Lie algebra

For

sl_{n} (C)

κ (X, Y) = 2 n tr (X Y) .

Adjoint irreducibility of a simple Lie algebra

Every invariant subspace of the Adjoint representation of a Lie algebra is an ideal. Hence the adjoint representation of a Simple Lie algebra is irreducible.

Trace form of a Lie algebra representation ( $(\cdot, \cdot)_{V}$ )

For a finite-dimensional Lie algebra representation

ρ : g \to gl (V)

, its trace form is the symmetric bilinear form

(x, y)_{V} = tr (ρ (x) ρ (y)) .

It is invariant:

([x, y], z)_{V} = (x, [y, z])_{V}

Trace trilinear form of a Lie algebra representation ( $B (X, Y, Z)$ )

For a finite-dimensional representation

d

, the trilinear form

B (X, Y, Z) = Tr (d (X) d (Y) d (Z))

is invariant under simultaneous adjoint action. This follows by writing the sum of its three infinitesimal variations as the trace of a commutator.

Heisenberg Lie algebra

The three-dimensional Heisenberg Lie algebra has a basis

a, b, c

with

[a, b] = c

and

c

central. It is a two-step Nilpotent Lie algebra.

Polynomial representation of the Heisenberg Lie algebra

C [x]

, the operators

d / d x

, multiplication by

x

, and the identity satisfy

[d / d x, x] = 1

. Sending

a, b, c

to these operators gives a faithful irreducible infinite-dimensional representation of the Heisenberg Lie algebra.

Universal enveloping algebra ( $U (g)$ )

The universal enveloping algebra

U (g)

is the associative algebra generated by

g

subject to

x y - y x = [x, y]

. Its modules are the same as Lie algebra representations.

Poincare-Birkhoff-Witt theorem

For an ordered basis

(x_{i})

of a Lie algebra

g

, the ordered monomials

x_{1}^{a_{1}} \dots x_{n}^{a_{n}}

form a basis of

U (g)

. In particular, a triangular decomposition

g = n^{-} \oplus t \oplus n^{+}

gives

U (g) ≅ U (n^{-}) \otimes U (t) \otimes U (n^{+})

as vector spaces.

Semisimple Lie algebra

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

Borel subalgebra

For a choice of positive roots, the corresponding Borel subalgebra is

b = t \oplus n^{+}

, the sum of a Cartan subalgebra and all positive root spaces.

Simple Lie algebra

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

Special linear Lie algebra ( $sl_{n}$ )

The special linear Lie algebra

sl_{n}

consists of the trace-zero

n \times n

matrices with the commutator bracket.

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

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})^{*} .

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

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} .

sl2 Lie algebra ( $sl_{2}$ )

The Lie algebra

sl_{2}

has generators

e, h, f

with

[h, e] = 2 e

[h, f] = - 2 f

, and

[e, f] = h

Classification of finite-dimensional sl2 representations

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.

Injectivity of sl2 lowering above weight zero

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 $λ \in Z_{\geq 0}$ )

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} .

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}

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

For a nondegenerate alternating matrix

J

, the symplectic Lie algebra is

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

Cn root system ( $C_{n}$ )

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}

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

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

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

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} .

Chiral decomposition of the complex Lorentz algebra

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

sl_{2} (C)

, giving

so (1, 3)_{C} ≅ sl_{2} (C) \oplus sl_{2} (C) .

Finite-dimensional irreducible representations are labelled

(j_{L}, j_{R})

Parity action on a Lorentz representation

Parity fixes

J_{i}

, negates

K_{i}

, and therefore exchanges the two chiral factors. It sends

(j_{L}, j_{R})

(j_{R}, j_{L})

so5 Lie algebra ( $so_{5}$ )

The Lie algebra

so_{5}

has dimension ten and root system

B_{2}

SO5 to SO4 branching ( $SO (5) ↓ SO (4)$ )

Under the standard block-diagonal subgroup, the vector and adjoint representations branch as

5 \to 4 \oplus 1, 10 \to 6 \oplus 4 .

Using

so_{4} ≅ su_{2} \oplus 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})

B2 root system ( $B_{2}$ )

The

B_{2}

root system consists of

\pm ε_{1}

\pm ε_{2}

, and

\pm ε_{1} \pm ε_{2}

. Its vector representation has weights

ε_{1}, ε_{2}, 0, - ε_{2}, - ε_{1}

Isomorphism between so5 and sp4 ( $so_{5} (C) ≅ sp_{4} (C)$ )

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

so_{5} (C) ≅ sp_{4} (C)

Fundamental representations of B2 ( $[1, 0], [0, 1]$ )

For simple roots

(1, 0)

and

(- 1, 1)

, the fundamental weights are

ω_{1} = (1/2, 1/2)

and

ω_{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)

Cartan subalgebra

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.

Root system

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.

Positive root

A choice of positive roots selects exactly one of

α

and

- α

and is closed under addition whenever the sum is a root.

Simple root ( $α_{i}$ )

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.

Highest root ( $θ$ )

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

Half-sum of positive roots ( $ρ$ )

The Weyl vector is

ρ = \frac{1}{2} \sum_{α \in R^{+}} α

. It satisfies

⟨ ρ, α_{i}^{\lor} ⟩ = 1

for every simple root.

Coroot ( $α^{\lor}$ )

The coroot associated with a nonzero root is

α^{\lor} = 2 α / (α, α)

Root lattice ( $Q$ )

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

Dominance order ( $μ ⪯ λ$ )

For weights

μ, λ

, one writes

μ ⪯ λ

when

λ - μ

is a nonnegative integer combination of the simple roots.

Weight lattice ( $P$ )

The weight lattice is the set of vectors

λ

for which

⟨ λ, α^{\lor} ⟩

is an integer for every root

α

Dominant integral weight

An integral weight

λ

is dominant when

⟨ λ, α_{i}^{\lor} ⟩ \geq 0

for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.

Fundamental weight ( $ω_{i}$ )

The fundamental weights are dual to the simple coroots:

⟨ ω_{i}, α_{j}^{\lor} ⟩ = δ_{ij}

Fundamental representation

The fundamental representation associated with a fundamental weight

ω_{i}

is the irreducible highest-weight representation

V (ω_{i})

Weyl reflection ( $s_{α}$ )

The reflection associated with a root is

s_{α} (x) = x - ⟨ x, α^{\lor} ⟩ α

Reflection group of a root system ( $W (Φ)$ )

The reflection group of a root system

Φ

is the subgroup of the orthogonal group generated by the reflections

s_{α}

for

α \in Φ

. It permutes the roots and acts freely and transitively on the fundamental systems.

Weyl group ( $W$ )

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

Coxeter length ( $ℓ (w)$ )

The Coxeter length

ℓ (w)

is the smallest number of simple reflections whose product is

w

. Its parity gives the sign of a Weyl-group element.

Weyl stabilizer of a dominant point

λ

and

w λ

both lie in the closed dominant chamber, then

w

is a product of simple reflections whose walls contain

λ

. In particular,

w λ = λ

Inversion set of a Weyl-group element ( $N (w)$ )

For a chosen positive system of a root system

Π

, the inversion set is

N (w) = {β \in Π : w (β) \in - Π} .

For a finite Weyl group,

∣ N (w) ∣ = ℓ (w)

Positive-root criterion for Coxeter length

In the geometric representation of a Coxeter system,

ℓ (w s_{i}) > ℓ (w) ⟺ w (e_{i}) is a positive root,

and

ℓ (s_{i} w) > ℓ (w) ⟺ w^{- 1} (e_{i}) is a positive root .

(2)

Replacing “positive” by “negative” reverses either length inequality.

Dynkin diagram

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

Fundamental system of a root system ( $Δ$ )

A fundamental system

Δ

is a basis made of roots such that every root is a linear combination of

Δ

whose nonzero coefficients all have the same sign. Its members are the simple roots.

Positive system of a root system ( $Π$ )

The positive system associated with a fundamental system of a root system

Δ

consists of the roots whose coordinates in the basis

Δ

are nonnegative.

Fundamental chamber of a root system

The fundamental chamber is the connected component

C = {v : ⟨ v, α ⟩ > 0 for every α \in Δ}

of the complement of the reflecting hyperplanes. The Weyl group acts freely and transitively on its chambers.

Closed dominant Weyl chamber

For a root basis

Δ

, the closed dominant chamber is

\overline{C (Δ)} = {λ : ⟨ λ, α^{\lor} ⟩ \geq 0 for every α \in Δ} .

Every Weyl-group orbit meets it in exactly one point.

Highest-weight representation

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

Complete reducibility of semisimple Lie algebra representations

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

Generation by fundamental representations

λ = \sum_{i} m_{i} ω_{i}

is dominant integral, the tensor product

⨂_{i} V (ω_{i})^{\otimes m_{i}}

contains

V (λ)

as the irreducible summand generated by the tensor product of highest-weight vectors.

Weight multiplicity decreases away from a dominant weight

μ

and

λ

are weights of a finite-dimensional representation,

μ

is dominant, and

μ ⪯ λ

, then

mult (μ) \geq mult (λ) .

Successively subtract a simple root

α_{i}

for which

⟨ λ, α_{i}^{\lor} ⟩ > 0

and use injectivity of the corresponding

sl_{2}

lowering operator.

Singular vector

A singular vector is a nonzero weight vector annihilated by the positive nilpotent subalgebra

n^{+}

Verma module ( $M_{λ}$ )

For a Borel subalgebra

b = t \oplus n^{+}

and a weight

λ

, the Verma module is

M_{λ} = U (g) \otimes_{U (b)} C_{λ},

where

n^{+}

kills

C_{λ}

and

h \in t

acts by

λ (h)

Formal character of a weight module ( $ch V$ )

For a module with finite-dimensional weight spaces, its formal character is

ch V = \sum_{μ} (dim V_{μ}) e^{μ}

in the completed group algebra of the weight lattice.

q-character of a highest-weight representation ( $ch_{q} L_{λ}$ )

For the principal specialization, let

h_{pr} = 2 ρ^{\lor}

, so every simple root takes value two on

h_{pr}

. The q-character is

ch_{q} L_{λ} = \sum_{μ} (dim L_{λ} [μ]) q^{μ (h_{pr})} .

It is obtained from the Weyl character formula by substituting

e^{μ} = q^{μ (h_{pr})}

Crystal basis

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.

Kashiwara operator ( $e_{i}, f_{i}$ )

On a crystal,

f_{i}

follows an edge of color

i

and

e_{i}

follows that edge backwards.

Tensor product of crystals

The tensor product crystal uses the

i

-string data

ε_{i}

and

φ_{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.

Casimir element

The quadratic Casimir element is a central element of

U (g)

. On a highest-weight module of highest weight

λ

, one standard normalization makes it act by

∣ λ + ρ ∣^{2} - ∣ ρ ∣^{2}

Principal sl2 subalgebra ( $sl_{2}^{pr}$ )