^{-1})w{⟩}_K$. Thus $V=W\oplus W^{\perp }$. Restriction to a compact real form, followed by integration of a Lie-algebra representation, proves the Weyl complete reducibility theorem.

If a submodule $U$ of a Lie algebra representation $V$ admits an intertwining map $P:V\to U$ with $P{|}_U=I_U$, then $P^2=P$ and its kernel is an invariant complement. For a complex semisimple Lie algebra, the Hom representation on maps $V\to U$ whose restriction is a scalar identity reduces finding $P$ to splitting of a trivial quotient for a semisimple Lie algebra.

In a finite-dimensional short exact sequence $0\to N\to E\to \mathbb{C}\to 0$ of representations of a complex semisimple Lie algebra, the generalized zero eigenspace of the Casimir operator maps onto the trivial quotient. All its irreducible composition factors have zero Casimir eigenvalue, so they are trivial. The action is therefore strictly upper triangular and has solvable image; because a semisimple Lie algebra is a perfect Lie algebra, that image is zero. Every lift in this generalized zero eigenspace is invariant, giving a split short exact sequence.

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.

For $n\ge 2$ and $m\ge 0$, the representation ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of ${\mathfrak{sl}}_n(\mathbb{C})$ is irreducible with highest weight $m{\omega }_1$. Its monomial basis supplies weights by adding $m$ defining-representation weights, with repeated weights counted separately.

For the defining three-dimensional special linear Lie algebra representation $V$, contraction $(uv)\otimes f\mapsto f(u)v+f(v)u$ maps $(S^{2}V)\otimes V^{\ast }$ onto $V$. The kernel contains the highest-weight vector $e_1^2\otimes e_3^{\ast }$ of Dynkin labels $(2,1)$. The Weyl dimension formula gives its irreducible module dimension $15$, the entire kernel dimension; the Weyl complete reducibility theorem supplies the displayed splitting.

The formal character identity ${\chi }_{{\wedge }^{2}U}(x)=\frac{1}{2}({\chi }_U(x{)}^2-{\chi }_U(x^2))$ counts unordered pairs of distinct basis vectors. Substituting ${\chi }_{2,1}=h_2(x)h_1(x^{-1})-h_1(x)$ and the Weyl character formula gives the displayed multiplicity-free decomposition. Its dimensions sum to $21+42+24+15+3=105=\left(\genfrac{}{}{0ex}{}{15}{2}\right)$. The sl3 interlacing character formula provides an exact weight-enumeration check.

For the special linear Lie algebra $\mathfrak{sl}(3)$, with Dynkin labels indexing irreducible highest-weight representations,The summands have dimensions $60,24,15,15,6$, summing to $120=15\cdot 16/2$. The identity follows from the formal character of a weight module formula ${\chi }_{S^{2}W}(z)=({\chi }_W(z{)}^2+{\chi }_W(z^2))/2$ and the Weyl character formula.

Over a field of characteristic different from two, any Lie algebra ideal in ${\mathfrak{sl}}_n$ containing a nonscalar matrix is the whole special linear Lie algebra. Indeed, if $A_{ji}\ne 0$ for $i\ne j$, then $[E_{ij},[E_{ij},A]]=-2A_{ji}{E}_{ij}$ extracts a matrix unit. For a nonscalar diagonal matrix, $[E_{ij},A]=(A_{jj}-A_{ii})E_{ij}$ extracts one instead. Its bracket with $E_{ji}$ produces $H=E_{ii}-E_{jj}$, and $[H,E_{ji}]=-2E_{ji}$ extracts the reverse unit. Brackets $[E_{ij},E_{jk}]=E_{ik}$ and $[E_{ki},E_{ij}]=E_{kj}$ spread these units to every row and column; opposite pairs span all trace-zero diagonal matrices. Thus only ideals consisting of scalar matrices can be proper. This proof works even when restrictions of distinct roots of a root system to a Cartan subalgebra coincide in positive characteristic.

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

For the special linear Lie algebra ${\mathfrak{sl}}_3(\mathbb{C})$ with upper triangular positive roots, the simple coroots are $\mathrm{diag}(1,-1,0)$ and $\mathrm{diag}(0,1,-1)$. Their dual fundamental weights are ${\omega }_1(\mathrm{diag}(t_1,t_2,t_3))=t_1$ and ${\omega }_2(\mathrm{diag}(t_1,t_2,t_3))=t_1+t_2=-t_3$. In simple-root coordinates, ${\omega }_1=(2{\alpha }_1+{\alpha }_2)/3$ and ${\omega }_2=({\alpha }_1+2{\alpha }_2)/3$. The primitive vectors $e_1\in {\mathbb{C}}^3$ and $e_1\wedge e_2\in {\Lambda }^2{\mathbb{C}}^3$ realize these Fundamental representations of sl3.

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 $[e,f]=h$.

This is the $n=2$ instance of the special linear Lie algebra. Classifying its modules is a separate representation-theory problem.

The real traceless two-by-two matrices form a three-dimensional Lie algebra. A basis adapted to conformal mechanics obeys $\{D,E\}=-E$, $\{D,K\}=K$, $\{E,K\}=2D$.

The SU(2) Lie algebra is the compact real form of ${\mathfrak{sl}}_2(\mathbb{C})$. In the Hermitian physics convention its generators satisfy $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$.

In the anti-Hermitian basis $T^a=-i{\sigma }^a/2$, the SU(2) Lie algebra has $[T^a,T^b]={\epsilon }^{abc}{T}^c$ and Killing form $\kappa (T^a,T^b)=-2{\delta }^{ab}$. This is negative definite; the positive internal form for compact Yang-Mills theory is its negative.

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.

In the sl2 Lie algebra, if $Xv=0$ and $Hv=mv$, then the commutator relations give $H{Y}^{j}v=(m-2j)Y^{j}v$ and $X{Y}^{k+1}v=(k+1)(m-k)Y^{k}v$. Induct using $XY=YX+H$. This controls the raising operator on a string formed by the lowering operator and yields the classification of finite-dimensional sl2 representations.

In the irreducible sl2 Lie algebra module $V(m)$, use $v_j=f^j{v}_0$, with $0\le j\le m$. The Lie-invariant bilinear form $B(v_j,v_k)=(-1{)}^j{1}_{\{j+k=m\}}$ is nondegenerate: its anti-diagonal is nonzero. Invariance follows from $h{v}_j=(m-2j)v_j$, $f{v}_j=v_{j+1}$ and $e{v}_j=j(m-j+1)v_{j-1}$. Its transpose equals $(-1{)}^{m}B$, so it is a symmetric bilinear form when $m$ is an even number, and an alternating bilinear form when $m$ is an odd integer. The Weyl complete reducibility theorem gives a nondegenerate invariant form on any finite-dimensional module by taking the orthogonal direct sum of these forms.

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 $M_{-m-2}$.

An intermediate-series ${\mathfrak{sl}}_2$-module has one-dimensional weight spaces indexed by all integers. In the basis $(w_i{)}_{i\in \mathbb{Z}}$, one useful normalization isIt is irreducible exactly when every raising coefficient is nonzero.

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

The Lie algebra of the compact symplectic group consists of $X^{†}=-X$ and $XJ+J{X}^T=0$. Equivalently $X=\left(\begin{array}{cc}A& B\\ -\overline{B}& \overline{A}\end{array}\right)$ with $A^{†}=-A$ and $B^T=B$. Its real dimension is $n(2n+1)$, and its complexification is the symplectic Lie algebra.

In its block form, a matrix-unit basis of the unitary Lie algebra spans the anti-Hermitian block $A$, with the corresponding conjugate block below. Independent real and imaginary symmetric matrix units span $B$, with lower block $-\overline{B}$. These give $n^2+n(n+1)$ independent real generators, including diagonal symmetric entries.

For the symplectic Lie algebra $\mathfrak{sp}(4)$ preserving $J=\left(\begin{array}{cc}0& I_2\\ -I_2& 0\end{array}\right)$, write $h_1=E_{11}-E_{33}$ and $h_2=E_{22}-E_{44}$. Its positive roots of a root system have sl2 subalgebra associated with a root triplesThe matrix unit commutator identity verifies $[X_{\alpha },Y_{\alpha }]=H_{\alpha }$ and the two eigenvalue relations $[H_{\alpha },X_{\alpha }]=2X_{\alpha }$, $[H_{\alpha },Y_{\alpha }]=-2Y_{\alpha }$.

Over $\mathbb{C}$, the symplectic Lie algebra on a four-dimensional symplectic vector space acts faithfully on its five-dimensional primitive exterior square, preserving its symmetric bilinear form. Faithfulness follows because an operator killing all wedges $e_i\wedge e_j$ has all off-diagonal coefficients zero and satisfies $A_{ii}+A_{jj}=0$ for every pair, hence is zero. The source and target Lie algebras both have dimension ten, so this faithful map is an isomorphism to the Special orthogonal Lie algebra ${\mathfrak{so}}_5$.

For the defining four-dimensional representation $V$ of the symplectic Lie algebra ${\mathfrak{sp}}_4$, with the short simple root numbered first, its highest weight is ${\omega }_1$. Its tensor square decomposes aswith dimensions $10,5,1$. The first summand is the symmetric square; the other two form the exterior square, split by symplectic contraction of an exterior square.

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

In an orthonormal basis $({\epsilon }_1,{\epsilon }_2)$, the roots are $\pm ({\epsilon }_1-{\epsilon }_2)$, $\pm ({\epsilon }_1+{\epsilon }_2)$, $\pm 2{\epsilon }_1$, and $\pm 2{\epsilon }_2$. Choose simple roots ${\alpha }_1={\epsilon }_1-{\epsilon }_2$ and ${\alpha }_2=2{\epsilon }_2$, with ${\alpha }_1$ a short root. The fundamental weights are ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$.

The $C_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm 2{\epsilon }_i$ for $1\le i\le 3$. One root basis is ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3=2{\epsilon }_3$. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is six-dimensional, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3$.

The C3 root system consists of $\pm 2e_i$ and $\pm e_i\pm e_j$ in ${\mathbb{R}}^3$. Its Weyl group is the signed symmetric group on three coordinates, of order $48$. The reflecting planes are $x_i=0$ and $x_i=\pm x_j$. The open Weyl chamber $x_1>x_2>x_3>0$ has closure generated by $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. Sign choices and absolute-coordinate orderings give its $48$ translates. On the unit sphere the chamber is a spherical triangle with angles $\pi /2$, $\pi /3$, $\pi /4$, and hence area $\pi /12$; the areas of all $48$ triangles sum to $4\pi$.

For the C3 root system, take ${\alpha }_1=e_1-e_2$, ${\alpha }_2=e_2-e_3$, ${\alpha }_3=2e_3$. With columns indexing coroots, the Cartan matrix isThe convention with rows indexing coroots gives $A^T$. Both describe a double bond between the short root ${\alpha }_2$ and long root ${\alpha }_3$, with arrow toward ${\alpha }_2$.

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

Differentiating $R^{T}R=I$ gives $\mathfrak{so}(3)=\{A\in M_3(\mathbb{R}):A^T=-A\}$. The generators $L_{a}v=e_a\times v$ satisfy $[L_a,L_b]={ϵ}_{abc}{L}_c$ by the cross product identity. This identifies it with the SU(2) Lie algebra by matching a skew-Hermitian Pauli basis.

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

For $n\ge 2$, the Bn root system has simple roots ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n={\epsilon }_n$. Its Dynkin diagram is a chain with a double terminal bond pointing toward short ${\alpha }_n$. The highest root is $\theta ={\epsilon }_1+{\epsilon }_2$. In the Extended Dynkin diagram, ${\alpha }_0=-\theta$ attaches by a single bond to ${\alpha }_2$ for $n\ge 3$. For $n=2$, both long nodes $0,1$ attach by double bonds to short node $2$. Rank one is the rank-one root system; its affine diagram has two equal-length nodes joined by a double bond without a short-root arrow.

The $B_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm {\epsilon }_i$ for $1\le i\le 3$. Its root basis ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3={\epsilon }_3$ exhibits it as the dual root system of the C3 root system. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is the seven-dimensional vector representation, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3,0$.

The complexified so4 Lie algebra consists of four-by-four complex skew-symmetric matrices and has dimension six. Its root system is $D_2=A_1\times A_1$, reflecting its decomposition into two simple rank-one ideals.

The self-dual and anti-self-dual combinations of the skew generators form two commuting copies of ${\mathfrak{sl}}_2(\mathbb{C})$. Consequentlyand its Adjoint representation decomposes as $(3,1)\oplus (1,3)$.

The spin group $\mathrm{Spin}(n)$ is the connected double cover of $SO(n)$ constructed inside the units of a Clifford algebra.

Two unit quaternions act on Euclidean four-space by $q\mapsto aq{b}^{-1}$. This norm-preserving action maps onto the SO(4) group. Its kernel is precisely $(1,1)$ and $(-1,-1)$: a kernel pair must have equal entries commuting with every quaternion. The two simply connected SU(2) factors therefore form the universal spin cover.

A spinor representation is a representation of a Spin group, or infinitesimally of a Special orthogonal Lie algebra, constructed from a Clifford algebra. In even dimensions it can split into two chiral spinor representations.

An irreducible complex Clifford algebra module in even spacetime dimension $2k$ has dimension $2^k$ and splits under the connected Spin group into two Weyl spinor modules of dimension $2^{k-1}$. In odd dimension $2k+1$, an irreducible spinor still has dimension $2^k$ but has no Weyl splitting. Reality conditions depend on signature and are additional to these complex counts.

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

For the Lorentz algebra convention $[M_{\mu \nu },M_{\rho \sigma }]=i(M_{\mu \sigma }{\eta }_{\nu \rho }+M_{\nu \rho }{\eta }_{\mu \sigma }-M_{\mu \rho }{\eta }_{\nu \sigma }-M_{\nu \sigma }{\eta }_{\mu \rho })$, choose $\eta =\mathrm{diag}(s,-s,-s,-s)$, $s=\pm 1$, ${ϵ}_{123}=1$, $J_i={ϵ}_{ijk}{M}_{jk}/2$, and $K_i=M_{0i}$. The resulting rotation bracket is $[J_i,J_j]=si{ϵ}_{ijk}{J}_k$ as well as the two displayed brackets. Changing the Minkowski metric signature while holding this generator convention fixed reverses all coefficients. A simultaneous redefinition of the generators can instead preserve the familiar form. Conventions must be specified before comparing algebra signs.

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 $(j_L,j_R)$.

For Hermitian rotations $J_i$ and boosts $K_i$, the complex combinations $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ are adjoints of one another. Although their complex brackets split into two commuting ${\mathfrak{sl}}_2$ factors, they are not two independent Hermitian compact rotation algebras. The real Lorentz spin group is $SL(2,\mathbb{C})$; $SU(2)\times SU(2)$ is a different compact real form.

The complexified Lorentz algebra splits into two commuting ${\mathfrak{sl}}_2$ factors. Extending two homogeneous polynomial representations of SU2 therefore gives every finite-dimensional irreducible complex group representation of the Lorentz spinor double cover. The kernel sign permits descent precisely when $j_L+j_R$ is an integer. These field group representations are generally nonunitary for a positive-definite inner product; infinite-dimensional unitary Lorentz group representations belong to a different construction.

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 $(1,0)\oplus (0,1)\oplus (\frac{1}{2},\frac{1}{2})$.

Identify ${\mathbb{R}}^4$ with the quaternions. Unit quaternions act by left multiplication, embedding the SU(2) group in $SO(4)$. Under this subgroup, the complexified vector representation branches as $4\to 22$, and the Adjoint representation as $6\to 3\oplus 31$. Right multiplication supplies the commuting second factor.

The long roots of a root system $\pm (1,1)$ and $\pm (1,-1)$ give the commuting $SU(2)$ factors of the standard $SO(4)$ subgroup. Choosing $(1,1)$ for the left factor pairs a weight $(x,y)$ with its coroot as $x+y$. The five-dimensional vector then gives two doublets and a singlet, whereas a short-axis root would give a triplet and two singlets.