A preprint has established an upper bound for the index of a broad class of Borel subalgebras and has proved zero-index statements for several associated modules. The main theorem covers Borel subalgebras of gl(m|n) and of basic classical Lie superalgebras, excluding psl(2|2). Its module results include the nilradical, every ideal inside it, and the dual of every abelian ideal contained in it.
The central inequality is ind(b) ≤ rank(g) − |K(∆+)|, where K(∆+) is the Kostant cascade of the positive roots. In plain terms, the index cannot exceed the rank of the algebra after the size of this cascade has been subtracted. When the cascade has as many members as the rank, the result becomes ind(b)=0. The paper gives g=osp(m,2n), with m not congruent to 2 modulo 4 and the standard Borel subalgebra, as an example of that zero-index condition.
A root-system method extended
The proof strategy extends strongly orthogonal roots and the Kostant cascade to generalized root systems, a broader root-system framework used for the study’s algebraic arguments. For a non-trivial irreducible generalized root system, the paper states that the cascade is strongly orthogonal, giving the root-level structure used in the later dimension and index calculations.
The same theorem places a limit on the size of the construction: |K(∆+)| ≤ dim(V), where V is the ambient space in the generalized-root-system setting. A parallel construction starts from each non-empty abelian ideal A and forms a set denoted A(A). The paper states that this constructed set is strongly orthogonal, extending the cascade-style structure to the abelian-ideal problem.
The authors also extend the index to g-modules through a definition based on orbit codimension. In this formulation, an ideal i of a Borel subalgebra b is treated as a b-module under the adjoint action. That translation enables questions about ideals to be expressed in the same module-index framework as the broader results.
Zero index reaches related modules
The vanishing results go beyond the Borel subalgebra itself. One corollary states that ind(b,n)=0 for the nilradical module and that ind(b,i)=0 for every ideal i contained in the nilradical. The result therefore covers both the nilradical and all ideals inside it within the stated algebraic setting.
A separate theorem treats abelian ideals contained in the nilradical. It states that ind(b,a∗)=0 for the dual module of every such abelian ideal. The proof reaches this conclusion after establishing the bracket identity [b,eA(a)]=a, which supplies the key step in the argument.
The broader claim remains open
The upper bound is not presented as the final answer in every case. The authors suggest that ind(b) ≤ rank(g) − |K(∆+)| may generally be an equality, but they do not prove that stronger statement. The established result is the inequality itself, while equality remains an open question within the paper’s programme.
The main results do not cover psl(2|2). The stated reason is that this basic classical case has some two-dimensional root spaces, while the other cases discussed have root spaces that are one-dimensional. Because the arguments do not cover those higher-dimensional root spaces, the theorem does not settle psl(2|2).
The conclusions have a defined mathematical boundary. Theorem 1.1 is limited to Borel subalgebras of gl(m|n) and the specified basic classical Lie superalgebras, so the bound should not be read as a result for structures outside that scope. The osp example is one stated instance of the zero-index condition, rather than a blanket assertion that every algebra in the study has index zero.
The supplied front matter identifies the manuscript as arXiv version 1, dated 25 August 2026. It presents formal mathematical conclusions for the stated Lie-superalgebra classes, Borel subalgebras and related modules, with the possible equality in the upper bound still unproved.
Paper data and sources
Original title: On the Index of Borel Subalgebras of Lie Superalgebras
Authors: Simon M. Goodwin, Samuel Renforth
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text