Simple Lie algebra

, a nonnegative integer. The nonzero terminal vector is essential: choosing a later index after the ladder has already vanished gives no constraint. Applied to the Adjoint representation of a semisimple Lie algebra, this argument proves integrality of Cartan integers.

Symmetric powers of the defining sln representation ( $Sym^{m} C^{n} = V (m ω_{1})$ )

 0  0

For

n \geq 2

and

m \geq 0

, the representation

Sym^{m} C^{n}

sl_{n} (C)

is irreducible with highest weight

m ω_{1}

. Its monomial basis supplies weights by adding

m

defining-representation weights, with repeated weights counted separately.

Contraction splitting of a defining module tensor a dual symmetric square ( $V \otimes Sym^{2} V^{} = ker C \oplus ι (V^{})$ )

 0  0

For the defining

n

-dimensional special linear Lie algebra module, with

n \geq 2

, write

T_{i} (p) = e_{i} \otimes p (y_{1}, \dots, y_{n})

. The equivariant contraction is

C (T_{i} (p)) = \partial_{y_{i}} p

. The map

ι (f) = \sum_{i} e_{i} \otimes y_{i} f

is also equivariant, since

\sum_{i} e_{i} \otimes y_{i}

is an invariant identity tensor. Direct differentiation gives

C ι = (n + 1) I

, so the displayed direct sum follows. Its trace-free part is an irreducible representation of highest weight

ω_{1} + 2 ω_{n - 1}

. Its highest-weight vector is

e_{1} \otimes y_{n}^{2}

; the Weyl dimension formula gives dimension

n (n - 1) (n + 2) /2

, exactly

dim ker C

// Target: semisimple-lie-algebra.bigb

sl3 decomposition of the symmetric-square dual tensor product ( $(S^{2} V) \otimes V^{*} ≅ Γ_{2, 1} \oplus Γ_{1, 0}$ )

 0  0

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^{*}

onto

V

. The kernel contains the highest-weight vector

e_{1}^{2} \otimes e_{3}^{*}

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.

Weight basis of the trace-free symmetric-square dual tensor module ( $ker (S^{2} W \otimes W^{*} \to W)$ )

 0  0

Let

t_{ij} = (e_{i} \otimes e_{j} + e_{j} \otimes e_{i}) /2

, and let

f_{i}

be the dual basis. The equivariant contraction sends

t_{ij} \otimes f_{k}

δ_{ik} e_{j} + δ_{jk} e_{i}

. Its kernel has a weight basis consisting of

t_{ii} \otimes f_{k}

for

i \neq = k

t_{ij} \otimes f_{k}

for three distinct indices with

i < j

, and

t_{ij} \otimes f_{j} - \frac{1}{2} t_{ii} \otimes f_{i}

for

i \neq = j

. The three families have sizes six, three and six. This is the irreducible fifteen-dimensional highest-weight representation with highest weight

2 L_{1} - L_{3}

Exterior square of the sl3 representation of highest weight (2,1) ( $⋀^{2} Γ_{2, 1} ≅ Γ_{5, 0} \oplus Γ_{2, 3} \oplus Γ_{3, 1} \oplus Γ_{1, 2} \oplus Γ_{0, 1}$ )

 0  0

The formal character identity

χ_{\land^{2} U} (x) = \frac{1}{2} (χ_{U} (x)^{2} - χ_{U} (x^{2}))

counts unordered pairs of distinct basis vectors. Substituting

χ_{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 = (2 15)

. The sl3 interlacing character formula provides an exact weight-enumeration check.

Symmetric square of the sl3 representation of highest weight (2,1) ( $S^{2} Γ_{2, 1}$ )

 0  0

For the special linear Lie algebra

sl (3)

, with Dynkin labels indexing irreducible highest-weight representations,

S^{2} Γ_{2, 1} ≅ Γ_{4, 2} \oplus Γ_{3, 1} \oplus Γ_{0, 4} \oplus Γ_{1, 2} \oplus Γ_{2, 0} .

(1)

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

χ_{S^{2} W} (z) = (χ_{W} (z)^{2} + χ_{W} (z^{2})) /2

and the Weyl character formula.

Matrix-unit extraction lemma for special linear ideals

 0  0

Over a field of characteristic different from two, any Lie algebra ideal in

sl_{n}

containing a nonscalar matrix is the whole special linear Lie algebra. Indeed, if

A_{ji} \neq = 0

for

i \neq = j

, then

[E_{ij}, [E_{ij}, A]] = - 2 A_{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}] = - 2 E_{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.

Fundamental representations of sl3 ( $V (ω_{1}) = C^{3}, V (ω_{2}) = Λ^{2} C^{3}$ )

 0  0

The two fundamental representations of

sl_{3}

are the defining representation

V (ω_{1}) = C^{3}

and its dual

V (ω_{2}) = Λ^{2} C^{3} ≅ (C^{3})^{*} .

(1)

Two conjugate SU(3) triplets with a triplet ( $\overline{3} \otimes \overline{3} \otimes 3 = 2 \overline{3} \oplus 6 \oplus V_{(1, 2)}$ )

 0  0

Split the two dual factors into antisymmetric and symmetric tensors. The invariant Levi-Civita symbol identifies the antisymmetric pair with

3

, giving

6 \oplus \overline{3}

after multiplying by the remaining triplet. On the symmetric part, contraction

T^{a}_{ba}

maps onto

\overline{3}

; the equivariant splitting is

T^{a}_{b c} = (δ_{b}^{a} v_{c} + δ_{c}^{a} v_{b}) /4

. Its trace-free kernel has dimension

18 - 3 = 15

and highest weight with Dynkin labels

(1, 2)

. The Weyl dimension formula gives dimension

15

, so this is exactly the irreducible kernel. It is conjugate to the other fifteen-dimensional module with labels

(2, 1)

; dimension alone does not distinguish the two.

Tensor cube of the defining SU(3) representation ( $3^{\otimes 3} = 10 \oplus 8 \oplus 8 \oplus 1$ )

 0  0

The tensor product of Lie algebra representations

3 \otimes 3

splits into the symmetric

6

and antisymmetric

\overline{3}

. Then

6 \otimes 3 = 10 \oplus 8

and

\overline{3} \otimes 3 = 8 \oplus 1

, as highest-weight character subtraction verifies. The decuplet is fully symmetric under quark permutations, the singlet fully antisymmetric, and the two octets constitute the two-dimensional mixed permutation multiplicity space. Applying the Pauli constraint on three-quark flavour multiplets gives one ground-state spin-one-half octet and one spin-three-halves decuplet.

// Target: physics.bigb

Fundamental weights of sl3 ( $ω_{1} = ε_{1}, ω_{2} = - ε_{3}$ )

 0  0

For the special linear Lie algebra

sl_{3} (C)

with upper triangular positive roots, the simple coroots are

diag (1, - 1, 0)

and

diag (0, 1, - 1)

. Their dual fundamental weights are

ω_{1} (diag (t_{1}, t_{2}, t_{3})) = t_{1}

and

ω_{2} (diag (t_{1}, t_{2}, t_{3})) = t_{1} + t_{2} = - t_{3}

. In simple-root coordinates,

ω_{1} = (2 α_{1} + α_{2}) /3

and

ω_{2} = (α_{1} + 2 α_{2}) /3

. The primitive vectors

e_{1} \in C^{3}

and

e_{1} \land e_{2} \in Λ^{2} C^{3}

realize these Fundamental representations of sl3.

Triple tensor decomposition for the defining sl3 representation ( $V \otimes V \otimes V^{*}$ )

 0  0

For the defining representation

V = V (ω_{1})

sl_{3}

V \otimes V \otimes V^{*} ≅ V (2 ω_{1} + ω_{2}) \oplus V (2 ω_{2}) \oplus V (ω_{1})^{\oplus 2} .

(1)

sl2 Lie algebra ( $sl_{2}$ )

 0  0

The Lie algebra

sl_{2}

has generators

e, h, f

with

[h, e] = 2 e

[h, f] = - 2 f

, 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.

Weyl reflection lift in an sl2 representation ( $s = exp (e) exp (- f) exp (e)$ )

 0  0

In a finite-dimensional sl2 Lie algebra representation, raising and lowering operators are nilpotent operators, so their exponentials are finite sums. In the defining two-dimensional representation, the displayed product is

(0 - 1 10)

. It sends

h

- h

by conjugation, lifting the nontrivial Weyl group reflection. Its square acts on the irreducible highest-weight module

L (n) = Sym^{n} C^{2}

(- 1)^{n}

. Thus a Weyl reflection can have a representative of order four on a representation.

// Target: linear-operator-theory.bigb

Explicit weight basis of the exterior square of Sym4 for sl2 ( $Λ^{2} (Sym^{4} C^{2}) = L (6) \oplus L (2)$ )

 0  0

Let

e_{i} = x^{4 - i} y^{i}

and

w_{ij} = e_{i} \land e_{j}

for

i < j

. For the sl2 Lie algebra, the raising and lowering operators are

E = x \partial_{y}

F = y \partial_{x}

, and

w_{ij}

has weight

8 - 2 (i + j)

. A weight basis of the irreducible representation

L (6)

w_{01}, w_{02}, w_{03} + 2 w_{12}, w_{04} + 8 w_{13}, w_{14} + 2 w_{23}, w_{24}, w_{34} .

(1)

A weight basis of the complementary

L (2)

w_{03} - 3 w_{12}, w_{04} - 2 w_{13}, w_{14} - 3 w_{23} .

(2)

The first vector in each list is a highest-weight vector. Applying

F

along the first list gives successive coefficients

3, 2, 1, 12, 5, 2

; along the second it gives

1, 2

. The final vectors are killed by

F

. At the overlapping weights

2, 0, - 2

, the two listed vectors are independent, proving the direct sum.

// Target: semisimple-lie-algebra.bigb

sl2R Lie algebra ( $sl (2, R)$ )

 0  0

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} = 2 D

SU(2) Lie algebra ( $su (2)$ )

 0  0

The SU(2) Lie algebra is the compact real form of

sl_{2} (C)

. In the Hermitian physics convention its generators satisfy

[J_{i}, J_{j}] = i ϵ_{ijk} J_{k}

Killing form of the SU(2) Lie algebra ( $κ (T^{a}, T^{b}) = - 2 δ^{ab}$ )

 0  0

In the anti-Hermitian basis

T^{a} = - i σ^{a} /2

, the SU(2) Lie algebra has

[T^{a}, T^{b}] = ε^{ab c} T^{c}

and Killing form

κ (T^{a}, T^{b}) = - 2 δ^{ab}

. This is negative definite; the positive internal form for compact Yang-Mills theory is its negative.

Killing-normalized SU(2) quadratic Casimir ( $C_{κ} = \frac{1}{2} j (j + 1) I$ )

 0  0

With Hermitian angular-momentum generators

[J_{a}, J_{b}] = i ϵ_{ab c} J_{c}

, the Killing form is

2 δ_{ab}

and its inverse contraction is

C_{κ} = \frac{1}{2} \sum_{a} J_{a}^{2}

. Equivalently the anti-Hermitian basis

T_{a} = - i J_{a}

has Killing form

- 2 δ_{ab}

and gives the same operator. In the complex highest-weight basis,

C_{κ} = H^{2} /8 + (E^{+} E^{-} + E^{-} E^{+}) /4

. It acts on spin

j

j (j + 1) /2

, so it differs in normalization from the usual angular-momentum quadratic Casimir operator.

Classification of finite-dimensional sl2 representations

 0  0

For every

n \geq 0

there is one irreducible

sl_{2} (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.

Clebsch-Gordan splitting of a defining module and its cubic symmetric power ( $L (1) \otimes L (3) = L (4) \oplus L (2)$ )

 0  0

For the defining sl2 Lie algebra module with basis

x, y

, put

A_{k} = x \otimes x^{3 - k} y^{k}

and

B_{k} = y \otimes x^{3 - k} y^{k}

. The raising operator has

e A_{k} = k A_{k - 1}

and

e B_{k} = A_{k} + k B_{k - 1}

; the lowering operator has

f A_{k} = B_{k} + (3 - k) A_{k + 1}

and

f B_{k} = (3 - k) B_{k + 1}

. The highest-weight vectors

A_{0}

and

B_{0} - A_{1}

generate irreducible submodules of highest weights four and two. Bases are respectively

(A_{0}, 3 A_{1} + B_{0}, A_{2} + B_{1}, A_{3} + 3 B_{2}, B_{3})

and

(B_{0} - A_{1}, B_{1} - A_{2}, B_{2} - A_{3})

. Their vectors are independent in every weight space, proving the displayed direct sum without only comparing dimensions.

Parity unimodality of an sl2 character ( $c_{j} \geq c_{j + 2} (j \geq 0)$ )

 0  0

For a finite-dimensional sl2 Lie algebra representation, write

χ (q) = \sum_{j} c_{j} q^{j}

. Each irreducible summand has character

q^{m} + q^{m - 2} + \dots + q^{- m}

, so

c_{j} = c_{- j}

and

c_{j} \geq c_{j + 2}

for nonnegative

j

. Thus each parity subsequence is symmetric and unimodal. If all weights have one parity, removing the unused exponents gives an ordinarily unimodal coefficient sequence. The full sequence including every integer exponent need not be unimodal:

q + q^{- 1}

has coefficients

1, 0, 1

sl2 highest-weight lowering formula ( $X Y^{k + 1} v = (k + 1) (m - k) Y^{k} v$ )

 0  0

In the sl2 Lie algebra, if

X v = 0

and

H v = m v

, then the commutator relations give

H Y^{j} v = (m - 2 j) Y^{j} v

and

X Y^{k + 1} v = (k + 1) (m - k) Y^{k} v

. Induct using

X Y = Y X + H

. This controls the raising operator on a string formed by the lowering operator and yields the classification of finite-dimensional sl2 representations.

Trace of a raising-lowering product in an irreducible sl2 module ( $tr (X Y) = k (k + 1) (k + 2) /6$ )

 0  0

For a highest-weight vector

v

of weight

k

in an irreducible finite-dimensional sl2 Lie algebra module, the basis

v, Y v, \dots, Y^{k} v

diagonalizes

X Y

, with eigenvalue

(j + 1) (k - j)

Y^{j} v

. Summing those eigenvalues gives

tr (X Y) = k (k + 1) (k + 2) /6

. The formula

X Y^{n} v = n (k - n + 1) Y^{n - 1} v

requires

v

to be a highest-weight vector, not an arbitrary vector of the module.

Invariant form on an irreducible sl2 module

 0  0

In the irreducible sl2 Lie algebra module

V (m)

, use

v_{j} = f^{j} v_{0}

, with

0 \leq j \leq 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 - 2 j) 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.

Injectivity of sl2 lowering above weight zero

 0  0

On any finite-dimensional

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.

Reducibility of an sl2 Verma module ( $M_{λ}$ reducible iff $λ \in Z_{\geq 0}$ )

 0  0

The Verma module

M_{λ}

has basis

(f^{r} v_{λ})_{r \geq 0}

and

e f^{r} v_{λ} = r (λ - r + 1) f^{r - 1} v_{λ} .

(1)

It is reducible exactly when

λ = m \in Z_{\geq 0}

; then its unique proper nonzero submodule is generated by

f^{m + 1} v_{λ}

and is isomorphic to

M_{- m - 2}

Nonsplit zero-weight Verma extension for sl2 ( $0 \to M (- 2) \to M (0) \to C \to 0$ )

 0  0

The Verma module

M (0)

has basis

v_{n} = f^{n} v_{0}

, with

h v_{n} = - 2 n v_{n}

f v_{n} = v_{n + 1}

and

e v_{n} = n (1 - n) v_{n - 1}

. Its submodule spanned by

v_{n}

for

n \geq 1

M (- 2)

, and the quotient is the trivial one-dimensional module. A complementary submodule would be trivial and hence lie in the zero-weight space

C v_{0}

, but

f v_{0} = v_{1} \neq = 0

. Therefore the displayed extension does not split. This explains why the finite-dimensional hypothesis in the Weyl complete reducibility theorem is essential.

Intermediate-series sl2 module

 0  0

An intermediate-series

sl_{2}

-module has one-dimensional weight spaces indexed by all integers. In the basis

(w_{i})_{i \in Z}

, one useful normalization is

h w_{i} = (a + 2 i) w_{i}, f w_{i} = w_{i - 1}, e w_{i} = (b - ia - i (i + 1)) w_{i + 1} .

(1)

It is irreducible exactly when every raising coefficient is nonzero.

Symplectic Lie algebra ( $sp_{2 n}$ )

 0  0

For a nondegenerate alternating matrix

J

, the symplectic Lie algebra is

sp_{2 n} = {A : A J + J A^{T} = 0}

Matrix root basis of the symplectic Lie algebra ( $sp_{2 l} = H \oplus ⨁_{r \in C_{l}} C e_{r}$ )

 0  0

Relative to

J = (0 - I I 0)

, a symplectic Lie algebra matrix has blocks

(P R Q - P^{T})

with

Q, R

symmetric. Its diagonal Cartan subalgebra has basis

E_{ii} - E_{l + i, l + i}

. Write

ε_{i} (h) = λ_{i}

. A root-space decomposition is obtained from

e_{ε_{i} - ε_{j}} e_{ε_{i} + ε_{j}} e_{- ε_{i} - ε_{j}} e_{2 ε_{i}} = E_{ij} - E_{l + j, l + i} = E_{i, l + j} + E_{j, l + i} = E_{l + i, j} + E_{l + j, i} = E_{i, l + i}, e_{- 2 ε_{i}} = E_{l + i, i} . (i \neq = j), (i < j), (i < j),

(1)

The matrix unit rule

[h, E_{ab}] = (h_{aa} - h_{bb}) E_{ab}

proves

[h, e_{r}] = r (h) e_{r}

. These root vectors and the diagonal basis span all the permitted blocks and are independent. There are

2 l^{2}

roots and

l

diagonal basis elements, giving dimension

l (2 l + 1)

Tensor-square decomposition of the defining symplectic representation ( $V \otimes V = L (2 ω_{1}) \oplus L (ω_{2}) \oplus L (0)$ )

 0  0

For

n \geq 2

, the defining Lie algebra representation

V

sp_{2 n}

has the displayed tensor product decomposition. The crystal of the defining symplectic representation and the crystal tensor-product rule give exactly these three highest vertices, each occurring once. The symmetric square is

L (2 ω_{1})

of dimension

n (2 n + 1)

. The exterior square splits into its invariant inverse-form line and the primitive exterior square

L (ω_{2})

of dimension

n (2 n - 1) - 1

. Their dimensions sum to

4 n^{2}

. At

n = 1

, the exterior square is only the invariant line, giving

L (2 ω_{1}) \oplus L (0)

Compact symplectic Lie algebra ( $usp (2 n) ≅ sp (n)$ )

 0  0

The Lie algebra of the compact symplectic group consists of

X^{†} = - X

and

X J + J X^{T} = 0

. Equivalently

X = (A - \overline{B} B \overline{A})

with

A^{†} = - A

and

B^{T} = B

. Its real dimension is

n (2 n + 1)

, and its complexification is the symplectic Lie algebra.

Matrix-unit basis of the compact symplectic Lie algebra

 0  0

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.

Symplectic root sl2 triple ( $(X_{α}, H_{α}, Y_{α})$ )

 0  0

For the symplectic Lie algebra

sp (4)

preserving

J = (0 - I_{2} I_{2} 0)

, 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 triples

α ε_{1} - ε_{2} ε_{1} + ε_{2} 2 ε_{1} 2 ε_{2} X_{α} E_{12} - E_{43} E_{14} + E_{23} E_{13} E_{24} H_{α} h_{1} - h_{2} h_{1} + h_{2} h_{1} h_{2} Y_{α} E_{21} - E_{34} E_{41} + E_{32} E_{31} E_{42}

(1)

The matrix unit commutator identity verifies

[X_{α}, Y_{α}] = H_{α}

and the two eigenvalue relations

[H_{α}, X_{α}] = 2 X_{α}

[H_{α}, Y_{α}] = - 2 Y_{α}

Exceptional isomorphism between sp4 and so5 ( $sp_{4} ≅ so_{5}$ )

 0  0

Over

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} \land 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

so_{5}

Tensor-square decomposition of the defining sp4 representation

 0  0

For the defining four-dimensional representation

V

of the symplectic Lie algebra

sp_{4}

, with the short simple root numbered first, its highest weight is

ω_{1}

. Its tensor square decomposes as

V \otimes V ≅ V (2 ω_{1}) \oplus V (ω_{2}) \oplus V (0),

(1)

with 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.

Cn root system ( $C_{n}$ )

 0  0

In an orthonormal basis

(ε_{i})

, the

C_{n}

roots are

\pm ε_{i} \pm ε_{j}

for

i < j

and

\pm 2 ε_{i}

Cn Dynkin diagrams ( $C_{n}, C_{n}^{(1)}$ )

 0  0

For

n \geq 2

, the finite Dynkin diagram has a chain of short roots

α_{1}, \dots, α_{n - 1}

with single bonds, followed by a double bond to the long root

α_{n}

, pointing toward

α_{n - 1}

. The Extended Dynkin diagram adds the long root

α_{0} = - 2 ε_{1}

, doubly bonded to

α_{1}

with arrow toward

α_{1}

. All other bonds remain. For

n = 1

, the finite diagram is a single vertex; the affine diagram has two equal-length vertices joined by a double bond without an arrow, with Cartan matrix

(2 - 2 - 2 2)

Positive-root data for Cn ( $α_{i} = ε_{i} - ε_{i + 1}, α_{n} = 2 ε_{n}$ )

 0  0

Choose positive roots of a root system

ε_{i} - ε_{j}

ε_{i} + ε_{j}

for

i < j

, and

2 ε_{i}

. The simple roots are

α_{i} = ε_{i} - ε_{i + 1}

for

i < n

and

α_{n} = 2 ε_{n}

. Their coroots are

α_{i}^{\lor} = ε_{i} - ε_{i + 1}

and

α_{n}^{\lor} = ε_{n}

. Therefore the fundamental weights and Weyl vector are

ω_{i} = \sum_{j = 1}^{i} ε_{j}, ρ = \sum_{j = 1}^{n} (n - j + 1) ε_{j} .

(1)

The highest root is

2 ε_{1} = 2 α_{1} + \dots + 2 α_{n - 1} + α_{n}

. These formulas include rank one.

Cn Weyl group ( $W (C_{n}) ≅ (Z /2 Z)^{n} ⋊ S_{n}$ )

 0  0

The Weyl group of the Cn root system is the signed symmetric group. A Weyl reflection in

2 ε_{i}

changes the sign of coordinate

i

; one in

ε_{i} - ε_{j}

interchanges coordinates

i, j

; one in

ε_{i} + ε_{j}

interchanges and negates both coordinates. These generate all independent sign changes and all permutations, giving order

2^{n} n!

C2 root system ( $C_{2}$ )

 0  0

In an orthonormal basis

(ε_{1}, ε_{2})

, the roots are

\pm (ε_{1} - ε_{2})

\pm (ε_{1} + ε_{2})

\pm 2 ε_{1}

, and

\pm 2 ε_{2}

. Choose simple roots

α_{1} = ε_{1} - ε_{2}

and

α_{2} = 2 ε_{2}

, with

α_{1}

a short root. The fundamental weights are

ω_{1} = ε_{1}

and

ω_{2} = ε_{1} + ε_{2}

C3 root system ( $C_{3}$ )

 0  0

The

C_{3}

roots are

\pm ε_{i} \pm ε_{j}

for

1 \leq i < j \leq 3

and

\pm 2 ε_{i}

for

1 \leq i \leq 3

. One root basis is

α_{1} = ε_{1} - ε_{2}

α_{2} = ε_{2} - ε_{3}

α_{3} = 2 ε_{3}

. With

A_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩

, its Cartan matrix is

2 - 1 0 - 1 2 - 2 0 - 1 2 .

(1)

The fundamental representation of highest weight

[1, 0, 0]

is six-dimensional, with weights

\pm ε_{1}, \pm ε_{2}, \pm ε_{3}

Rank-two subsystems of a C3 root system ( $A_{2} and C_{2}$ )

 0  0

In the realization

Φ (C_{3}) = {\pm 2 e_{i}, \pm e_{i} \pm e_{j} : i < j}

, each coordinate two-plane cuts out a C2 root system. Each plane

\pm x_{1} \pm x_{2} \pm x_{3} = 0

cuts out an A2 root system; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two root subsystems comprise three of type

C_{2}

and four of type

A_{2}

. For an

A_{2}

simple pair

p, q

, equal lengths and angle

120

degrees give roots

\pm p, \pm q, \pm (p + q)

. For a mixed-length simple pair

s, l

with

∥ l ∥^{2} = 2∥ s ∥^{2}

and

s \cdot l = - ∥ s ∥^{2}

, the B2 root system has roots

\pm s, \pm l, \pm (s + l), \pm (2 s + l)

. These lists also verify that the selected pairs are simple, rather than simply independent.

Weyl chamber geometry of C3 ( $x_{1} > x_{2} > x_{3} > 0$ )

 0  0

The C3 root system consists of

\pm 2 e_{i}

and

\pm e_{i} \pm e_{j}

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

π /2

π /3

π /4

, and hence area

π /12

; the areas of all

48

triangles sum to

4 π

Cartan matrix convention for C3 ( $A_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩$ )

 0  0

For the C3 root system, take

α_{1} = e_{1} - e_{2}

α_{2} = e_{2} - e_{3}

α_{3} = 2 e_{3}

. With columns indexing coroots, the Cartan matrix is

A_{ij} = \frac{2 ( α _{i} , α _{j} )}{( α _{j} , α _{j} )}, A = 2 - 1 0 - 1 2 - 2 0 - 1 2 .

(1)

The convention with rows indexing coroots gives

A^{T}

. Both describe a double bond between the short root

α_{2}

and long root

α_{3}

, with arrow toward

α_{2}

Special orthogonal Lie algebra ( $so_{n}$ )

 0  0

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

Exterior square realization of the orthogonal adjoint representation ( $u \land v \mapsto (w \mapsto ⟨ v, w ⟩ u - ⟨ u, w ⟩ v)$ )

 0  0

For a real inner product space

V

, the displayed map identifies

Λ^{2} V

with its Special orthogonal Lie algebra. It sends a wedge of distinct orthonormal basis vectors to the corresponding elementary skew-symmetric matrix, so it is an isomorphism. For an orthogonal transformation

g

, invariance of the inner product gives

Φ (gu \land gv) = g Φ (u \land v) g^{- 1}

. Thus the exterior square of the defining representation is the Adjoint representation of the orthogonal group. Complexification identifies their weight spaces, allowing the root system to be read from the exterior-square weights and the zero-weight multiplicity.

Exterior square of the defining orthogonal representation ( $Λ^{2} V ≅ so (V)$ )

 0  0

For a complex vector space with a nondegenerate symmetric bilinear form

Q

, the map

u \land v \mapsto [w \mapsto Q (v, w) u - Q (u, w) v]

identifies the exterior square of the defining representation with the Adjoint representation of a Lie algebra of

so (V)

. Equivariance follows from invariance of

Q

; an orthogonal basis verifies bijectivity. It is irreducible when

so (V)

is simple. In dimension four it splits into the adjoint modules of the two commuting sl2 Lie algebras.

Dn root system ( $D_{n} = {\pm e_{i} \pm e_{j} : i < j}$ )

 0  0

For the complexified Special orthogonal Lie algebra

so_{2 n}

n \geq 2

, choose positive roots of a root system

e_{i} - e_{j}

and

e_{i} + e_{j}

for

i < j

. Its Weyl vector is

ρ = (n - 1, n - 2, \dots, 0)

. Dominant coordinates satisfy

λ_{1} \geq \dots \geq λ_{n - 1} \geq ∣ λ_{n} ∣

. The Weyl dimension formula becomes

\prod_{i < j} ((λ_{i} + ρ_{i})^{2} - (λ_{j} + ρ_{j})^{2}) / (ρ_{i}^{2} - ρ_{j}^{2})

. The case

D_{2}

is two independent A1 root systems, while

D_{3}

is isomorphic to the A3 root system.

Weyl group of Dn ( $W (D_{n}) ≅ (Z /2 Z)^{n - 1} ⋊ S_{n}$ )

 0  0

The Weyl group of the Dn root system consists of coordinate permutations together with an even number of sign changes. Reflection in

ε_{i} - ε_{j}

swaps coordinates

i, j

; reflection in

ε_{i} + ε_{j}

swaps them and negates both. These generate all even signed permutations, giving order

2^{n - 1} n!

Fundamental weights of Dn ( $ω_{1}, \dots, ω_{n}$ )

 0  0

With simple roots

α_{i} = ε_{i} - ε_{i + 1}

for

i < n

and

α_{n} = ε_{n - 1} + ε_{n}

, the fundamental weights are

ω_{i} = ε_{1} + \dots + ε_{i}

for

i \leq n - 2

, followed by

ω_{n - 1} = (ε_{1} + \dots + ε_{n - 1} - ε_{n}) /2

and

ω_{n} = (ε_{1} + \dots + ε_{n}) /2

. All roots have squared length two, so their pairings with coroots verify the dual-basis identities. The last two weights are the half-spin highest weights; for

n = 2

, the defining highest weight is

ω_{1} + ω_{2}

Matrix root basis of the even orthogonal Lie algebra

 0  0

For the antidiagonal symmetric matrix

J

in dimension

2 n

, put

\overset{ˉ}{i} = 2 n + 1 - i

and

H_{i} = E_{ii} - E_{\overset{ˉ}{i} \overset{ˉ}{i}}

. The diagonal Cartan subalgebra has coordinates

ε_{i} (H) = t_{i}

. The root spaces of weights

ε_{i} - ε_{j}

are spanned by

E_{ij} - E_{\overset{ˉ}{j} \overset{ˉ}{i}}

; those of weights

ε_{i} + ε_{j}

E_{i \overset{ˉ}{j}} - E_{j \overset{ˉ}{i}}

, with negative-root spaces given by transposes. The condition

A_{ij} = - A_{\overset{ˉ}{j} \overset{ˉ}{i}}

gives these generators directly and excludes roots

2 ε_{i}

Real four-dimensional rotation algebra splitting ( $so (4) ≅ so (3) \oplus so (3)$ )

 0  0

For antisymmetric Maurer–Cartan forms with

d λ_{ij} = λ_{ik} \land λ_{kj}

, let

α^{\pm} = (λ_{12} \pm λ_{43}, λ_{23} \pm λ_{41}, λ_{31} \pm λ_{42})

. Direct expansion gives

d α_{1}^{\pm} = - α_{2}^{\pm} \land α_{3}^{\pm}

and cyclic analogues, with no mixed terms. The dual triples form two commuting ideals, each with the SO(3) Lie algebra brackets. This is a real compact splitting, distinct from the splitting of the complexified algebra into two copies of

sl_{2} (C)

SO(3) Lie algebra ( $so (3)$ )

 0  0

Differentiating

R^{T} R = I

gives

so (3) = {A \in M_{3} (R) : A^{T} = - A}

. The generators

L_{a} v = e_{a} \times v

satisfy

[L_{a}, L_{b}] = ϵ_{ab c} L_{c}

by the cross product identity. This identifies it with the SU(2) Lie algebra by matching a skew-Hermitian Pauli basis.

Bn root system ( $B_{n}$ )

 0  0

In an orthonormal basis

(ε_{i})

, the

B_{n}

roots are

\pm ε_{i} \pm ε_{j}

for

i < j

and

\pm ε_{i}

Bn Dynkin diagram and affine extension

 0  0

For

n \geq 2

, the Bn root system has simple roots

α_{i} = ε_{i} - ε_{i + 1}

for

i < n

and

α_{n} = ε_{n}

. Its Dynkin diagram is a chain with a double terminal bond pointing toward short

α_{n}

. The highest root is

θ = ε_{1} + ε_{2}

. In the Extended Dynkin diagram,

α_{0} = - θ

attaches by a single bond to

α_{2}

for

n \geq 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.

B3 root system ( $B_{3}$ )

 0  0

The

B_{3}

roots are

\pm ε_{i} \pm ε_{j}

for

1 \leq i < j \leq 3

and

\pm ε_{i}

for

1 \leq i \leq 3

. Its root basis

α_{1} = ε_{1} - ε_{2}

α_{2} = ε_{2} - ε_{3}

α_{3} = ε_{3}

exhibits it as the dual root system of the C3 root system. With

A_{ij} = ⟨ α_{i}, α_{j}^{\lor} ⟩

, its Cartan matrix is

2 - 1 0 - 1 2 - 1 0 - 2 2 .

(1)

The fundamental representation of highest weight

[1, 0, 0]

is the seven-dimensional vector representation, with weights

\pm ε_{1}, \pm ε_{2}, \pm ε_{3}, 0

so4 Lie algebra ( $so_{4}$ )

 0  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.

Chiral decomposition of the complexified so4 Lie algebra ( $so_{4} (C) ≅ sl_{2} (C) \oplus sl_{2} (C)$ )

 0  0

The self-dual and anti-self-dual combinations of the skew generators form two commuting copies of

sl_{2} (C)

. Consequently

so_{4} (C) ≅ sl_{2} (C) \oplus sl_{2} (C),

(1)

and its Adjoint representation decomposes as

(3, 1) \oplus (1, 3)

Spin group

 1  0

The spin group

Spin (n)

is the connected double cover of

SO (n)

constructed inside the units of a Clifford algebra.

Exterior-square double cover of SO0(3,2) ( $Sp (4, R) / {\pm I} ≅ S O_{0} (3, 2)$ )

 0  0

Fix a volume form on a four-dimensional real symplectic vector space. Wedge pairing on its exterior square has signature

(3, 3)

. The symplectic group fixes the bivector inverse to the symplectic form. Its orthogonal complement is five-dimensional with signature

(3, 2)

after choosing the appropriate overall sign. The induced representation has kernel

{\pm I}

: identity action on the entire exterior square preserves every two-plane and thus every line, forcing a scalar transformation with square one. Its derivative is injective and both group dimensions are ten, so its image is the identity component of

SO (3, 2)

Hermitian-matrix double cover of SO0(3,1) ( $S L (2, C) / {\pm I} ≅ S O_{0} (3, 1)$ )

 0  0

On two-by-two Hermitian matrices, the determinant of

X = t I + x \cdot σ

t^{2} - ∣ x ∣^{2}

. The action

X \mapsto A X A^{†}

preserves it for

A \in S L (2, C)

. A kernel element is unitary and commutes with every Hermitian matrix, hence is

\pm I

. The derivative is injective and both real dimensions are six, giving the full proper orthochronous Lorentz group as the connected image.

Symmetric-matrix double cover of SO0(2,1) ( $S L (2, R) / {\pm I} ≅ S O_{0} (2, 1)$ )

 0  0

Congruence

X \mapsto A X A^{T}

on real symmetric two-by-two matrices preserves the quadratic form

- det X

of signature

(2, 1)

when

det A = 1

. Its kernel is

{\pm I}

and its derivative is injective:

a X + X a^{T} = 0

for every symmetric

X

implies

a = 0

. Both Lie groups have dimension three, so the connected image is the full identity component. Also

A^{T} J A = (det A) J

identifies

Sp (2, R)

with

S L (2, R)

Exterior-square double cover of SO(3,3) ( $S L (4, R) / {\pm I} ≅ S O_{0} (3, 3)$ )

 0  0

A fixed volume form makes wedge pairing on

Λ^{2} R^{4}

a symmetric form of signature

(3, 3)

. The exterior-square action of special linear group matrices preserves it. Its kernel is

{\pm I}

: a transformation acting trivially on every bivector preserves every two-plane and hence every line, so is scalar, with scalar square one. The derivative is injective and both groups have dimension fifteen, so the connected image is the full identity component. This is a concrete real spin-group cover.

Spin(4) double cover ( $Spin (4) ≅ S U (2) \times S U (2)$ )

 0  0

Two unit quaternions act on Euclidean four-space by

q \mapsto a q 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.

Spin representation

 1  0

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.

Spinor dimension in arbitrary spacetime dimension ( $dim_{C} S_{D} = 2^{⌊ D /2 ⌋}$ )

 0  0

An irreducible complex Clifford algebra module in even spacetime dimension

2 k

has dimension

2^{k}

and splits under the connected Spin group into two Weyl spinor modules of dimension

2^{k - 1}

. In odd dimension

2 k + 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.

Lorentz algebra ( $so (1, 3)$ )

 0  0

The Lorentz algebra has rotation generators

J_{i}

and boost generators

K_{i}

satisfying

[J_{i}, J_{j}] = ϵ_{ijk} J_{k}, [J_{i}, K_{j}] = ϵ_{ijk} K_{k}, [K_{i}, K_{j}] = - ϵ_{ijk} J_{k} .

(1)

Lorentz generator ( $M_{μν} = - M_{νμ}$ )

 0  0

Lorentz generators produce infinitesimal Lorentz transformations. In four dimensions they split into three rotation and three boost generators. Their action on supercharges uses the appropriate spinor representation, while their action on translations is the mixed Poincare algebra commutator. Signs in component formulas must agree with whether the generators act on states, operator columns or dual rows.

Rotation and boost commutators with a fixed metric signature ( $[K_{i}, K_{j}] = - s i ϵ_{ijk} J_{k}, [J_{i}, K_{j}] = s i ϵ_{ijk} K_{k}$ )

 0  0

For the Lorentz algebra convention

[M_{μν}, M_{ρ σ}] = i (M_{μ σ} η_{ν ρ} + M_{ν ρ} η_{μ σ} - M_{μ ρ} η_{ν σ} - M_{ν σ} η_{μ ρ})

, choose

η = diag (s, - s, - s, - s)

s = \pm 1

ϵ_{123} = 1

J_{i} = ϵ_{ijk} M_{jk} /2

, and

K_{i} = M_{0 i}

. The resulting rotation bracket is

[J_{i}, J_{j}] = s i 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.