Rank-one calibration for the supertannakian attack

Date: 6 September 2026
Role: exact setup and route guard for the next ATTACK
Status: elementary deductions and obligations; no answer to Question 5.16

1. The canonical local algebra

Let $\mathfrak g=\mathfrak g_{\bar0}\oplus\mathfrak g_{\bar1}$ be a Lie superalgebra over a field of characteristic zero and choose $q\in\mathfrak g_{\bar1}$. Define

$$z=\frac12[q,q].$$

The super Jacobi identity applied to $(q,q,q)$ gives $3[q,[q,q]]=0$, hence $[q,z]=0$. Therefore

$$\mathfrak h_q=\mathbf k z\oplus\mathbf k q$$

is a Lie subsuperalgebra (with either summand allowed to vanish), and its enveloping relation is

$$q^2=z,\qquad qz=zq.$$

This construction is basis-free up to rescaling $q$; it is attached to one odd tangent direction rather than to a presentation of $G$.

2. The solved boundary

If $z=0$, the rank-one enveloping algebra is exterior in $q$. The usual odd-translation example $D(\theta)=1$ on $\Lambda(\theta)$ belongs here. It illustrates why an ordinary Jacobson radical need not be stable under odd symmetry, but it does not probe the missing case in Coulembier–Sherman Question 5.16.

The paper’s partial theorem applies when the whole odd Lie part is an ideal. Because $[\mathfrak g_1,\mathfrak g_1]$ is even, that hypothesis forces the odd bracket to vanish. The first genuinely new local datum is therefore $z=q^2\ne0$.

3. Two functorial regimes to separate

The even one-parameter closure generated by $z$ has a Jordan decomposition. On each finite rational representation this yields:

• a weight decomposition for the diagonalizable part;
• a finite filtration for the unipotent part.

Since $[q,z]=0$, the odd action respects this decomposition/filtration. On an algebra object, the even action is a derivation, so weights add under multiplication and the unipotent filtration interacts with multiplication by the Leibniz rule.

The next attack must make this statement at the level of $A$-modules and relative tensor products. Merely proving it for the underlying super vector space is insufficient.

4. Exact proof obligations

1. Integration. Specify when $\mathfrak h_q$ integrates to a closed rank-one subsupergroup. Where it does not, state the restriction theorem in the distribution-algebra category and prove that it detects the relevant exact sequences.
2. Rank-one flatness. Relate the $z$-filtered associated graded algebra to the square-zero odd case and determine whether rigidity/flatness lifts through one filtration step.
3. Detection. Prove that nonzero $\operatorname{Tor}_1^A$, or the corresponding failure of bimodule rigidity, survives some rank-one restriction.
4. Internal versus ordinary radical. Never substitute the ordinary radical after forgetting the supergroup action. Only invariant ideal subobjects enter internal semisimplicity.
5. Counterexample standard. A negative answer requires explicit $G$-equivariant $A$-modules $M,N$ and a nonzero internal Tor class, not merely a nonflat underlying ordinary module.

5. Route-killing test

If a nonzero obstruction class restricts to zero for every odd line $\mathfrak h_q$, document that example and stop the rank-one route. A move to higher-dimensional odd subspaces is justified only if a bounded intrinsic detection rank follows from the example. No low-dimensional algebra census is authorized by this packet.

Sources

• Kevin Coulembier and Alexander Sherman, Absolutely flat algebras in tensor categories, especially Theorems 3.17, 5.15, 5.18 and Question 5.16.
• Kevin Coulembier, Mateusz Stroiński, and Tony Zorman, Simple algebras and exact module categories.
• Alexander Sherman, On symmetries of the Duflo–Serganova functor, methodological precedent for odd-direction reduction.
• Dave Benson and Julia Pevtsova, Group schemes and their Lie algebras over a symmetric tensor category, background for group schemes, distributions, and tangent Lie objects.