Lie-rings

David Pierce Athens, summer, 2005

(From Plato’s door, according to Tzetzes [12th c.]:

Μηδε�ς �γεωμ�τρητος ε�σ�τω μου τ�ν στ�γην. Why did the AMS choose a version of this as its motto?)

E is an abelian group, in signature {+, −, 0}.

End(E) is its abelian group of endomorphisms.

Mult(E) is the abelian group of multiplications of E: bi-additive maps

(x, y) ↦ m(x, y) : E × E → E.

If m ∈ Mult(E), then (E, m) is a ring; also, Mult(E) contains m^{•}, the converse of m, given by

m^{•}(x, y) = m(y, x).

Mult(End(E)) contains composition, c, and so

⟨c, c^{•}⟩ ⩽ Mult(End(E));

in particular, the bracket, c − c^{•} or b, is in Mult(End(E)).

A ring (E, m) is:

• associative, if m(m(x, y), z) = m(x, m(y, z));

• commutative, if associative, and m − m^{•} = 0;

• a Lie-ring, if m + m^{•} = 0 and

m(m(x, y), z) = m(x, m(y, z)) − m(y, m(x, z))

(the Jacobi identity).

It is obvious that (End(E), c) is associative;

to see that (End(E), b) is a Lie-ring requires a computation.

Theorem A. Among non-associative rings, Lie-rings are the most “natural” in a precise sense.

Theorem B. There are Lie-rings (E, m) with t in End(E) such that Th(E, m, t) is model-complete and ω-stable.

There is an isomorphism m ↦ λ^{m} : Mult(E) → Hom(E, End(E)),

• where λ^{m} is x ↦ λ^{m}(x) : E → End(E),

• where λ^{m}(x) is y ↦ m(x, y) : E → E.

We can now recast the Jacobi identity:

m(m(x, y), z) = m(x, m(y, z)) − m(y, m(x, z)) becomes

λ^{m}(m(x, y)) = λ^{m}(x) ∘ λ^{m}(y) − λ^{m}(y) ∘ λ^{m}(x)
= b(λ^{m}(x), λ^{m}(y)),

This says λ^{m} is a ring-homomorphism from (E, m) to (End(E), b). Theorem A. Let (p, q) ∈ ℤ × ℤ, and let m be pc − qc^{•}. Call a ring (E, ∗) an m-ring if λ^{∗} is a ring-homomorphism from (E, ∗) to (End(E), m). The following are equivalent:

• (End(E), m) is an m-ring for all E.

• m is c or b or 0.

A derivation of a ring (E, m) is an element D of End(E) such that

D(m(x, y)) = m(Dx, y) + m(x, Dy);

—rearranged,

D(m(x, y)) − m(x, Dy) = m(Dx, y);

—with y removed,

D ∘ λ^{m}(x) − λ^{m}(x) ∘ D = λ^{m}(Dx),

that is,

b(D, λ^{m}(x)) = λ^{m}(Dx);

—with x removed,

λ^{b}(D) ∘ λ^{m} = λ^{m} ∘ D,

that is, the following commutes:

E ⟶^{λ^{m}} End(E)
D↓  ↓^{λ^{b}(D)}
E ⟶_{λ^{m}} End(E)

The derivations of (E, m) compose a subgroup

Der(E, m)

of End(E); this subgroup is closed under b.

For any abelian group E, there is a commutative diagram

\xymatrix{(End(E), b) \ar[r]^{λ^{b}} \ar[dr]_{λ^{b}} & (Der(End(E), c), b) \ar[d]^{⊆} \\  & (End(End(E)), b)}

Suppose now (E, m) is a Lie-ring. Then λ^{m} is the adjoint representation: there is a commutative diagram

\xymatrix{(E, m) \ar[r]^{λ^{m}} \ar[dr]_{λ^{m}} & (Der(E, m), b) \ar[d]^{⊆} \\  & (End(E), b)}

Combining gives

\xymatrix{(E, m) \ar[dr]_{λ^{m}} \ar[drr]^{λ^{b}∘λ^{m}} \\  & (End(E), b) \ar[r]_{λ^{b}} & (Der(End(E), c), b)}

This means that every D in E is now a derivation f ↦ Df of (End(E), c) by the rule

(Df)x = m(D, fx) − f(m(D, x)).

Still (E, m) is a Lie-ring, mapping into (Der(End(E), c), b).

Let t ∈ End(E). Call the structure

(E, m, t)

a vector Lie-ring if:

• {Dt : D ∈ E} is the universe of a sub-field K of (End(E), c), and

• E acts on K as a vector-space (over K) of derivations:

(gD)f = g(Df).

Theorem B.

• The class of vector Lie-rings is elementary.

• If n < ω, then the theory of vector Lie-rings of dimension n is companionable.

• the model-companion of this theory becomes complete and ω-stable when characteristic 0 is specified.

(Related results are being worked out independently by Martin Bays.)