Super-Chevalley Restriction and Relative Lie Algebra Cohomology over the 2|3 Algebra
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| Format: | Preprint |
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2026
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| _version_ | 1866908993741389824 |
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| author | Chang, Chi-Ming |
| author_facet | Chang, Chi-Ming |
| contents | Let $A:=\mathbb{C}[z_+,z_-]\otimes Λ(θ_1,θ_2,θ_3)$, with $z_\pm$ even and $θ_1,θ_2,θ_3$ odd. For a reductive Lie algebra $\mathfrak g$, let $\mathfrak g[A]:=\mathfrak g\otimes A$ be the corresponding current Lie superalgebra. Motivated by the Chang--Yin description of weak-coupling $1/16$-BPS cohomology in $\mathcal N=4$ super-Yang--Mills, we study the relative Lie algebra cohomology $H^\bullet(\mathfrak g[A],\mathfrak g;\mathbb{C})$. We isolate three finite-rank phenomena. First, the natural $3|2$ super-commuting restriction map, viewed as a super analogue of Chevalley restriction and its commuting-scheme variants, already fails to be an isomorphism for $\mathfrak g=\mathfrak{so}_7$; the obstruction is a non-Cartan class. Second, the same algebra produces explicit fortuitous classes for $\mathfrak{sl}_2$ and $\mathfrak{so}_7$, giving concrete counterexamples to naive stable-image expectations suggested by the type-A Loday--Quillen--Tsygan theorem and its current-algebra refinements. Third, the classical relative cohomologies for the Langlands-dual pair $(\mathfrak{so}_7,\mathfrak{sp}_6)$ are not isomorphic. We then record the conjectural quantum deformation of the differential expected to restore duality, together with first-order evidence pairing the fortuitous and non-Cartan $\mathfrak{so}_7$ classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23549 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Super-Chevalley Restriction and Relative Lie Algebra Cohomology over the 2|3 Algebra Chang, Chi-Ming Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra 17B56 (Primary), 13A50, 14L30, 17B45, 17B81 (Secondary) Let $A:=\mathbb{C}[z_+,z_-]\otimes Λ(θ_1,θ_2,θ_3)$, with $z_\pm$ even and $θ_1,θ_2,θ_3$ odd. For a reductive Lie algebra $\mathfrak g$, let $\mathfrak g[A]:=\mathfrak g\otimes A$ be the corresponding current Lie superalgebra. Motivated by the Chang--Yin description of weak-coupling $1/16$-BPS cohomology in $\mathcal N=4$ super-Yang--Mills, we study the relative Lie algebra cohomology $H^\bullet(\mathfrak g[A],\mathfrak g;\mathbb{C})$. We isolate three finite-rank phenomena. First, the natural $3|2$ super-commuting restriction map, viewed as a super analogue of Chevalley restriction and its commuting-scheme variants, already fails to be an isomorphism for $\mathfrak g=\mathfrak{so}_7$; the obstruction is a non-Cartan class. Second, the same algebra produces explicit fortuitous classes for $\mathfrak{sl}_2$ and $\mathfrak{so}_7$, giving concrete counterexamples to naive stable-image expectations suggested by the type-A Loday--Quillen--Tsygan theorem and its current-algebra refinements. Third, the classical relative cohomologies for the Langlands-dual pair $(\mathfrak{so}_7,\mathfrak{sp}_6)$ are not isomorphic. We then record the conjectural quantum deformation of the differential expected to restore duality, together with first-order evidence pairing the fortuitous and non-Cartan $\mathfrak{so}_7$ classes. |
| title | Super-Chevalley Restriction and Relative Lie Algebra Cohomology over the 2|3 Algebra |
| topic | Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra 17B56 (Primary), 13A50, 14L30, 17B45, 17B81 (Secondary) |
| url | https://arxiv.org/abs/2604.23549 |