Seminar Sophus Lie 3 (1993) 107–115

Constant Yang-Mills potentials

Elke Mundt

The sourceless Yang-Mills equations on a manifold (E, g) for a potential A = A_{α}dx^{α} with values in some Lie algebra L read

D_{α}F^{αβ} = ∂_{α}F^{αβ} + [A_{α}, F^{αβ}] = 0

where [., .] denotes the commutator in L, F_{αβ} = ∂_{α}A_{β} − ∂_{β}A_{α} + [A_{α}, A_{β}] are the field strength components, ∂_{α} = \frac{∂}{∂x^{α}} the partial derivative with respect to the local coordinates x^{α} of x ∈ E, D_{α} the gauge-covariant derivative and g = g_{αβ}dx^{α}dx^{β} a Riemannian metric. We want to discuss Yang-Mills potentials A with constant components A_{α} in some gauge and choice of coordinates. The Yang-Mills equations D_{α}F^{αβ} = 0 then collapse to the algebraic equations

[F_{αβ}, A^{β}] = [[A_{α}, A_{β}], A^{β}] = 0 (YM)

A general solution of (YM) is not available. It depends on the structure of L and the signature of g whether or not there exists a non-trivial solution of (YM). We can decide this problem for special types of Lie algebras - Abelian, nilpotent, compact - and for all Lie algebras L of a dimension ≤ 5. If there exists a nontrivial solution of (YM) at all, then the problem arises to find all solutions of (YM).

Let E be an n-dimensional vector space and E^{∗} its dual. Let further {e_{α}} = {e_{1}, …, e_{n}} be a base of E and {e^{β}} = {e^{1}, …, e^{n}} the dual base of E^{∗}. A scalar product g on E has components g_{αβ} := g(e_{α}, e_{β}); we set (g^{αβ}) := (g_{αβ})^{−1}. The pair (E, g) is interpreted as the (flat) physical space and the spacetime if g is Euclidean and Lorentzian respectively. Let L be an N -dimensional real Lie algebra and {X_{i}} = {X_{1}, …, X_{N}} a base of L, c_{ij}^{k} the structure constants of L with respect to {X_{i}}, that means [X_{i}, X_{j}] = c_{ij}^{k}X_{k} for i, j, k = 1, …, N. Physically interpreted, L is the Lie algebra of some Lie group G of gauge symmetries.

An element A = a_{α}^{i}e^{α} ⊗ X_{i} ∈ E^{∗} ⊗ L, a_{α}^{i} ∈ R, is called a one-form or a potential on E with values in L. The field strength F ∈ (∧^{2}E^{∗}) ⊗ L^{1} and the current J ∈ E^{∗} ⊗ L^{2} of A are defined by

F (x, y) := [A(x), A(y)]
J(x) := g^{αβ}[A(e_{α}), F (x, e_{β})]

for x, y ∈ E. Note the antisymmetry: F (x, y) = −F (y, x). There is a component representation with Greek indices α, β, γ, … = 1, …, n:

A = e^{α} ⊗ A_{α}, where A_{α} ≡ A(e_{α}) := a_{α}^{i}X_{i} ∈ L,

F = (e^{α} ∧ e^{β}) ⊗ F_{αβ}, where F_{αβ} ≡ F (e_{α}, e_{β}) := [A_{α}, A_{β}],

J = e^{α} ⊗ J_{α}, where J_{α} ≡ J(e_{α}) := [A^{β}, F_{αβ}],

and another representation with Latin indices i, j, k, … = 1, …, N:

A = a^{i} ⊗ X_{i}, where a^{i} := a_{α}^{i}e^{α} ∈ E^{∗},

F = f^{k} ⊗ X_{k}, where f^{k} := c_{ij}^{k}a^{i} ∧ a^{j},

J = j^{m} ⊗ X_{m}, where j^{m} := c_{kl}^{m}f^{k}^{}a̸^{l}.,

The Greek indices are raised and lowered by means of (g^{αβ}) and (g_{αβ}) respectively. The inner multiplication with respect to g of a tensor by a one-form is denoted by . The potential A is called flat if F = 0; it is called a Yang-Mills (abbreviated YM) potential if J = 0. A flat potential is trivial in the sense that it can be gaugetransformed to zero. We search for potentials A such that F ≠ 0 and J = 0.

Now we specialize the type of the Lie algebra L in order to make the problem tractable. The following three structural theorems are essential.

Theorem 1. Let L = L_{I} ⊕ L_{II} be the direct sum of Lie algebras L_{I}, L_{II} and let A = A_{I} + A_{II} be the corresponding decomposition of an L-valued potential into an L_{I} -valued potential A_{I} and an L_{II} -valued potential A_{II}. The YM equations for A are equivalent to both the YM equations for A_{I} and A_{II}.

Theorem 2. Let L = L_{I} [figure] L_{II} be the semidirect sum of an ideal L_{I} and a subalgebra L_{II} and let A = A_{I} + A_{II} be the corresponding decomposition of a potential. The YM equations for A imply the YM equations for A_{II}.

Theorem 3. If a Lie subalgebra M of L admits a non-flat YM potential then so does L.

The proofs are easy and omitted here.[1]

Theorem 4. Every YM potential A with values in a 3-nilpotent Lie algebra L is a constant YM potential.

Proof. (YM) holds identically: [[L, L], L] = 0 ⇒ [[A_{α}, A_{β}], A^{β}] = 0.

Example . The Heisenberg algebra H(2m + 1) has the only non-vanishing structure relations [X_{i}, Y_{i}] = Z for i = 1, …, m with respect to some base {X_{1}, …, X_{m}, Y_{1}, …, Y_{m}, Z}. It is 3-nilpotent.

Corollary . Every non-Abelian nilpotent Lie algebra L admits a non-flat YM potential A.

Proof. It is known that the 3-dimensional Heisenberg algebra appears as a subalgebra of every nilpotent Lie algebra L: H(3) ⊆ L.

Theorem 5. If L′ = [L, L] is Abelian and of codimension 1 in L then L admits only flat YM potentials.

Proof. Let a base {X_{1}, …, X_{N−1}} of L′ be completed to a base {X_{0}, X_{1}, …, X_{N−1}} of L. Since L′ is Abelian, the only non-trivial commutator relations are [X_{p}, X_{0}] = c_{p0}^{q}X_{q}, where p, q = 1, …, N − 1. Since L′ has dimension N − 1, the matrix c = (c_{p0}^{q}) is regular and so is its square b = (b_{p}^{q}) := c^{2}. Now F and J reduce to the components f^{r} = 2c_{q0}^{r}(a^{q} ∧ a^{0}) and j^{p} = b_{q}^{p}(a^{q} ∧ a^{0})^{}a̸^{0} respectively. The YM equations become equivalent to (a^{q} ∧ a^{0})^{}a̸^{0} = 0. Hence each pair (a^{q}, a^{0}), p = 1, …, N − 1, is linearly dependent and F = 0.

Theorem 6. A constant YM potential on a Euclidean or Minkowski space with values in a compact Lie algebra is flat.[1]

Corollary . Let L = L_{I} [figure] L_{II} be the semidirect sum of an ideal L_{I} and a compact subalgebra L_{II} and let A = A_{I} + A_{II} be the corresponding decomposition of a potential. There exists a gauge in which the YM equations for A reduce to the YM equations for A_{I} and to A_{II} = 0.

Concisely expressed: Compact right summands in a semidirect decomposition of a Lie algebra can be ignored. The situation may appear as a special case to the Levi-Malcev theorem which states that every Lie algebra is the semidirect sum L = I [figure] S of a solvable ideal I and a semisimple subalgebra S.

Now we want to consider low values of dim L ≤ 5 and Euclidean (E, g). We make use of the classification of the isomorphy types of Lie algebras. The Levi-Malcev theorem allows to reduce the problem of classification of all Lie algebras to the following subproblems:

Classification of the solvable Lie algebras. Solvable Lie algebras are completely classified in the literature up to dimension 6. ([2], [3], [4], [5], [6], [7])

Classification of all semisimple, or rather, simple Lie algebras. Simple Lie algebras are completely classified nowadays.

Classification of the derivations of solvable Lie algebras. This problem is solved up to dimension 9. [8]

The number of isomorphy types of N -dimensional Lie algebras rapidly increases with N:

dim L = N Number of  Number of
isomorphy types indecomposable types
1  1  1
2  2  1
3  8  6
4  19  10
5  –  40
6  –  > 100

Let us follow Mubarakzyanov’s notation for the isomorphy types of indecomposable Lie algebras: L_{N,k}^{h,...} means the k-th type of the Lie algebra of dimension N. Eventual superscripts h, … stand for the continous parameters on which L depends. Mubarakzyanov further abbreviates the direct sum of m copies of a Lie algebra L by mL.

We use the Levi-Malcev theorem and, moreover, the following structure results to reduce the big number of isomorphy types of low-dimensional Lie algebras:

– Mubarakzyanov classifies solvable Lie algebras with respect to their maximal nilradical NR, where dim NR ≥ \frac{N}{2} . Furthermore he classifies with respect to the dimension of the centre Z(L) of L, where dim Z(L) ≤ 2 dim NR − N. Only the fact whether Z(L) = 0 or Z(L) ≠ 0 matters here.

– Every indecomposable solvable Lie algebra with a non-vanishing centre Z(L) ≠ 0 contains the 3-dimensional Heisenberg algebra as a subalgebra. According to Theorem 3 the corresponding (YM) has a non-flat solution.

In view of the above theorems we should work inductively with respect to N.

The existence of non-flat YM potentials remaines open only for

- non-compact simple Lie algebras and

- indecomposable solvable Lie algebras with Abelian nilradical of dim NR ≤ N − 2 and with Z(L) = 0.

Lie algebra L (dim L = N)

decomposable THEOR.1 indecomposable semidirect sum

THEOR.2

solvable simple compact THEOR.6

nilpotent THEOR.4

dim NR = N − 1

dim NR = N − 2 non-compact

?

Abelian non-A.

Abelian

non-A.

THEOR.3

THEOR.3

Z(L) = 0 Z(L) ≠ 0 (Z(L) = 0) / ? Z(L) ≠ 0

THEOR.5 THEOR.3

THEOR.3

Let us now go through all isomorphy types of Lie algebras with N ≤ 4 and the open cases for N = 5 in the following tables. dim L ≤ 3

Non-vanishing

L_{N,k}^{h}  commutator relations Remarks
L_{1}  Abelian ⇒ Every A is flat.
2L_{1}  decomposable ⇒ THEOR.1
L_{2} [X_{1}, X_{2}] = X_{1} (YM) ⇒ (a^{1} ∧ a^{2})^{}a̸^{2} = 0

⇒ a^{1}, a^{2} linearly dependent ⇒ (a^{1} ∧ a^{2}) = 0, hence F = 0, i. e. A is flat

3L_{1}  decomposable ⇒ THEOR.1
L_{2} ⊕ L_{1} [X_{1}, X_{2}] = X_{1}  decomposable ⇒ THEOR.1
L_{3,1} [X_{2}, X_{3}] = X_{1} L_{3,1} ≅ H(3)

nilpotent ⇒ THEOR.4

L_{3,2} [X_{1}, X_{3}] = X_{1} NR = 2L_{1}, Z(L) = 0
[X_{2}, X_{3}] = X_{2}  ⇒ THEOR.5
L_{3,3}^{h} [X_{1}, X_{3}] = X_{1} NR = 2L_{1}, Z(L) = 0
[X_{2}, X_{3}] = hX_{2}  ⇒ THEOR.5
h ≤ |1|, h ≠ 0
L_{3,4}^{p}  [X_{1}, X_{3}] = pX_{1} − X_{2} NR = 2L_{1}, Z(L) = 0
[X_{2}, X_{3}] = X_{1} + pX_{2} ⇒ THEOR.5
p ≥ 0
L_{3,5} [X_{1}, X_{2}] = X_{3} L_{3,5} ≅ so(3)
[X_{2}, X_{3}] = X_{1}  compact ⇒ THEOR.6
[X_{3}, X_{1}] = X_{2}
L_{3,6} [X_{1}, X_{2}] = −X_{3} L_{3,6} ≅ sl(2, R)
[X_{2}, X_{3}] = X_{1}  (YM) ⇒ f^{3}^{}a̸^{2} = f^{2}^{}a̸^{3},

f^{3}^{}a̸^{1} = −f^{1}^{}a̸^{3}, f^{2}^{}a̸^{1} = f^{1}^{}a̸^{2} ⇒ a^{1}, a^{2}, a^{3} linearly dependent, hence F = 0, i. e. A is flat

If L has dimension N ≤ 3 and admites a non-flat YM potential then L ≅ H(3). dim L = 4 (indecomposable)

Non-vanishing

L_{N,k}^{h} commutator relations  Remarks
L_{4,1} [X_{2}, X_{4}] = X_{1},[X_{3}, X_{4}] = X_{2}  nilpotent ⇒ THEOR.4
L_{4,2}^{α} [X_{1}, X_{4}] = αX_{1},[X_{2}, X_{4}] = X_{2} NR = 3L_{1}, Z(L) = 0
[X_{3}, X_{4}] = X_{2} + X_{3}, α ≠ 0  ⇒ THEOR.5
L_{4,3} [X_{1}, X_{4}] = X_{1}  NR = 3L_{1}, Z(L) = 0
[X_{2}, X_{4}] = X_{1} + X_{2}  ⇒ THEOR.5

[X_{3}, X_{4}] = X_{2} + X_{3}

L_{4,4}^{β,γ} [X_{1}, X_{4}] = X_{1},[X_{2}, X_{4}] = βX_{2} NR = 3L_{1}, Z(L) = 0
[X_{3}, X_{4}] = X_{2} + X_{3}, β, γ ≠ 0 ⇒ THEOR.5
L_{4,5}^{α,p} [X_{1}, X_{4}] = αX_{1}  NR = 3L_{1}, Z(L) = 0
[X_{2}, X_{4}] = pX_{2} − X_{3}  ⇒ THEOR.5

[X_{3}, X_{4}] = X_{2} + pX_{3} α ≠ 0, p ≥ 0

L_{4,6} [X_{1}, X_{4}] = X_{1},[X_{3}, X_{4}] = X_{2}  NR = 3L_{1}, Z(L) ≠ 0
⇒ THEOR.3
L_{4,7} [X_{2}, X_{3}] = X_{1},[X_{1}, X_{4}] = 2X_{1} NR = H(3)
[X_{2}, X_{4}] = X_{2}  ⇒ THEOR.3

[X_{3}, X_{4}] = X_{2} + X_{3}

L_{4,8}^{h} [X_{2}, X_{3}] = X_{1}  NR = H(3)
[X_{1}, X_{4}] = (1 + h)X_{1}  ⇒ THEOR.3

[X_{2}, X_{4}] = X_{2},[X_{3}, X_{4}] = hX_{3} h ≤ |1|,h ≠ 0

L_{4,9}^{p} [X_{2}, X_{3}] = X_{1},[X_{1}, X_{4}] = 2pX_{1} NR = H(3)
[X_{2}, X_{4}] = pX_{2} − X_{3}  ⇒ THEOR.3

[X_{3}, X_{4}] = X_{2} + pX_{3}, p ≥ 0

L_{4,10} [X_{1}, X_{3}] = X_{1},[X_{2}, X_{3}] = X_{2}  NR = 2L_{1}, Z(L) = 0
[X_{1}, X_{4}] = −X_{2},[X_{2}, X_{4}] = X_{1} ∃ a non-flat A*

* A = (e^{1} + e^{2}) ⊗ (X_{1} + X_{2}) + e^{3} ⊗ X_{3} + e^{4} ⊗ X_{4} on a Euclidean space (E, g) with orthonormal base {e^{1}, …, e^{4}} of E^{∗} is a non-flat YM potential: F_{13} = F_{23} = X_{1} + X_{2}, F_{14} = F_{24} = X_{1} − X_{2}, but J = 0.

114 Mundt

If L has dimension N ≥ 4 then the number of Lie algebras which admits a non-flat YM potential are predominant. dim L = 5 with L indecomposable, solvable, NR = 3L_{1}, Z(L) = 0

\begin{aligned}L_{5,33}^{αβ}: & [X_{1}, X_{4}] = X_{1}, [X_{3}, X_{4}] = αX_{3}, [X_{2}, X_{5}] = X_{2}, [X_{3}, X_{5}] = βX_{3}, \\ & (α, β) ≠ (0, 0) \end{aligned}

(YM) ⇒ (a^{1} ∧ a^{4})^{}a̸^{4} = 0, (a^{2} ∧ a^{5})^{}a̸^{5} = 0,

(a^{3} ∧ b)^{}b̸ = 0, b := αa^{4} + βa^{5}

⇒ (a^{1}, a^{4}), (a^{2}, a^{5}), (a^{3}, b) linearly dependent
⇒ F = 2(a^{1} ∧ a^{4}) ⊗ X_{1} + 2(a^{2} ∧ a^{5}) ⊗ X_{2} + 2(a^{3} ∧ b) ⊗ X_{3} = 0,

i. e., A is flat

\begin{aligned}L_{5,34}^{α}: & [X_{1}, X_{4}] = αX_{1}, [X_{2}, X_{4}] = X_{2}, [X_{3}, X_{4}] = X_{3}, [X_{1}, X_{5}] = X_{1}, \\ & [X_{3}, X_{5}] = X_{2} \end{aligned}

H(3) is a subalgebra ⇒ THEOR.3

\begin{aligned}L_{5,35}^{αβ}: & [X_{1}, X_{4}] = αX_{1}, [X_{2}, X_{4}] = X_{2}, [X_{3}, X_{4}] = X_{3}, [X_{1}, X_{5}] = βX_{1}, \\ & [X_{2}, X_{5}] = −X_{3}, [X_{3}, X_{5}] = X_{2}, (α, β) ≠ (0, 0) \end{aligned}

L_{4,10} is a subalgebra ⇒ THEOR.3

References

[1] Schimming, R. and E. Mundt, Constant potential solutions of the Yang-Mills equation, J. Math. Phys. 33 (1992), 1–5.

[2] Morozov, V. V., Classificaziya nilpotentnych algebrach Lie shestovo poryadka, Izv. Vyss. Uchebn. Zaved. Mat. 5 (1958), 161–171 (Russian).

[3] Mubarakzyanov, G. M., O rasreshimych algebrach Lie, Izv. Vyss. Uchebn. Zaved. Mat. 32 (1963), 114–123 (Russian).

[4] Mubarakzyanov, G. M., Classificaziya veshestvennych structur algebr Lie pyatovo poryadka, Izv. Vyss. Uchebn. Zaved. Mat. 34 (1963), 99–105 (Russian).

[5] Mubarakzyanov, G. M., Classificaziya rasreshimych algebr Lie shestovo poryadka s odnim nenilpotentnym bazisnym elementom, Izv. Vyss. Uchebn. Zaved. Mat. 35 (1963), 104–116 (Russian).

[6] Mubarakzyanov, G. M., Necotorye theoremy o rasreshimych algebrach Lie, Izv. Vyss. Uchebn. Zaved. Mat. 55 (1966), 95–98 (Russian).

[7] Turkowski, P., Solvable Lie algebras of dimension six, J. Math. Phys. 31 (1990), 1344–1350.

[8] Turkowski, P., Low-dimensional real Lie algebras, J. Math. Phys. 29 (1988), 2139–2144.

Fachbereich Mathematik/Informatik Ernst-Moritz-Arndt-Universität Greifswald Friedrich-Ludwig-Jahn-Str.15a O-2200 Greifswald, Germany

Received March 31, 1993 and in final form April 14, 1993