Knowlpedia

Definition

Formal group law

An n-dimensional, not necessarily commutative group multiplication encoded by tuples of formal power series.

Definition

Let RR be a and write X=(X1,…,Xn)X=(X_1,\ldots,X_n), Y=(Y1,…,Yn)Y=(Y_1,\ldots,Y_n). An nn-dimensional formal group law over RR is a tuple

F(X,Y)=(F1(X,Y),…,Fn(X,Y))∈R[[X,Y]]nF(X,Y)=(F_1(X,Y),\ldots,F_n(X,Y)) \in R[[X,Y]]^n

satisfying the identities

F(X,0)=X,F(0,Y)=Y,F(X,0)=X,\qquad F(0,Y)=Y,
F(F(X,Y),Z)=F(X,F(Y,Z)),F(F(X,Y),Z)=F(X,F(Y,Z)),

and admitting a tuple i(X)∈(X)R[[X]]ni(X)\in(X)R[[X]]^n with

F(X,i(X))=0=F(i(X),X).F(X,i(X))=0=F(i(X),X).

All identities are identities of , interpreted using .

Linear and higher-order terms

The unit identities force

F(X,Y)=X+Y+terms of total degree at least 2.F(X,Y)=X+Y+\text{terms of total degree at least \(2\)}.

They also make the inverse tuple exist uniquely by recursive degree. Including the inverse axiom in the definition emphasizes the group structure.

No commutativity axiom is imposed here. A law is commutative when F(X,Y)=F(Y,X)F(X,Y)=F(Y,X). In particular, the is a narrower use of “formal group law.”

Geometric meaning

The tuple FF is the coordinate expression for multiplication on a pointed . Choosing coordinates turns a suitable into a formal group law, and changing coordinates produces an isomorphic law. This relationship is made exact by .

Tangent bracket

Write

F(X,Y)=X+Y+B(X,Y)+O(3),F(X,Y)=X+Y+B(X,Y)+O(3),

where BB is bilinear. The antisymmetric part

[u,v]=B(u,v)−B(v,u)[u,v]=B(u,v)-B(v,u)

is the bracket on the . Over a characteristic-zero field this tangent algebra determines the isomorphism class of the finite-dimensional law. More precisely, every specified homomorphism of tangent integrates to a unique formal-group-law homomorphism; an isomorphism of tangent Lie algebras therefore integrates to a unique isomorphism of laws.

References
  1. Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 1–2, one- and higher-dimensional formal group laws.
  2. Jean-Pierre Serre, Lie Algebras and Lie Groups, second edition, Springer, 1992. Publisher record. Relevant: Part I, formal groups and Lie algebras.