MODEL THEORY OF LIE ALGEBRAS

TRISTRAM DE PIRO

Abstract. We are concerned with proving certain model theoretic properties for simple finite dimensional Lie algebras over an algebraically closed field of characteristic 0.It is hoped that the techniques used apply similarily for the theory of simple non abelian algebraic groups using just the group language. It is shown the underlying theory in the language of rings is ℵ_{1}-categorical. The automorphisms are classified and from this is deduced the existence of strongly minimal sets having a field like structure, and the structure of the countable models. Using the method of coordinatisation, we make progress towards finding a natural language for eliminating quantifiers. Finally, the question of adding an automorphism and non commuting derivations is discussed.

1. Preliminaries

In what follows we consider a finite dimensional Lie algebra L over an algebraically closed field F of characteristic 0. This possesses a bracket operation [ ] which is bilinear with respect to + and the action of F on L, and satisfies the Lie axioms;

(L1) [xx] = 0 ∀x ∈ L
(L2) [[xy]z] + [[yz]x] + [[zx]y] = 0 ∀x, y, z ∈ L

Much of the basic theory of Lie algebras can be found in [1]. We consider L as a first order structure in the language of rings, and

𝔏 = ⟨L, +, −, [ ], 0⟩

is the Lie algebra considered as a structure for this language. We make the algebraic assumption that L is simple; that is non-abelian and contains no non-trivial ideals. Before proceeding to the main body of the proof, we assemble some known facts which apply generally to finite dimensional algebras over algebraically closed fields in any characteristic. The reader may find a proofs of these results in [2].

Date: December 1, 2001.

Suppose A is an n− dimensional algebra over F. Given a basis for A we can specify the multiplication table for A by an n^{3} tuple \overline{c} of elements from F. Again consider A as a structure for the language of rings and let

𝔄 = ⟨A, +, −, ., 0⟩

Let L and L^{∗} denote the underlying ring languages of A and its field F. Then

Theorem 1.1 (Finite-Dimensional Transfer). There is a map depending on \overline{c}; σ : Ł_{A} formulae ⟶ Ł_{F}^{∗} formulae such that if ϕ ∈ Sent(L_{A}) then σ(ϕ) ∈ Sent(L_{F}^{∗}) and

𝔄 ⊧ ϕ ⟺ 𝔉 ⊧ σ(ϕ)

We can also prove using an ultraproduct argument the following important theorem;

Theorem 1.2. If A is an n-dimensional algebra over F and if 𝔅 ≡ 𝔄, then there exists ℭ ≻ 𝔅 such that C is an n-dimensional algebra over a field G with 𝔊 ≡ 𝔉

The above theorems can be used to prove the following;

Theorem 1.3. A finite dimensional algebra over a κ-stable field is κstable. In particularly, a finite dimensional algebra over an algebraically closed field is ω-stable.

For a complete ω-stable theory T ,we define its Morley rank to be;

MR(T) = sup{MR(ρ) : ρ ∈ S^{1}(𝔐), 𝔐 ⊧ T }

Then Theorem(1.3) may be improved to;

Theorem 1.4. If A is a finite dimensional algebra over an algebraically closed field F, then MR(T h(𝔄)) ≤ dim_{F}A

For a background to the theory of ultrapowers, see [3], and for a discussion of Morley rank and stable theories, see [4].

This means the underlying ring theory of finite dimensional Lie algebras over algebraically closed fields has finite Morley rank. In particular, the following theorem, usually used in connection with groups of finite Morley rank applies, the reader may consult [5] for more information;

Theorem 1.5 (Zilber’s Indecomposibility Theorem). If (A_{i}) is a family of indecomposable subsets of a group G of finite Morley rank (possibly with added ring structure), and if each A_{i} contains the identity 0 of G , then the subgroup generated by the subsets A_{i} is definable and connected. Moreover, there are finitely many indices such that

⟨⋃_{i} A_{i}⟩ = A_{i_{1}} + … A_{i_{m}}

for some m < 2^{MR(G)+1} − 2, where ⟨⋃_{i} A_{i}⟩ is the subgroup generated by the A_{i}.

2. ℵ_{1}-categoricity

We now return to the case of simple Lie algebras L over algebraically closed fields of characteristic 0 and proceed to show the underlying ring theory is ℵ_{1}-categorical. Suppose then that 𝔏′ ⊧ T h(𝔏). As L has characteristic 0, it is clear that 𝔏′ is a Lie ring of characteristic 0. We will show later that 𝔏′ is in fact a simple Lie ring. Assuming this for now, we show how to represent elements of 𝔏′ by certain special words in the underlying ring language. The idea behind this construction stems from [6].

Fix a_{0} ∈ 𝔏′, and consider the subgroup of 𝔏′ given by

[𝔏′a_{0}] = {[xa_{0}] : x ∈ 𝔏′}

As 𝔏′ has characteristic 0, its subgroups are connected, hence indecomposable, and [𝔏′a_{0}] is definable by

ϕ(x) ≡ ∃y(x = [ya_{0}])

Now let

I_{1} = ⟨[𝔏′a_{0}], [x[𝔏′a_{0}]] : x ∈ 𝔏′⟩

Again, the subgroups [x[𝔏′a_{0}]] are connected, so applying Theorem (1.5), gives

I_{1} = [𝔏′a_{0}] + [b_{1}[𝔏′a_{0}]] + … + [b_{n}[𝔏′a_{0}]]

which is definable by

ϕ_{1}(x) ≡ ∃y_{0} … ∃y_{n}∃x_{1} … ∃x_{n}(x = [y_{0}a_{0}] + [x_{1}[y_{1}a_{0}]] + … + [x_{n}[y_{n}a_{0}]])

Inductively, define

I_{m+1} = ⟨I_{m}, [xI_{m}] : x ∈ 𝔏′⟩

Then, again, I_{m+1} is a definable connected subgroup of 𝔏′. Hence, we obtain an increasing sequence of definable connected subgroups of 𝔏′. As |I_{j+1} : I_{j}| = ∞ we must have MR(I_{j+1}) > MR(I_{j}). So by the fact MR(T h(𝔏)) is finite, the construction terminates to give I_{m} = I_{m+1} for some m. Then I_{m} must be an ideal of 𝔏′ . However, we are supposing 𝔏′ is simple, hence 𝔏′ = I_{m} . By the definition of I_{m} we have that

𝔏′ = {w(a_{0}) : a_{0} ∈ 𝔏′, w(y) is a special word of bounded length nk}

where

w(y) = w_{1}(y) + … + w_{n}(y) with n ≤ (2^{m+1} − 2)^{m}, m = MR(T h(𝔏))

and each w_{j}(y) is of the form

w_{j}(y) = [x_{j1}[… [x_{jk}y] …], x_{ji} ∈ 𝔏′, k ≤ m = MR(T h(𝔏))

Now define

J = {a ∈ 𝔏′ :  ∀w, w_{1}, w_{2} special words of length ≤ nk;
(1) w(a_{0}) = w_{1}(a_{0}) → w(a) = w_{1}(a),
(2) ∀x([x[w_{1}(a_{0})]] = w(a_{0}) → [x[w_{1}(a)]] = w(a)),
(3) (w_{1}(a_{0}) + w_{2}(a_{0}) = w(a_{0}) → w_{1}(a) + w_{2}(a) = w(a))}

This is a definable subset of 𝔏′ in T h(𝔏)_{a_{0}}. In particular, when 𝔏 is the original algebra, J is easily seen to be nothing more than span_{F}(a_{0}). We now proceed to interpret End_{𝔏′}(𝔏′) in 𝔏′. The claim is that

End_{𝔏′}(𝔏′) = {γ_{a} : a ∈ J}

where

γ_{a}(w(a_{0})) = w(a)

This is because if θ ∈ End_{𝔏′}(𝔏′), then θ(a_{0}) satisfies the definition of J, so θ = γ_{θ(a_{0})}. Conversely, by part (1) of the definition of J, the γ_{a} are well defined, and parts (2),(3) show they are bilinear and preserve [ ].Now we can interpret

{γ_{a} : a ∈ J}

as a ring with 0, 1 in 𝔏′ as follows. For the underlying set take

ψ(x, y) ≡ ∃t∃w( w is a special word of length ≤ nk
∧x = w(a_{0}) ∧ y = w(t) ∧ t ∈ J)

Define an equivalence relation on this set by

ψ′(x_{1}, y_{1}, x_{2}, y_{2}) ≡ ∀z([zx_{1}] = [zx_{2}] → [zy_{1}] = [zy_{2}])

This equivocates the tuples (x, γ_{a}(x)) and (y, γ_{a}(y)), as if [x − y, z] = 0 then [γ_{a}(x) − γ_{a}(y), z] = 0 because γ_{a} ∈ End_{𝔏′}(𝔏′). Therefore, the equivalence classes represent the graphs of each γ_{a}. To interpret the ring structure of End_{𝔏′}(𝔏′), for example relations such as γ_{a_{1}} ∘ γ_{a_{2}} = γ_{a_{3}},we just take as the underlying set

ψ′′(x, y, z) ≡ ψ(x, y) ∧ ψ(y, z) ∧ ψ(x, z)

Then, we define an equivalence relation on this set by

ψ′′′(x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}) ≡ ψ′(x_{1}, y_{1}, x_{2}, y_{2})∧ψ′(y_{1}, z_{1}, y_{2}, z_{2})∧ψ′(x_{1}, z_{1}, x_{2}, z_{2})

A similar construction works for γ_{a_{1}} + γ_{a_{2}} = γ_{a_{3}}. We have now interpreted End_{𝔏′}(𝔏′) in 𝔏′ as a ring. We now observe that End_{𝔏′}(𝔏′) is a skew field. As if θ ∈ End_{𝔏′}(𝔏′), we have Ker(θ) and Im(θ) are both ideals in 𝔏′. Hence, as 𝔏′ is simple, for θ ≠ 0, we must have that θ is invertible. Moreover, the fact that End_{𝔏′}(𝔏′) is interpretable in 𝔏′ which has finite Morley rank, means that End_{𝔏′}(𝔏′) is an algebraically closed field, by a result due to Macintyre. Therefore, we conclude

Theorem 2.1. If 𝔏′ ⊧ T h(𝔏) and 𝔏′ is simple as a Lie ring, then 𝔏′ is a simple Lie algebra over an algebraically closed field G of characteristic 0

We have yet to establish that 𝔏′ is finite dimensional over G. In order to see this, note first that we can easily interpret the action of G on 𝔏′ in 𝔏′ using the methods above. Hence we can interpret the the expanded two-type algebra structure

𝔐 = ⟨G, 𝔏′, +_{G}, ▪_{G}, +_{𝔏′}, [ ]_{𝔏′}, Action_{G}𝔏′⟩

We then have that 𝔐 has finite Morley rank. If 𝔏′ has infinite dimension over G, let {x_{1}, …, x_{n}, … } be an infinite sequence of linearly independent elements in 𝔏′. Then define

A_{n} = span_{G}{x_{1}, …, x_{n}} for n < ω

{A_{n}}_{n<ω} is an infinite sequence of definable groups in 𝔐. Moreover, as Char(G) =0, |A_{i+1} : A_{i}| is infinite. Therefore, MR(A_{i+1}) > MR(A_{i}) in 𝔐. This implies 𝔐 cannot have finite Morley rank, which is a contradiction. So 𝔏′ is finite dimensional over G. We may also specify a bound on the dimension of any model of T h(𝔏). For if 𝔏′ satisfies the hypotheses of Theorem (2.1), then dim_{G}(𝔏′) ≤ MR(T h(𝔐)), where 𝔐 is the expanded algebra structure above. However MR(T h(𝔐)) ≤ r▪MR(T h(𝔏)) where r is the tuple length of elements of 𝔐 interpreted in 𝔏′. These inequalities combine to give dim_{G}(𝔏′) ≤ 2dim_{F}(𝔏). We therefore have

Theorem 2.2. If 𝔏′ ⊧ T h(𝔏) and 𝔏′ is simple as a Lie ring, then 𝔏′ is a finite dimensional simple Lie algebra over an algebraically closed field G of characteristic 0, and moreover dim_{G}(𝔏′) ≤ 2dim_{F}(𝔏).

We now turn to the problem of proving that if 𝔏′ ⊧ T h(𝔏) then 𝔏′ is simple as a Lie ring. It is relatively straightforward to show that 𝔏 itself is simple as a Lie ring. Now by Theorem(1.2), if 𝔏′ ⊧ T h(𝔏) then 𝔏′ ≺ 𝔏′′ where 𝔏′′ is an n−dimensional Lie algebra over a field G ≡ F. We claim if L′′ is simple as a Lie ring then so must L′ be. If not, then choosing a_{0} ∈ 𝔏′, with a_{0} in a non trivial ideal I of 𝔏′, and using exactly the argument as before Theorem(2.1), we can define I in 𝔏′ by a formula ϕ(x, a_{0}).Then

𝔏′ ⊧ ∃x(ϕ(x, a_{0}) ∧ x ≠ o)
𝔏′ ⊧ ¬∀x(ϕ(x, a_{0}))
𝔏′ ⊧ ∀x∀y(ϕ(x, a_{0}) ∧ ϕ(y, a_{0}) → ϕ(x + y, a_{0}))
𝔏′ ⊧ ∀x∀y(ϕ(x, a_{0}) → ϕ([yx], a_{0}))

Hence, as 𝔏′ ≺ 𝔏′′, ϕ(x, a_{0}) defines a non trivial ideal in 𝔏′′ which is a contradiction. Hence, we may suppose w.l.o.g that 𝔏′ ≡ 𝔏 is a finite dimensional algebra over G ≡ F with dim_{G}(𝔏′) = dim_{F}(𝔏) = n. Let us call this class of structures Σ. We can now find a formula ϕ(x, y) which uniformly defines ideals generated by subspaces of the form [𝔏′a] for any a ∈ 𝔏′ where 𝔏′ ∈ Σ. To see this, let a ∈ 𝔏′. As usual, [𝔏′a] is a definable subspace of 𝔏′. Then

⟨[x[𝔏′a]] : x ∈ 𝔏′⟩ = [𝔏′a] + [x_{1}[𝔏′a]] + … + [x_{m}[𝔏′a]]
where m ≤ dim_{G}(𝔏′) = n

Note the bound holds for all structures in Σ. Now using the fact that the structures we are considering all have dimension n, this process of forming the ideal generated by [𝔏′a] terminates in at most n steps. Hence we obtain a canonical formula for structures in Σ that define such ideals given by;

ϕ(x, a) ≡ ∃x_{1}^{1}∃ \underbrace{w_{1}^{1}∃w_{2}^{1} … w_{1}^{t}w_{2}^{t}}_{t=m} … ∃ \underbrace{z_{1}^{1} … z_{n}^{1} … z_{1}^{t} … z_{n}^{t}}_{t=m^{n}}(x = [x_{1}^{1}a] + [w_{1}^{1}[w_{2}^{1}a]] +
… +[w_{1}^{t}[w_{2}^{t}a]] + … + [z_{1}^{1} … [z_{n}^{1}a] …] + … + [z_{1}^{t} … [z_{n}^{t}a] …])

We now let σ(a) be;

∃x(x ≠ 0 ∧ ϕ(x, a)) ∧ ¬∀x(ϕ(x, a)) ∧ ∀x∀y(ϕ(x, a) ∧ ϕ(y, a) → ϕ(x + y, a))
∧ ∀x∀y(ϕ(x, a) → ϕ([yx], a))

Then if 𝔏′ ∈ Σ is not simple, it has a non trivial proper ideal I. Choose a ∈ I, then 𝔏′ ⊧ σ(a),therefore 𝔏′ ⊧ ∃x(σ(x)). Conversely, we have that if 𝔏′ ⊧ ∃x(σ(x)), then as ϕ(x, a) defines the ideal generated by [𝔏′a], 𝔏′ is not simple. We then have

𝔏′ is not simple ⇔ 𝔏′ ⊧ ∃x(σ(x)) for 𝔏′ ∈ Σ

Hence, as by hypothesis, we have 𝔏 ⊧ ¬∃x(σ(x)), all the 𝔏′ ≡ 𝔏 for 𝔏′ ∈ Σ are simple as Lie rings, so therefore all 𝔏′ ≡ 𝔏 are. Combining this result with the previous theorems of this section and the fact that the theory of algebraically closed fields in characteristic zero is ℵ_{1}- categorical we obtain;

Theorem 2.3. If 𝔏′, 𝔏′′ ⊧ T h(𝔏) and Card(𝔏′) = Card(𝔏′′) = ℵ_{1}, then 𝔏′, 𝔏′′ are simple finite dimensional Lie algebras over algebraically closed fields G′ ≅ G′′ and dim_{G′}(𝔏′) ≤ 2dim_{F}(𝔏)

We are now close to proving ℵ_{1}-categoricity for T h(𝔏). First observe that by the classification of root systems for simple Lie algebras, there can only be a finite number of simple Lie algebras of a given dimension over an algebraically closed field of cardinality ℵ_{1}. Hence, due to the bound in dimension provided by Thm(2.3) for 𝔏′ ⊧ T h(𝔏), there can only be finitely many models of T h(𝔏) with cardinality ℵ_{1}; that is I(ℵ_{1}, T h(𝔏)) < ∞. This implies by a result due to Lachlan-Shelah, that if T h(𝔏) is not ℵ_{1}-categorical, then T h(𝔏) is ℵ_{0}-categorical. Hence it is sufficient to prove T h(𝔏) is not ℵ_{0}−categorical to obtain the desired result. In order to prove this fact , it is necessary to invoke the root space decomposition for semisimple Lie algebras over an algebraically closed field of characteristic !0! , given by;

L = H ⊕ ∐_{α∈Φ_{≠0}} L_{α}

Here H is a maximal toral subalgebra, and is in fact the zero root space in the decomposition, and for α ∈ Φ(Φ ⊂ H^{∗}), α ≠ 0

L_{α} = {x ∈ L : [hx] = α(h)x, ∀h ∈ H}

Now returning to the proof, by Ryll-Nardzewski’s Theorem it is sufficient to show that T h(𝔏) has a countable model which realises infinitely many distinct types. Using the fact that T h(𝔏) is ω-stable, we can find a countable model 𝔐 which realises all types in S(T h(𝔏)). Such a model is clearly ℵ_{o}-saturated and hence ℵ_{0}-homogenous. We have 𝔐 is a simple f.d Lie algebra over an algebraically closed field F with Char(F) = 0. Hence, it posesses non trivial root spaces M_{α}(α ≠ 0).

Consider the 3 tuples;

{(nx, ny, [xy])}; n ∈ Z_{+}, x ∈ M_{α}, y ∈ M_{−α}}

We claim the types realised by these tuples are distinct, which gives the result. Otherwise by homogeneity of 𝔐, there exists an automorphism Θ of 𝔐 for which;

Θ : (nx, ny, [xy]) ↦ (mx, my, [xy]) for m, n ∈ Z_{+}, m ≠ n

Then we have;

Θ([nx, ny]) = [Θ(nx), Θ(ny)] = [mx, my] = m^{2}h
Θ([nx, ny]) = n^{2}Θ([xy]) = n^{2}[xy] = n^{2}h

Hence, as h ≠ 0,m = n, a contradiction.

3. Chevalley Bases

The results established in the last section imply by Morley’s Theorem that T h(𝔏) is κ-categorical in all uncountable cardinals κ, and T h(𝔏) has ℵ_{0} countable models. It is to the algebraic structure of these models that we now turn. Given that we are now dealing exclusively with algebras, I will use L to denote all models of the theory. We now require some further results about simple Lie algebras L. Given a root space decomposition Φ(L, H) for L, it follows, by the classification of representations of 𝔰𝔩_{2}(C) and the existence of a base Δ(Φ) = {α_{i_{1}}, …, α_{i_{n}}} for a chosen set of positive roots, that we can find a basis h_{α_{1}}, … h_{α_{n}} for H such that α(h_{α_{i}}) ∈ Z for 1 ≤ i ≤ n. This of course means that for α ∈ Φ_{≠0}, [h_{a_{i}}, L_{α}] ∈ Z. We can in fact find ν_{α} ∈ L_{α} such that

[ν_{α}ν_{β}] = m_{αβ}ν_{α+β} with m_{αβ} ∈ Z

The set

{h_{α_{i}} : α_{i} ∈ Δ(Φ)} ∪ {ν_{α} : α ∈ Φ_{≠0}}

is known as a Chevalley basis for L. I will denote the Z-Lie-module spanned by this basis as L_{Z}. As this integral structure is completely formalisable in the language of rings, it follows that L_{Z} is embedded in all models of T h(L).We now want to show that the linear independence of L_{Z} is preserved on taking elementary substructures of models of T h(L). So suppose for contradiction L ≺ L′ and the Chevalley basis of L_{Z} is linearly dependent in L′. Then choose a minimal dependent set S. We consider three possible cases; \underline{Case 1} S contains h_{α_{i}} and ν_{α} with α ∈ Φ_{≠0}.

As Φ spans H^{∗}, choose h_{α_{j}} with α(h_{α_{j}}) = m and m ∈ Z_{≠0}. We have that h_{α_{j}} acts semisimply on the Chevalley basis with integer eigenvalues and therefore preserves S. Then we have;

λh_{α_{i}} + μν_{α} + … = 0
⇒  [h_{α_{j}}, λh_{α_{i}} + μν_{α} + …] = 0
⇒  λ[h_{α_{j}}, h_{α_{i}}] + μα(h_{α_{j}})ν_{α} + … = 0
⇒  μmν_{α} + … = 0 (as H is abelian)
⇒  μm = 0 (minimality of S) ⇒ μ = 0

This is a contradiction. We now turn to;

\underline{Case 2} S contains no h_{α_{i}}

Then choose ν_{α} ∈ S. Given a basis Δ(Φ) for the positive roots Φ^{+} and assuming that α is positive, it follows by the geometry of the root system that we can find α_{i_{1}}, … α_{i_{k}} ∈ Δ(Φ) such that;

α = α_{i_{1}} + … + α_{i_{k}} and α_{i_{1}} + … + α_{i_{j}} ∈ Φ^{+} for 1 ≤ j ≤ k

Then apply the sequence ν_{α_{i_{k}}} , … ν_{α_{i_{1}}} to the dependence relation on S. This preserves or lowers the cardinality of the dependence relation on S, and produces a new minimal dependence relation S′ containing h_{α_{i_{1}}}. Now use Case 1. Finally, we have

\underline{Case 3} S contains no ν_{α}

Using the fact Φ spans H^{∗} ,we can find α such that α(h_{α_{i_{j}}}) ∈ Z_{≠0}. Now apply ν_{α} to the relation and reduce to Case 2.

We can now use this result to prove the following theorem, where k_{∞} denotes an algebraically closed field of transcendence degree ℵ_{1}.

Theorem 3.1. If L′, L′′ ⊧ T h(L) with Card(L′) = Card(L′′) = ℵ_{1}, then L′ ≅ L′′ as Lie algebras, not merely rings, and L′ = L′′ = L_{Z} ⊗k_{∞}

For this, choose a prime model L_{0} ≺ L′, L′′. Let L′_{Z} be the integral structure attached to a Chevalley basis in L′. Then L′_{Z} is present in L_{0} and in L′′. By the preceding remarks, the passage from L′ to L′′ via L_{0} preserves the linear independence of L′_{Z}. So L′ is a subalgebra of L′′. Now repeat for a Chevalley basis in L′′ to get the result. We can now characterise the countable models of T h(L). Let k_{n} denote an algebraically closed field of transcendence degree n. We claim the following theorem is true;

Theorem 3.2. The countable models of T h(L) are given by

L_{Z} ⊗ k_{0} ⊂ L_{Z} ⊗ k_{1} ⊂ … ⊂ L_{Z} ⊗ k_{n} ⊂ …

where ⊂ denotes substructure only.

We first have to show that L_{Z} ⊗ k_{n} ⊧ T h(L). Using Theorem(1.2), we can find L′, L′′ in some uncountable cardinal κ such that L_{Z} ⊗ k_{n} ≺ L′ and L ≺ L′′. Then L′ and L′′ have the same dimension over the given Chevalley basis L_{Z}, so L′ ≅ L′′ and therefore L_{Z} ⊗ k_{n} ≡ L. Next suppose M ⊧ T h(L) with M countable. We know that M is a simple Lie algebra over an algebraically closed field k_{n} say. Then taking a prime model again, we see that M contains L_{Z}, and the Chevalley basis of L_{Z} is linearly independent in M. Hence, L_{Z} ⊗ k_{n} is a subalgebra of M Now, as M is simple, the result follows easily.

4. Strongly Minimal Sets

In order to obtain the strongly minimal sets, we first classify the automorphisms of L′ ⊧ T h(L). So fix L_{Z} ⊗ k wher k is an algebraically closed field of characteristic 0. We first have all the algebra automorphisms are realised as a an affine algebraic group. To see this in the case where k = C, let G(C) be a Lie group having L(C) as its Lie algebra. Then the adjoint map Ad : G(C) → Aut(L(C)) has the property that

e^{ad(X)} = Ad(expX) in Aut(L(C)) (∀X ∈ L(C))

Using the fact that all derivations of simple Lie algebras are inner, it follows that the connected component Aut(L(C))^{0} is generated by Ad(expX) for X ∈ L(C), and hence

Aut(L(C))^{0} ≅ G(C)/Z(G(C))

which is a simple Lie group. We also have the following class of automorphisms of L_{Z} ⊗ k given by;

Θ(∑_{α∈Φ} k_{α}ν_{α}) = ∑_{α∈Φ} \overline{Θ}(k_{α})ν_{α}

where \overline{Θ} ∈ Aut(k) and I have used \{ν_{α}\} to denote the full Chevalley basis associated to L, and the sum over the zero root means taking the sum over the {h_{α_{i}} : α_{i} ∈ Δ(Φ)}. These are not algebra automorphisms but are isomorphisms in the language of rings,as;

Θ([∑_{α∈Φ} k_{α}nu_{α}, ∑_{β∈Φ} k_{β}ν_{β}])

= Θ( ∑_{α,β∈Φ} k_{α}k_{β}Γ_{αβ}^{γ}ν_{γ}) (Γ_{αβ}^{γ} ∈ Z)
= ∑_{α,β∈Φ} \overline{Θ}(k_{α})\overline{Θ}(k_{β})Γ_{αβ}^{γ}ν_{γ} (as \overline{Θ} fixes prime subfield of k)
= [Θ(∑_{α∈Φ} k_{α}ν_{α}), Θ(∑_{β∈Φ} k_{β}ν_{β})]

We now prove the following theorem; Theorem 4.1.

Aut(L_{Z} ⊗ k) ≅ G(k) × Aut(k)

where G(k) is an algebraic group corresponding to L_{Z} ⊗ k which can be written as a product G^{o}(k) ▪ Γ(Φ) where G^{0}(k) is the connected component and simple as an algebraic group and Γ(Φ) is the finite group of automorphisms of the root system corresponding to L_{Z} ⊗ k.

So let Ψ ∈ Aut(L_{Z} ⊗ k). Then the arguments of the last chapter show that {Ψ(ν_{α}) : α ∈ Φ} form a Chevalley basis for L_{Z} ⊗ k. Hence we can define an algebra automorphism Λ by;

Λ(∑_{α∈Φ} k_{α}Ψ(ν_{α})) = ∑_{α∈Φ} k_{α}ν_{α}

So Λ ∈ G(k) and ΛΨ fixes {ν_{α} : α ∈ Φ}. We can define each 1dimensional space span_{k}(ν_{α}) in the ring language by a formula ϕ(x, ν_{α}), (see Section 2). It therefore follows that ΛΨ fixes span_{k}(ν_{α}). Hence,

ΛΨ(∑_{α∈Φ} k_{α}ν_{α}) = ϵ_{α}(k_{α})ν_{α}

where ϵ_{α} : k → k. Now fix a regular element h_{0} in the Cartan of L_{Z}, so h_{0} ∈ span_{Z}{h_{α_{i}} : α_{i} ∈ Δ(Φ)} and α(h_{0}) ∈ Z_{≠0} for all α ∈ Φ_{≠0}. We have that ΛΨ fixes h_{0} and therefore span_{k}(h_{0}). So given α ∈ Φ_{≠0}, with α(h_{0}) = n ≠ 0 we have

ΛΨ([kh_{α_{0}}, ν_{α}]) = ϵ_{α}(kn)ν_{α} = ϵ_{α}(k)nν_{α}

whereas

[ΛΨ(kh_{0}), ν_{α}] = ϵ_{h_{0}}(k)nν_{α}

which shows that ϵ_{α} = ϵ_{h_{0}} on k for all α ∈ Φ_{≠0}. Moreover

ΛΨ([kh_{0}, k′ν_{α}]) = ϵ_{h_{0}}(kk′)nν_{α} = [ΛΨ(kh_{0}), ΛΨ(k′ν_{α})] = ϵ_{h_{0}}(k)ϵ_{h_{0}}(k′)nν_{α} which shows that ϵ_{h_{0}}(kk′) = ϵ_{h_{0}}(k)ϵ_{h_{0}}(k′) and hence that ϵ_{h_{0}} ∈ Aut(k), and so ϵ_{α} ∈ Aut(k) for α ∈ Φ_{≠0}. This means that ΛΨ = Θ where Θ is one of the ring automorphisms described above, and Ψ = Λ^{−1}Θ. It is not true that the Λ and the Θ commute. However we can show that the field automorphisms normalise the algebra automorphisms. That is if Λ is an algebra automorphism, and Θ is a field automorphism then ΘΛΘ^{−1} is still an algebra automorphism. In order to see this, we use the structural result given above, and we are reduced to showing that the field automorphisms normalise both the graph automorphisms and the automorphisms generated by e^{ad(X)} for X ∈ L.

We consider first the case when Λ is a graph automorphism in Γ(L). Let α_{i} ∈ Δ(Φ) be a set of simple roots corresponding to our fixed Chevalley basis, then the automorphism of the graph sends these to α_{Λ(i)} and induces a canonical automorphism of the corresponding Cartan subalgebra H by duality. We know that a representative of Λ maps each root space L_{α_{i}} onto L_{α_{Λ(i)}} for α_{i} ∈ Δ(Φ). and we denote the corresponding scalars involved relative to our fixed Chevalley basis by μ_{i}. Now conjugating Λ by a field automorphism, we have that

λν_{α_{i}} ↦ Θ̄(λ)ν_{α_{i}} ↦ Θ̄(λ)μ_{i}ν_{α_{Λ(i)}} ↦ λΘ̄^{−1}(μ_{i})ν_{α_{Λ(i)}}

for α_{i} ∈ Δ(Φ) and the effect on H is the same as the original Λ. By the isomorphism theorem for simple Lie algebras any algebra isomorphisms from the root spaces L_{α_{i}} to L_{α_{Λ(i)}} together with the given isomorphism of H extends uniquely to an isomorphism of the full algebra. As the map on L_{−α_{i}} is uniquely determined given the maps on H and L_{α_{i}}, which is a set of generators for the Chevalley basis (see below), it follows that Θ^{−1}ΛΘ is indeed an algebra automorphism.

For the case of Int(L) we may suppose that such automorphisms are realised faithfully by the adjoint action of G^{o} where G^{o} is a simple connected algebraic group having L as its Lie algebra. We have that G^{o} is defined over a field K and we first aim to descend the field of definition to the prime subfield Q. So let the tuple k̄ define G^{0} and let ā be a generic point of the irreducible set G^{o} over Q(k̄). Then consider

V = locus(āk̄/Q^{alg})

As multiplication μ on G^{o} is defined over Q, the statement ϕ(x̄) ⊂ pr(V) given by μ defines a multiplication on the fibre V (x̄) is algebraic, defined over Q and holds for the generic point k̄ of pr(V) over Q^{alg}. Hence the fibres of V are linear algebraic groups almost everywhere. It can be shown using Zilber’s decomposability theorem and definability of the dimension of fibres that the statement ϕ′(x̄) given by ”the fibre V (x̄) is a simple algebraic group of dimension n” is also definable over Q^{alg}. Again, it follows this holds almost everywhere on pr(V). Now using the fact that Q^{alg} ≺ K^{alg}, we can find a parameter f̄ ∈ ϕ′(Q^{alg}), and hence the fibre V (f̄) defines a simple algebraic group.

It remains to show that V (l̄) and V (f̄) are biregularly isomorphic as algebraic groups over K^{alg}. However, this follows from the fact that ϕ′(x̄) defines a continously varying family of simple algebraic groups of a given dimension. Using the isomorphism theorem for such groups, there can only be finitely many isomorphism types for the fibres over K^{alg} and hence the isomorphism type is constant.

We therefore have that G^{o} is defined over Q^{alg} and hence we may assume the field of definition is a finite Galois extension of Q. Now by classical arguments (Serre) we can descend the field of definition of G^{o} to Q. By standard Lie theory, we may assume that the action of G^{0} on L is given by conjugation. Then a simple matrix calculation shows that conjugating the action of Λ by a field automorphism Θ on L is equivalent to conjugating by Λ^{Θ}, which is the matrix obtained from Λ by applying Θ to the coefficients. As we assumed that G^{0} was defined over Q, it follows that Λ^{Θ} is still in G^{0} as required.

We can now prove

Theorem 4.2. The formulae ϕ(x, ν_{α}) for α ∈ Φ_{≠0} and ϕ(x, h_{α_{i}}) for α_{i} ∈ Δ(ϕ) define strongly minimal sets for T h(L)

In order to see this, let us fix a big model \overline{M} over k_{∞} for the theory. Let \overline{a} ∈ M, with ψ_{1}(x, \overline{a}), ψ_{2}(x, \overline{a}) mutually inconsistent. Suppose also that the ϕ(x, ν_{α}) ∧ ψ_{i}(x, \overline{a}) split span_{k}(ν_{α}) into 2 disjoint infinite sets in \overline{M}. By Downward Lowenheim-Skolem, we can find a countable elementary substructure M of \overline{M} containing \overline{a} and automatically the Chevalley basis {ν_{α}}. So

M = L_{Z} ⊗ k_{n} ≺ L_{Q} ⊗ k_{∞} = \overline{M}

Again ϕ(x, ν_{α})∧ψ_{i}(x, \overline{a}) split span_{k_{n}}(ν_{α}) into 2 disjoint infinite subsets in M. Using the fact T h(L) is ℵ_{1}-categorical, hence has no Vaughtian pairs, we can find b_{1}, b_{2} in \overline{M} \ M satisfying ϕ(x, v_{α}) ∧ ψ_{i}(x, \overline{a}). However, then b_{1}, b_{2} ∈ k_{∞} \ k_{n} within span_{k_{∞}}(ν_{α}), so tp(b_{1}|k_{n}) = tp(b_{2}|k_{n}) and assuming k_{∞} is sufficiently saturated, there is an automorphism \overline{Θ} of k_{∞} fixing k_{n} and taking b_{1} to b_{2}. By the above, we can find Θ ∈ Aut(\overline{M}) fixing M and taking b_{1} to b_{2}, which contradicts the fact that the ψ_{i}(x, \overline{a}) are mutually inconsistent. Hence, one of the ϕ(x, ν_{α}) ∧ ψ_{i}(x, \overline{a}) is algebraic, as required. We now want to examine the geometry of the strongly minimal sets ϕ(x, v_{α}) and prove they have a field like structure structure. This follows by showing the pregeometry associated to span_{k_{∞}}(ν_{α}) is that of an algebraically closed field. We divide the proof into the following 2 cases;

\underline{Case 1} span_{k_{∞}}(h_{α_{i}}) with α_{i} ∈ Δ(Φ)

We can find ν_{α} in the Chevalley basis with α(h_{α_{i}}) = n ≠ 0. We claim that acl(h_{α_{i}}) is just k_{0}h_{α_{i}} in \overline{M}. In order to see this, suppose λ ∈ acl(∅) in k_{∞}. Then λ satisfies p(x) ∈ Q[x]. Now let q(x) = xp(x\n) and

ψ(x, h_{α_{i}}) ≡ ∃y([h_{α_{i}}, y] = ny ∧ [q(x), y] = 0 ∧ x, y ≠ 0)

Here, [q(x), y] is short for q(ad(x)) ▪ y. This is satisfied by λh_{α_{i}} in \overline{M}. Then if \overline{M} ⊧ ψ(μh_{α_{i}}, h_{α_{i}}) we must have μ ≠ 0 and

[q(μh_{αi}), y] = 0 ⇔ q(nμ) = 0 ⇔ nμp(nμ\n) = 0 ⇔ p(μ) = 0

So ϕ(x, h_{α_{i}}) ∧ ψ(x, h_{α_{i}}) is algebraic as required. Therefore k_{0}h_{α_{i}} ⊂ acl(h_{α_{i}}). Now suppose λh_{α_{i}} with λ ∉ k_{0} satisfies an algebraic formula ψ(x, h_{α_{i}}). By the above we can find infinitely many conjugate elements to λh_{α_{i}} in \overline{M}, by automorphisms fixing L_{Z}⊗k_{0} and therefore h_{α_{i}} which is impossible. Hence acl(h_{α_{i}}) = k_{0}h_{α_{i}} relative to the strongly minimal set span_{k_{∞}}(h_{α_{i}}).

\underline{Case 2} span_{k_{∞}}(ν_{α}) α ∈ Φ_{≠0}

For root spaces not in the Cartan it is harder to come up directly with algebraic formulae, so we use the following indirect method. The second part of the above argument shows that acl(ν_{α}) ⊂ k_{0}ν_{α}. Now in order to show that every element in k_{0}ν_{α} satisfies an algebraic formula over ν_{α}, we prove for λν_{α} ∈ k_{0}ν_{α} that tp(λν_{α}|ν_{α}) is algebraic. Then by the fact \overline{M} is ω-saturated, there must be a formula ψ(x, ν_{α}) which is algebraic for λν_{α}. If tp(λν_{α}|ν_{α}) is not algebraic, by homogeneity of \overline{M}, we can find Ψ ∈ Aut(\overline{M}) taking λν_{α} to μν_{α}, with tp^{k_{0}}(λ) ≠ tp^{k_{0}}(μ), and fixing ν_{α}. However, by Theorem(4.1), we have Ψ = ΘΛ where Λ is an algebra automorphism fixing ν_{α}

Theorem 4.3. The formulae ϕ(x, ν_{α}) for α ∈ Φ_{≠0} and ϕ(x, h_{α_{i}}) for α_{i} ∈ Δ(Φ) define strongly minimal sets with a field like associated pregeometry.

An obvious extension of these arguments shows that a basis for span_{k_{∞}}(ν_{α}) is just {λ_{i}ν_{α} : λ_{i} independent in k_{∞}}. Moreover if we take a countable sequence of independent elements {λ_{1}, … λ_{n}, … } in k_{∞} then it is clear that the models L_{Z} ⊗k_{n} are prime and minimal over the parameters ν_{α} and λ_{1}ν_{α}, …, λ_{n}ν_{α}. Hence, by the classic argument of Baldwin and Lachlan, it follows that

Theorem 4.4. The countable models of T h(L) are given by

L_{Z} ⊗ k_{0} ≺ L_{Z} ⊗ k_{1} ≺ … ≺ L_{Z} ⊗ k_{n} ≺ …

where all of the L_{Z} ⊗ k_{n} are strictly nonisomorphic.

Another consequence of the preceding results is the fact that all the strongly minimal sets defined over a fixed Chevalley basis are nonorthogonal. In order to see this, let us take for example two elements ν_{α} and ν_{β} in our finite basis, the parameters of which we may denote by c̄. Let L_{α} and L_{β} denote the strongly minimal sets defined by the formulae ϕ(x, ν_{α}) and ϕ(x, ν_{β}) in a saturated model L_{Z} ⊗ k_{∞}. Finally, choose a, b ∈ k_{∞} with a, b not independent over ∅ in k_{∞}. It is now straightforward to see that aν_{α} and bν_{β} are not independent over c̄. For we have seen that aν_{α} ∉ acl(c̄). On the other hand, if tp(av_{α}/c̄ ∪ bν_{β}) were non algebraic; aν_{α} would have infinitely many conjugates under automorphisms fixing c̄∪ bν_{β}; by ! the classification of automorphisma above, this is impossible. Hence, we have

Theorem 4.5. The strongly minimal sets L_{α} are all non orthogonal over the defining parameters c̄ of a Chevalley basis.

A more interesting question would be to ask to what extent the above remains true when we replace c̄ by a smaller subset containg the parameters necessary to define two strongly minimal sets L_{α} and L_{β}. First observe that given simple roots {α_{i} ∈ Δ(Φ)}, we have that the elements {x_{α_{i}}} ∪ {y_{α_{i}}} form a generating set for our Chevalley basis. This is an easy consequence of the geometry of root systems attached to Lie algebras and the fact that all root spaces are 1-dimensional, so all representations of sl_{2}(C) inside C are irreducible.

[x_{α_{i}}y_{α_{i}}] = h_{α_{i}}(x_{α_{i}} ∈ L_{α_{i}}, y_{α_{i}} ∈ L_{−α_{i}})

[x_{α_{1}} … [x_{α_{i_{k−1}}}x_{α_{i_{k}}}] …] = m_{α}x_{α} (m_{α} ∈ Z_{≠0}α ∈ Φ_{+}, α_{i_{1}}, …, α_{i_{k}} ∈ Δ(Φ))

[y_{α_{i_{1}}} … [y_{α_{i_{k−1}}}y_{α_{i_{k}}}] …] = n_{α}y_{α} (n_{α} ∈ Z_{≠0}, α ∈ Φ_{−}, α_{i_{1}}, …, α_{i_{k}} ∈ Δ(Φ))

This just means that our Chevalley basis c̄ is in dcl({x_{α_{i}}} ∪ {y_{α_{i}}}). Hence, Theorem 4.5 remains true over this smaller set. In fact, we can prove the following result which relies on general properties of strongly minimal sets with a field like structure.

Theorem 4.6. Given roots α, β ∈ Φ, the strongly minimal sets L_{α} and L_{β} are non-orthogonal over the defining parameters {ν_{α}, ν_{β}}.

In order to see this, working in L_{Z} ⊗ k_{∞} as usual, choose b ∈ L_{β} with b ∉ acl(ν_{β}). Now choose a prime model L_{Z} ⊗ k_{1} over {ν_{β}, b}, and let a ∈ L_{Z} ⊗ k_{1} ∩ L_{α} with a ∉ acl(ν_{α}, ν_{β}). We just have to show that a ∈ acl(ν_{α}, ν_{β}, b). If not, then tp(a/ν_{α}, nu_{β}, b) is non algebraic, and is the unique non-algebraic type over {ν_{α}, ν_{β}, b} containing ϕ(x, ν_{α}) which defines the strongly minimal set L_{α}. As a ∈ L_{Z} ⊗k_{1}, we must have that tp(a/ν_{α}, ν_{β}, b) is isolated by a formula ψ(x, ν_{α}, ν_{β}, b), with infinite realisations in L_{α} However, no a′ ∈ acl(ν_{α}) can satisfy this formula, hence as this constitutes an infinite set within !L_{α}, we have a contradiction to strong minimality of L_{α} and the theorem is proved. Combining this with the previous result, we see that the equivalenve relation defining non-orthogonality for strongly minimal sets L_{α} and L_{β} holds precisely when we add enough of the generators d̄⊂ {x_{α_{i}}}∪{y_{α_{i}}} to ensure that ν_{α}, ν_{β} ∈ dcl(d̄). This is a purely geometric question question depending on the relationships of the respective roots to a given basis Δ(Φ).

5. Adjoint Orbits

An adjoing orbit is defined to be G(k).x, where x ∈ L and G(k) acts by the adjoint map on L. If x is nilpotent we denote such orbits by V_{x} and if x is semisimple by S_{x}. By a standard calculation, one sees that

G(k).x = \frac{G(k)}{Iso(x)}

where Iso(x) is the isotropy group of x. As G(k) acts algebraically on L, all orbits are affine constructible sets. It can be shown that if H is a subgroup of G containing a Borel subgroup, then \frac{G}{H} carries the structure of a projective variety, hence if Iso(x) is contained in a Borel subgroup, then it must be strictly contained in that group. We first obtain a characterisation of x ∈ L whose isotropy group has connected component H ⊂ G. Given y ∈ L, we have that Ad(exp(y)).x = e^{ad(y)}.x, and hence Ad(exp(y)).x = x iff [y, x] = 0. Therefore {x ∈ L : Iso(x)^{0} = H} = {x ∈ L : [H, x] = 0} where H = Lie(H)

6. Elimination of Quantifiers

We turn to the problem of finding a natural language for which to eliminate quantifiers. The method uses the classical technique of coordinatisng definable sets over a strongly minimal one. As usual, we use the decomposition;

L = H ⊕ ∐_{α∈Φ_{≠0}} L_{α}

We first need the following lemmas.

Lemma 6.1. There exists a regular element h ∈ H such that h lies in the rational span of our fixed Chevalley basis, and if Γ is any automorphism of the root system fixing h then Γ is the identity.

First observe that the condition of regularity is an open condition on the fundamental Weyl chamber associated to our fixed set of simple roots α_{i} ∈ Δ(ϕ). Let us write a candidate h in the form

Σ_{i}λ_{i}α_{i} (λ_{i} ∈ Q_{≥0})

If Γ is any automorphism fixing h, then as Δ(Φ) = {α : (α, γ) > 0 indecomposable}, it follows that Γ must permute the set of simple roots, so Γ is a graph automorphism. Therefore, we must have that

λ_{Γ(i)} = λ_{i}

The union of these conditions for the finite group of graph automorphisms cuts out a finite union of hyperplanes on H^{∗}. It therefore follows that the conditions of the lemma are still an open condition on the fundamental Weyl chamber attached to the simple roots as required

Lemma 6.2. There exists a basis h_{i} for H consisting of regular elements, with each h_{i} lying in the rational span of our fixed Chevalley basis.

Consider the space V spanned by the orbit of h under the action of the Weyl group W in H^{∗}. This is clearly W invariant, and we have that H^{∗} = V ⊕V^{⊥}. Now if α is a root not in V, then V ⊂ P_{α} where P_{α} is the orthogonal hyperplane to α, as V is W invariant. It follows that α ∈ V^{⊥}, so that the root system Φ can be written as a disjoint union of 2 orthogonal subspaces, contradicting irreducibility of L. Hence V = H^{∗} and we can find w_{i} ∈ W such that w_{i}(h) is a basis for H^{∗}. As W preserves the inner product, we still have that the w_{i}(h) are regular elements in H. Finally, we need to check that each w_{i}(h) is still in the rational span of our basis. As W is generated by the reflections s_{α_{i}} in the simple roots, it is sufficient to prove that if h_{α_{i}} is part of our fixeds Chevalley basis, then s_{α_{j}}(α_{i})! is still in its rational span. We have that

s_{α_{j}}(α_{i}) = α_{i} − \frac{2(α_{i},α_{j})}{(α_{j},α_{j})}α_{j}

If the roots α_{i} and α_{j} are orthogonal, we are done as the coeffieient vanishes. Otherwise, the relevent calculation for our basis elements in H shows that;

s_{α_{j}}(h_{α_{i}}) = s_{α_{j}}(\frac{2t_{α_{i}}}{(α_{i},α_{i})}) = h_{α_{i}} − \frac{2(α_{i},α_{j})}{(α_{j},α_{j})}.\frac{(α_{j},α_{j})}{(α_{i},α_{i})}h_{α_{j}}

By standard results we have that \frac{2(α_{i},α_{j})}{(α_{j},α_{j})} is integer valued and \frac{(α_{j},α_{j})}{(α_{i},α_{i})} is rational if α_{i} and α_{j} are not orthogonal roots. This proves the lemma

We now have a field structure on the reduct K = ϕ(x, h)^{L};

Let p(x_{1}, …, x_{m}) ∈ Q[x_{1}, …, x_{m}]. Choose ν_{α} ∈ L_{α} such that h.ν_{α} = nν_{α} with n ≠ 0.

Consider

q(x̄) = x_{1} … x_{m}p(\frac{x_{1}}{n} , …, \frac{x_{m}}{n})

and let

q^{L}(x̄) ≡ ad(x_{1}) … ad(x_{m})p(\frac{ad(x_{1}}{n} , …, ad(\frac{x_{m})}{n}).ν_{α} = 0 ∧ ∧_{i=1}^{m}K(x_{i}) ∧ x_{i} ≠ o Note that as H is abelian the terms in ad(x_{i}) commute. Then,

q(k_{1}h, …, k_{m}h) iff ad(k_{1}h) … ad(k_{m}h)p(\frac{ad(k_{1}h}{n}, …, \frac{ad(k_{n}h)}{n})).ν_{α} = 0 iff k_{1} … k_{m}n^{m}p(k_{1}, …, k_{m}) = 0 iff p(k_{1}, …, k_{m}) = 0

Hence, after naming a set of simple roots, the reduct K carries the structure of an algebraically closed field. Moreover this field structure is definable without quantifiers. It follows that K admits elimination of quantifiers as if ϕ defines a subset of K^{n} then by finite dimensional transfer ϕ is constructible and hence equivalent to a Boolean combination of formula of the form q^{L}(x̄). We now want to decompose arbitrary definable subsets in L.

So choose a set of roots x_{α_{1}}, …, x_{α_{n}}, y_{α_{1}}, …, y_{α_{n}} generating our Chevalley basis, as in the remarks following Theorem 4.5. As h was chosen to be regular, for each positive root α ∈ Φ_{+}, the formula

ψ_{α}^{+}(xy) = [xx_{α_{i_{1}}}]x_{α_{i_{2}}}] … x_{α_{i_{k}}}] = y

puts the root space L_{α} in definable bijection with span(h). Similarily, for each negative root β ∈ Φ_{−}, the formula

ψ_{β}^{−}(xy) = [xy_{α_{i_{1}}}y_{α_{i_{2}}}] … y_{α_{i_{k}}}] = y

puts the root space L_{β} in definable bijection with span(h).

Finally for the w_{i}(h) found in Lemma 5.1, we have by the same arguments that there exists formulae ψ_{i,α}^{+}(xy) and ψ_{i,β}^{−}(xy) depending on i, which put the root spaces L_{α} for α ∈ Φ_{+} and L_{β} for β ∈ Φ_{−} in definable bijection with span(w_{i}(h)). It then follows that the formulae ψ_{0}(xz) obtained by concatenating ψ_{α}(xy) and ψ_{i,α}(yz) for some choice of α put the spaces span(h) and span(w_{i}(h)) in definable bijection.

We now obtain a simple description of definable sets in L by coordinatising with respect to the strongly minimal set span(h) over the generators {x_{α_{i}}, y_{α_{i}}} of our Chevalley basis. . Given a an arbitrary element in L, we have that a is interdefinable with its coordinates (a_{1} … a_{n}) relative to the Chevalley basis over {x_{α_{i}}, y_{α_{i}}}. By the above, we then have that each a_{i} is interdefinable with an element h_{i} ∈ span(h), again over {x_{α_{i}}, y_{α_{i}}}, which gives that a is interdefinable with the sequence (h_{1} … h_{n}) ∈ span(h). Now let ϕ(x̄) be a definable set (possibly with parameters) in L. As L has finite rank, we may divide ϕ(x) into the union ∨ϕ_{i}(x̄) of definable sets of Morley degree 1. For each ϕ_{i}(x̄), choose a realisation ā_{i} generic over the defining parameters of ! ϕ_{i} and the generators of our Chevalley basis. Then it follows we can find tuples h̄_{i} ∈ span(h) interdefinable with ā_{i} in span(h). As span(h) carries a field like structure, we let

V_{i}(x̄) = locus(h̄_{i}/acl(x_{α_{i}}, y_{α_{i}})), V_{i} ⊂ K^{m}

If f_{i}(x̄_{i}, ȳ_{i}) witnesses this interdefinability, then the statements;

{x̄ : ∃!ȳ(f_{i}(x̄, ȳ) ∧ ϕ_{i}(ȳ))}

{ȳ: ∃!x̄(f_{i}(x̄, ȳ) ∧ V_{i}(x̄))}

hold generically on V_{i} and ϕ_{i}, and therefore f_{i} witnesses a definable bijection between U_{i} ⊂ V_{i} and ϕ_{i}^{U} ⊂ ϕ_{i} where V_{i} \ U_{i} and ϕ_{i} \ ϕ_{i}^{U} has lower rank. Repeating the above argument for the lower dimensional sets, we finally exhaust each ϕ_{i}, after at most MR(ϕ_{i}) = n steps and it follows that

L ⊧ ϕ(ȳ) ↔ ⋁_{i_{1}...i_{n}} ∃x̄_{i_{1}...i_{n}}(U_{i_{1}i_{2}...i_{n}}(x̄_{i_{1}...i_{n}}) ∧ f_{i}(x̄_{i_{1}...i_{n}}, ȳ)))

where we have that the f_{i}(x̄, ȳ) are quantifier free and the U_{i}(x̄) are of the form q(x̄) introduced above.

To obtain quantifier elimination, we need to carry out this construction in a uniform way. We first need to find a way of associating elements of a Chevalley basis to arbitrary elements of L. Given x ∈ L we have the abstract Jordan decomposition

x = x_{s} + x_{n}

where ad(x) = ad(x_{s})+ad(x_{n}), ad(x_{s}) is semisimple, ad(x_{n}) is nilpotent and ad(x_{s}), ad(x_{n}) commute. As is easily verified such a decomposition is unique and in the simple case exists. We first add function symbols f_{s} and f_{n} to our language. Taking a saturated model L, we interpret f_{s} and f_{n} in the obvious way. We need to check that T h(L, f_{s}, f_{n}) has the same class of models and automorphisms.

First note that the statement Nilp(x) given by x is ad nilpotent is elementary for T h(L), as all models are of a given fixed dimension n, namely Nilp(x) ≡ ∀y(ad(x)^{n}y = 0)

Less obvious is the fact that SS(x) given by x is ad semisimple is also elementary. In order to see this, we invoke the result of Lemma 5.1. The existence of such a regular element is formalisable in the language of rings, for given a fixed Chevalley basis c̄ with multiplication table given by Mult(c̄ over Z, we have that h = ∑ λ_{i}c_{i} with λ_{i} ∈ Q, c_{i} some fixed ordering of our basis, and with the c̄_{i} characterised uniquely and definablywithinourChevalleybasis by the property SSC(c̄_{i}) that it acts semisimply with integer eigenvalues on that basis. Let Reg(x) be the statement

Reg(x) ≡ x = ∑ λ_{i}x_{i} ∧ Mult(x̄) ∧ SSC(x̄)

then by previous results on Chevalley bases, such a formula isolates the type of h over ∅. We claim that

SS(x) ≡ ∃y(Reg(y) ∧ [yx] = 0)

If x is semisimple, then by standard results we can find a Cartan subalgebra H′ containing it. We need the following lemma,

Lemma 6.1. Any two Cartan subalgebras H and H′in a simple Lie algebras L are conjugate.

The proof uses an argument from algebraic geometry;

It is sufficient to find regular elements h, h′ in H, H′ and an automorphism taking h to h′, as by regularity H = C_{o}(h) and H′ = C_{0}(h′). The set of regular elements H_{reg} and H′_{reg} are Zariski dense in H and we consider the map;

F : H_{reg} × Σ_{α∈Φ_{≠0}}L_{α}

given by

F (h, x_{α_{1}}, … x_{α_{n}}) = exp(ad(x_{α_{1}}) … exp(ad(x_{α_{n}}).h

As the x_{α_{i}} are nilpotent this is an algebraic function, and hence the image of F is constructible. Computing the derivative dF at (h, ¯0) we have,

dF_{h,¯0}(x_{α_{i}}) = \frac{d}{dt}_{t=0}(exp(tx_{α_{i}}).h) = α_{i}(h)

hence by regularity of h it follows easily that the matrix of dF has maximal rank at (h, ¯0), therefore the image of F has Zariski dense image inside L. Using the fact that the intersection of 2 Zariski dense open sets is Zariski dense we see that there exists automorphisms Φ and Ψ with Φ(h) = Ψ(h′) and h, h′ regular in H, H′. Then the automorphism ΦΨ^{−1} is as required. Note that an almost identical argument proves the stronger statement that any two cartan subalgebras H and H′ of a given Borel subalgebra B of L are conjugate under an automorphism fixing B; simply repeat the argument considering only the set of roots spanning B.

Using this Lemma, it follows that we can find h′ in H′_{reg} satisfying the formula Reg(x) and therefore as our original x commutes with h′, it must satisfy SS(x). For the converse, use the fact that if x commutes with h and Reg(h) holds then h must be regular for some Cartan in which x lies. Now it is clear that the statement

∀x(Nilp(f_{n}(x)) ∧ SS(f_{s}(x)) ∧ x = f_{n}(x) + f_{s}(x) ∧ [f_{n}(x), f_{s}(x)] = 0)

holds in L, so if L′ ⊧ T h(L), there exists a unique extension of L′ to a model of T h(L, f_{s}, f_{n}).

We need to finally check that the set of automorphisms is unchanged, which follows trivially from the uniqueness of the Jordan decomposition, and the fact that Nilp and SS are elementary for T h(L).

Having added Jordan functions, it is necessary to associate elements of a Chevalley basis to nilpotent and semisimple elements.

We define B_{x} to be the set of Borel subalgebras containing x and consider the following equivalence relation on L given by

E(x, y) iff B_{x} = B_{y}

It is trivial to see that this does in fact define an equivalence relation, less obvious is that this relation is definable in L.

Now given a Chevalley basis c̄, I claim that a set of simple roots x_{α_{i_{1}}}, …, x_{α_{in}} in that basis is characterised essentially uniquely over the subset of semisimple elements h_{α_{i}}, that is there exists a formula Simple′(x̄, h_{α_{i}}) in n variables over this subset such that if ȳ is any other n-tuple belonging to c̄ such that Simple′(ȳ, h_{α_{i}}) then ȳ corresponds to another set of simple roots in c̄. As there are only finitely many possible distinct n-subsets of c̄, it is sufficient to show that if ȳ is an n-tuple in c̄ not corresponding to a set of simple roots then ȳ realises a different type to x_{α_{i_{1}}}, …, x_{α_{in}} over h_{α_{1}}, … h_{α_{n}}. Otherwise there exists an isomorphism Ψ taking x_{α_{i_{1}}}, …, x_{α_{in}} to ȳ and fixing the corresponding Cartan subalgebra ! H, as it fixes a regular element in the rational span of the h_{α_{1}}, … h_{α_{n}}. We may assume that Ψ is an algebra automorphism by Theorem 4.1, and hence by standard theory maps x_{α_{i_{1}}}, …, x_{α_{in}} to another base Δ(Φ), which is a contradiction. We can easily remove the parameters h_{α_{i}} from our formula Simple′ and obtain a canonical formula Simple(x̄) for the statement that x̄_{1} … x̄_{n} is a set of simple roots for a Chevalley basis (of fixed integral type).

Simple(x_{1} … x_{n}) iff ∃ȳz_{1} … z_{n}(Chev(x̄_{1} … x̄_{n}ȳz_{1} … z_{n}))∧SS(z_{1} … z_{n})∧ Simple′(x_{1} … x_{n}, z_{1} … z_{n})

We are now in a position to prove the definability of the equivalence stated above. We proceed by in fact showing that ¬E(x, y) is definable. If ¬E(a, b) holds, then we can find a Borel subalgebra containing a but not b. By standard results we have that B = B(Δ) for some choice of simple roots associated to a given Cartan decomposition. By the result that all Cartan subalgebras are conjugate and the Weyl group acts transitively on the set of simple roots for a given root system, we may assume that our Chevalley basis spans the root spaces associated to B(Δ). Now we have that a must lie in the span of the semisimple elements and the root spaces generated by the simple roots corresponding to Δ, while b clearly does not. Therefore, we have that

¬E(x, y) ≡ ∃x_{1} … x_{n}ȳ∃z_{1} … z_{n}(Chev(x_{1} … x_{n}ȳz_{1} … z_{n})∧SS(z_{1} … z_{n})∧ Simple(x_{1} … x_{n})∧x ∈ span(x_{1} … x_{n}, [x_{α_{1}} … [x_{α_{i_{n−1}}}x_{α_{in}}] …], z_{1} … z_{n}))

as required. It therefore automatically follows that this relation is invariant under automorphisms of L. Again we add predicate symbols E_{B} to the language and we have that the models and automorphisms of

T h(L, f_{s}, f_{n}, E_{B}(x, y))

are unchanged.

One would like to also add an equivalence relation for the statement that x and y belong to the same set of parabolic subalgebras, as it seems plausible this statement is also first order definable, in progress....

We now want to formulate the notion of being generic with respect to Borel subalgebras, that is the statement characterising those elements belonging to a single Borel. More generally we show that the statement x belongs to n distinct Borel subalgebras is first order definable in L. Given a Borel subalgebra B, we let

B^{H} = {H ⊂ B, H a Cartan subalgebra}

and

E_{B}(H_{1}, H_{2}) iff H_{1}, H_{2} ∈ B

We then have the following lemma;

Lemma 6.2. Given a finite collection B_{1}, … B_{n} of distinct Borel subalgebras of L, there exist n distinct Cartan subalgebras H_{1}, …, H_{n} such that

∀1 ≤ i ≤ n, H_{i} ∈ B_{i} and ¬E_{B_{i}}(H_{j}, H_{k}) if j ≠ k

In order to prove this, we just have to show that given a Borel subalgebra B and a finite distinct set of Borels B_{1}, …, B_{n}, we can find a Cartan subalgebra T contained in B but not in the B_{i}. We have that ∪_{1≤i≤n}B ∩ B_{i} forms a proper Zariski closed set V inside B. By for example Lemma 6.1, we can find a semisimiple element x ∈ B \ V. Choose a maximal toral subalgebra T of B containing x, and consider the canonical projection π : B → H. We claim that π maps T onto H. For suppose that x ∈ B, then as T acts semisimply on B, we can write x = x_{0} + y where y is a sum of non-zero roots attached to T . As is easily checked, we must therefore have that y is a sum of ad−!nilpotent elements in B, hence π(y) = 0. and π(x) = π(x_{0}). Now splitting x_{0} = x_{0s} + x_{0n} into its semisimple and nilpotent parts, which are always contained inside any Borel subalgebra, we see that x_{0s} commutes with T and is semisimple, hence by definition must lie in T , and then π(x) = π(x_{0s}) as required. It follows that T must have dimension equal to H and is therefore in fact a Cartan subalgebra.

Now using the lemma, we divide the set B_{x} into disjoint classes, with each class attached to a single Cartan subalgebra, so B_{x} = ∪_{i≤n}(B, H_{i})_{x}. Letting s denote the order of the Weyl group or equivalently the number of Borels attached to a given Cartan decomposition, and < x̄ > the tuple obtained by succesive applications of the bracket operation, the statement thet x lies in 1 Borel with respect to that decomposition is given by

(B, H_{i})(x) ≡ ∃x̄_{1}∃x̄_{2} … ∃x̄_{s}∃ȳz̄(Chev(x̄_{1} … x̄_{s}ȳz̄)∧_{1≤j≤s}(Simple(x̄_{i})∧ SS(z̄) ∧ (x ∈ span(< z̄x̄_{1} >) ∧ r_{2} ≤ j ≤ s(x ∉ span(< z̄x̄_{j} >)

We can also formulate the notion of a set of Chevalley bases being attached to inequivalent Cartan subalgebras in the sense of Lemma 6.2, namely when the semisimple parts of each basis are not in the span of the relevant Borels;

Inequiv(c̄_{1}, c̄_{2}) ≡ SS(c̄_{2}) ∉ span(SS(c̄_{1}), Simple(c_{1}) ∧ SS(c̄_{1}) ∉ span(SS(c̄_{2}, Simple(c_{2})))

Therfore the statement that x belongs to at least n distinct Borels will be

B_{≥n}(x)∃c_{1} … ∃c_{n}(∧_{1≤i,j≤m}Inequiv(c̄_{i}, c̄_{j}) ∧ (B, SS(c_{i}))(x))

which is first order. Note that the statement B_{n}(x) that x belongs to n Borels is just B_{≥n} ∧ ¬B_{≥n+1} A simple consequence is that B_{1}(x) quotiented by E_{B} is a constructible set by elimination of imaginaries for algebraically closed fiels, hence the set of Borel subalgebras carries the structure of an algebraic variety. (in fact one can see that the quotient factors through the equivalence of spaces so lives inside the Grassmannian and is closed therefore projective.

Again we note that the formulae B_{1}, … B_{n} are automorphism invariant so the structure

T h(L, f_{s}, f_{n}, E_{B}, B_{i})

has the same class of models and automorphisms.

We now want to characterise algebraically the conditions B_{n} with respect to a fixed Cartan decomposition given by H. More specifically, let B be a fixed Borel subalgebra spanned by the elements h_{α_{i}} and x_{α} for α ∈ Φ_{>0} Suppose x ∈ B satisfies B_{≥2}, and choose B′ with x ∈ B∩B′. As any 2 Borel subalgebras intersect in a Cartan subalgebra, we find H′ ⊂ B ∩ B′. Now any 2 Cartan subalgebras in B are conjugate under an automorphism Λ fixing B, and we write Λ as a finite product of exponentials exp(y_{i}) with y_{i} ∈ B. Clearly B_{≥2}(exp(y_{i}).x), which forces exp(y_{i}.x) to be in the intersection of 2 Borels with respect to the Cartan decomposition for H. We need to show the following;

Lemma 6.3. x is in the intersection of 2 Borels with respect to a fixed Cartan decomposition

Working in a saturated model L, we add a predicate Reg to name the union of principal orbits under the action of algebra automorphisms in Aut(L) of the regular elements in span(h). Let us first observe that Reg is indeed automorphism invariant. So suppose that y ∈ Reg, therefore y = Λ(h′) with Λ an algabra automorphism and h′ ∈ span(h) is regular. If Ψ is an automorphism, Ψ(y) = ΦΛ′Λ(h′) where Φ is a field automorphism and Λ′ is an algebra automorphism. Using the results of Theorem 4.1, we may assume that Ψ(y) is of the form Λ′ΛΦ(h′). As h was chosen to lie in the rational span of our Chevalley basis, it follows easily that Φ fixes span(h) which gives the result.

Fixing h , we associate a single simple root x_{α_{i}} in our Chevalley basis necessary to put the field structure on span(h). Now if Λ is an automorphism fixing span(h) then it is straightforward to see that Λ fixes the centralizer C_{0}(h) which by regularity is precisely the corresponding Cartan subalgebra H. By the classification of automorphisms, we may assume Λ is induced either by the action of the Weyl group W or a graph automorphism as the field automorphisms fix our Chevalley basis. By Lemma 5.1, We set x(h) = Σ_{Λ∈Γ′}Λ(x_{α_{i}}), and we add a predicate Nilp to define the orbit of x(h) under Aut(L). By the classification of automorphisms this in fact corresponds to a single nilpotent orbit under the action of algebra automorphisms, as the set of diagonal automorphisms acts transitively on span(x(h)). We define a function f_{Reg,Nilp} by setting f_{Reg,Nilp}(span(h)) = x(!h) and f_{Reg,Nilp}(Λ(h′)) = Λ(x(h)) for h′ ∈ span(h) and Λ ∈ Aut(L). We claim this is indeed well defined. If Λ(h′) = Λ′(h′), than Λ′Λ^{−1} fixes h′. As h′ is regular, by the same arguments we may assume that Λ′Λ^{−1} permutes the set of simple roots and lies in Γ′. By definition of x(h) it then follows that Λ′Λ^{−1}(x(h)) = x(h) as required.

Finally, we add predicates f_{i}(xy) to name the possible bijections between span(h) and each root space, so we have added dim(L)+7 predicates in all. We claim that in this new language, T h(L, f_{s}, f_{n}, E_{B}, B_{i}, Reg, Nilp, f_{Reg,Nilp}, f_{i}) admits quantifier elimination and the class of models and their automorphisms is unchanged.

In progress....

References

[1] J.Humphreys, “Introduction to Lie Algebras and Representation Theory”, Springer-Verlag(1972) [2] B.Rose, “The Model Theory of Finite-Dimensional Algebras”, Proc. of L.M.S 40(1980),21-40 [3] C.Chang and H.Keisler, “Model Theory”, North-Holland Publishing Company(1973) [4] A.Pillay, “An Introduction to Stability Theory”, Oxford Logic Guides 8, Oxford Science Publications(1983)

[5] A.Nesin and A.Borovik, “Groups of Finite Morley Rank”, Oxford Science Publications(1994)

[6] A.Nesin, “Non Associative Rings of Finite Morley Rank”, in The Model Theory of Groups, Notre Dame Mathematical Lectures No 11(1989)

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

E-mail address: tdpdp@math.mit.edu