Is there a canonical Hamiltonian formalism for hypergraph rewriting? This question led us to the search for canonical Lie brackets and their generalizations. It turned out that gluing of two hypergraphs at a common sub-hypergraph generates various brackets on the vector space spanned by hypergraphs. We get Lie brackets if the common sub-hypergraph is a single vertex or a single n-edge. Each of these induces a Poisson bracket on the space of functions on hypergraphs. We can compute, for example, the Poisson bracket of the number of vertices and the number of edges. We elaborate on the idea that these brackets descend from a commutator bracket on the ruliad of rewriting rules. We propose graded versions. We propose some natural simultaneous gluings of multiple hypergraphs that may lead to higher arity algebras. Finally, we elaborate on what structural information can be read off from the multiway system .

Motivation from Hamiltonian formalism

Hamiltonian formalism consists of an algebra of observables
equipped with a Poisson bracket
. The Leibniz identity​
allows the interpretation of
as an infinitesimal generator (of a transformation of an underlying space). The Lie algebra relations Antisymmetry:
​ Jacobi identity:
​allow Lie integration of subalgebras of observables (to, e.g., symmetry groups). In classical mechanics,
is the algebra of functions on the phase space and
is the canonical Poisson bracket. The bracket is uniquely specified by the requirement that the phase space transformation induced by canonical momenta in certain direction is the linear translation of spatial coordinates in the same direction.

Insertion Lie bracket on rooted hypergraphs

The simplest Lie bracket we consider is a straightforward generalization of the insertion Lie bracket for rooted graphs from M. Marcolli, A. Port, Graph grammars, insertion lie algebras, and quantum field theory, 2015. The insertion Lie bracket of two rooted hypergraphs
G
1
and
G
2
is defined as the sum of differences
where
in the first term is glued by its root to a vertex
and
in the second term is glued by its root to a vertex
:
In[]:=
{g1,g2}={MarkRoot[PathHypergraph[2,2],1],MarkRoot[CorollaHypergraph[2,4],2]}​​RootedLieBracket[g1,g2]
Out[]=

,

Out[]=

1,
1,
1,
-1,
-1,
-1
The evolution of
y
along the transformation induced by
x
can be approximated by the Campbell identity:​
In[]:=
{g1,g2}={MarkRoot[CorollaHypergraph[1,1],1],MarkRoot[SimplexHypergraph[2,{3}],1]}​​CampbellIdentity[RootedLieBracket,<|g1->t|>,g2,3]
Out[]=

,

Out[]=

1,
t,
-t,

2
t
2
,
-
2
t
,
-
2
t
2
,

2
t
2
,

2
t
2
,

2
t
2
,

3
t
6
,
-
3
t
6
,
-
3
t
2
,
-
3
t
3
,

3
t
2
,

3
t
2
,

3
t
2
,

3
t
2
,

3
t
6
,

3
t
6
,
-
3
t
6
,
-
3
t
6
,
-
3
t
3
,
-
3
t
3
,
-
3
t
3
,
-
3
t
6

The insertion bracket extends bi-linearly to the vector space spanned by isomorphism classes of rooted hypergraphs
which we denote by
. Coefficients in a linear combination can be interpreted as values of a function on
so that
can be seen as a Lie bracket
on functions on
:
​It is well-defined even for non-finitary functions, like the number of vertices
or the number of
n
-edges
, since a hypergraph can become disconnected by removing one vertex in only finitely many ways:
In[]:=
g=RootedHypergraph[1,{{1,2},{2,3},{3,4},{4,2},{2,3,4}}]​​RootedPoissonBracket[Vf,Ef[2]][g]
Out[]=
Out[]=
3
If we interpret functions on
as linear functions on
, then non-linear functions on
can be understood as formal Taylor series whose monomials are symmetric products of linear functions. A Lie bracket on linear functions always extends to a Poisson bracket on non-linear functions:

Insertion Lie bracket on hypergraphs with tagged n-edge

Graded version

Commutator bracket on ruliad

Higher arity brackets

Higher arity relations

Structure on multiway system

Acknowledgments

Huge thanks to Carlos Zapata-Carratalá for setting the direction of the project, many discussions, and great enthusiasm. Huge thanks to Nik Murzin for pair programming, answering many technical questions, and helping visualizations until the last moment. Huge thanks to Xerxes Arsiwalla for sharing his insights and discussing topics from a higher perspective. All three were great inspiration for me. Huge thanks to all the staff of the Wolfram Winter School for the seamless organization, and huge thanks to Stephen Wolfram for supporting it. I loved the lectures and sessions and found the projects of the Wolfram Institute fascinating!

Further directions

◼
  • Construct the graded versions.
    • ◼
  • Can we have co-products and co-brackets by dissecting hypergraphs?
    • ◼
  • Does it make sense to classify the Lie algebras constructed above and compute some of their invariants?
    • ◼
  • Can any of this structure be used to make statements about dynamics of rewriting rules?
    • ◼
  • Can any of the Lie brackets be uniquely characterized by a dynamical property similar to the case of classical mechanics?

    • Code backbone

    CITE THIS NOTEBOOK

    Lie brackets on hypergraphs​
    by Pavel Hajek​
    Wolfram Community, STAFF PICKS, January 16, 2025
    ​https://community.wolfram.com/groups/-/m/t/3358052