### Self-Referential Algebraic Structures

#### Posted by John Baez

Any group acts as automorphisms of itself, by conjugation. If we differentiate this idea, we get that any Lie algebra acts as derivations of itself. We can then enhance this in various ways: for example a Poisson algebra is both a Lie algebra and a commutative algebra, such that any element acts as derivations of both these structures.

Why do I care?

In my paper on Noether’s theorem I got excited by how physics uses structures where each element acts to generate a one-parameter group of automorphisms of that structure. I proved a super-general version of Noether’s theorem based on this idea. It’s Theorem 8, in case you’re curious.

But the purest expression of the idea of a “structure where each element acts as an automorphism of that structure” is the concept of “rack”.

Even simpler than a rack is a “shelf”.

Alissa Crans defined a **shelf** to be a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a shelf endomorphism.

If you don’t like the circularity of this definition (which is the whole point), what I’m saying is that a shelf is a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that

$a \triangleright (b \triangleright c) = (a \triangleright b) \triangleright (a \triangleright c)$

This is called the **self-distributive law**, since it says the operation $\triangleright$ distributes over itshelf. (Sorry: “itself”.)

Actually this is a **left shelf**; there is also a thing called a “right shelf”. Any group is both a left shelf and a right shelf in a natural way, via conjugation.

We can similarly define a **rack** to be a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a rack *automorphism*.

In other words, a rack is a shelf $S$ where each map $a \triangleright - \colon S \to S$ is invertible.

It’s common to write the inverse of the map $a \triangleright -$ as $- \triangleleft a$. Then we can give a completely equational definition of a rack! It’s a set $S$ with two binary operations $\triangleright, \triangleleft \colon S \times S \to S$ obeying these identities:

$(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright (b \triangleright a), \qquad a \triangleleft(b \triangleleft c) = (a \triangleleft b) \triangleleft(a \triangleleft c)$

$(a \triangleleft b) \triangleright a = b , \qquad a \triangleleft(b \triangleright a) = b$

These axioms are slightly redundant, but nicely symmetrical.

If we have a shelf that’s a smooth manifold, and the operation $\triangleright$ is smooth, and an element $e \in S$ obeying $e \triangleright a = a$ for all $a \in S$, then I think we can differentiate the shelf operation at $e$ and get a “Leibniz algebra”.

A **Leibniz algebra** is a vector space $L$ with a bilinear operation $[-,-] \colon L \times L \to L$ such that each element $a \in L$ gives a map $[a,-] \colon L \to L$ that is a Leibniz algebra derivation. In other words, the **Jacobi identity** holds:

$[a, [b,c]] = [[a,b], c] + [b, [a,c]]$

Again, this is really a **left Leibniz algebra**, and we could also consider “right” Leibniz algebras. Any Lie algebra is naturally both a left and a right Leibniz algebra.

I’ve been trying to understand Jordan algebras, and here’s one thing that intrigues me: in a Jordan algebra $A$, any *pair* of elements $a,b \in A$ acts to give a derivation of the Jordan algebra, say $D_{a,b} \colon A \to A$, as follows:

$D_{a,b} (c) = a \circ (b \circ c) - b \circ (a \circ c)$

If $L_a$ means left multiplication by $a$, this says $D_{a,b} = L_a L_b - L_b L_a$.

This property of Jordan algebras seems nicer to me than the actual definition of Jordan algebra! So we could imagine turning it into a definition. Say a **pre-Jordan algebra** is a vector space $A$ equipped with a bilinear map $\circ : A \times A \to A$ such that for any $a,b \in A$, the map $D_{a,b} \colon A \to A$ defined above is a derivation, meaning

$D_{a,b}(c \circ d) = D_{a,b} (c) \circ d + c \circ D_{a,b}(d)$

for all $c, d \in A$.

**Puzzle.** Are there pre-Jordan algebras that aren’t Jordan algebras? Are any of them interesting?

## Re: Self-Referential Algebraic Structures

I don’t know about the puzzle, but self-actions have also been on my mind recently, so here are some random thoughts somewhat “off the shelf”. Let’s start with the definition of shelf:

$a \triangleright (b \triangleright c) = (a \triangleright b) \triangleright (a \triangleright c)$

Are you sure that this self-distributivity equation implies that every self-map $a \triangleright -$ is invertible? For example, it seems to me that the implication also makes every Heyting algebra into a shelf, because both sides then are equivalent to $(a \wedge b) \Rightarrow c$, but clearly the map $a \Rightarrow -$ is almost never invertible. So unless I’ve made a silly mistake, already implication on the Booleans defines a shelf, and in general the elements of a shelf only acts by endomorphisms rather than automorphisms. But building in inverses and hence invertibility shouldn’t be difficult.

Here are some further questions on the category theory of shelves:

Puzzle 2: What are the elements of the free shelf generated by one element? The free shelf generated by $n$ elements?Puzzle 3: Does the category of shelves have any extra structure which encodes the self-actions?Puzzle 4: If so, does the category of shelves have some kind of universal property, e.g. something like being terminal among all categories of self-acting objects?Perhaps Alissa has already looked into questions of this sort? I haven’t read the whole thesis yet.

I’m not familiar enough with the details to say much about this, but shelves also remind me of Wolfgang Rump’s cycloids and L-algebras. A cycloid is a set with a binary operation satisfying the equation $(a \triangleright b) \triangleright (a \triangleright c) = (b \triangleright a) \triangleright (b \triangleright c) ,$ and an L-algebra is essentially a cycloid together with a (suitably defined) unit element. I bet that the appearance of self-similarity and the Yang-Baxter equation in both Alissa’s thesis and Rump’s papers is no coincidence. Again implication in a Heyting algebra defines an L-algebra structure, and there also seem to be connections with quantum logic!