HEXAGONbeta

Class-two Lie algebras and quadratic-dual current constructions

A mix of human-written and AI-generated textHuman understanding: all partsmath.RT — Representation Theory

Contributed by Ammar Husain ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

For a reductive Lie algebra $\s$, a finite-dimensional $\s$-module RR and an $\s$-submodule K⊆⋀2RK\subseteq\bigwedge^2R, the class-two radical R⊕(⋀2R/K)R\oplus(\bigwedge^2R/K) carries an $\s$-action. For semisimple $\s$ the semidirect products $\s\ltimes(R\oplus(\bigwedge^2R/K))$ are, up to isomorphism, all Lie algebras whose radical has class at most two. We study the isomorphisms that fix the acting algebra pointwise, including their central mixing and shear terms. On the locus where central mixing vanishes, the resulting groupoid admits a description by invariant Grassmannians and equivariant linear groups. We express the wild matrix-pair construction as a functor into class-two Lie algebras and give its equivariant square in $\s$-modules. We work out single-isotypic slices, signed multiplicity-free tensor squares, and explicit low-dimensional multiplicity-space examples. The acting algebra may be zero and its central action need not be semisimple. Finally, we study pairs of Lie brackets on one object whose mixed composites vanish. Tensoring them with algebras carrying two independent commutative associative products, for instance pointwise multiplication and convolution exchanged by Fourier transform, produces further Lie algebras. For these pairs, in representations of $\s$, we define the groupoid of objects that are isomorphic to their bracket exchange σX\sigma X; we give the structure of the switches X→σXX\to\sigma X, the obstruction to an involutive switch, and its indecomposable objects in small dimension for semisimple $\s$.

Author comments: The choices of structures are on the author. So them being maximally unhelpful is due to them. The structures can be too weak or too strong to be of any use and that is retained as the author's responsibility.

Provenance statement

The categories/groupoids/operads for study were chosen by the author. Each section was then generated and revised between author, Opus 5.5 and Astra 6 with each playing roles of generator and reviewer at various stages.

Tools used

Anthropic
Claude Opus
OpenAI
ChatGPT
Anthropic
Claude Code
OpenAI
Codex

References

  1. D. Burde and F. Wagemann, ``Cohomology of perfect Lie algebras,'' preprint, arXiv:2411.14952 (2024). arXiv:2411.14952 (https://arxiv.org/abs/2411.14952).arXiv
  2. G. Bazzoni and V. Muñoz, ``Classification of minimal algebras over any field up to dimension 6,'' Transactions of the American Mathematical Society 364 (2012), 1007--1028. doi:10.1090/S0002-9947-2011-05471-1 (https://doi.org/10.1090/S0002-9947-2011-05471-1).DOI
  3. G. Bazzoni and J. Rojo, ``8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of characteristic 0,'' arXiv:2507.14907 (2025). arXiv:2507.14907 (https://arxiv.org/abs/2507.14907).arXiv
  4. M. A. Gauger, ``On the classification of metabelian Lie algebras,'' Transactions of the American Mathematical Society 179 (1973), 293--329. doi:10.1090/S0002-9947-1973-0325719-0 (https://doi.org/10.1090/S0002-9947-1973-0325719-0).DOI
  5. J. Scheuneman, ``Two-step nilpotent Lie algebras,'' Journal of Algebra 7 (1967), 152--159.
  6. G. Belitskii, R. Lipyanski, and V. V. Sergeichuk, ``Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild,'' Linear Algebra and its Applications 407 (2005), 249--262. doi:10.1016/j.laa.2005.05.007 (https://doi.org/10.1016/j.laa.2005.05.007).DOI
  7. J. R. Stembridge, ``Multiplicity-free products and restrictions of Weyl characters,'' Representation Theory 7 (2003), 404--439. doi:10.1090/S1088-4165-03-00150-X (https://doi.org/10.1090/S1088-4165-03-00150-X).DOI
  8. B. Ren and L. S. Zhu, ``Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center,'' Czechoslovak Mathematical Journal 67 (2017), 953--965. doi:10.21136/CMJ.2017.0253-16 (https://doi.org/10.21136/CMJ.2017.0253-16).DOI
  9. L. Yu. Galitski and D. A. Timashev, ``On the classification of metabelian Lie algebras,'' Journal of Lie Theory 9 (1999), 125--156. Journal of Lie Theory archive (https://jolt.centre-mersenne.org/articles/10.5802/jolt.163/).DOI
  10. P. Eberlein, ``The moduli space of 2-step nilpotent Lie algebras of type (p,q)(p,q),'' in Explorations in Complex and Riemannian Geometry , Contemporary Mathematics 332 , American Mathematical Society (2003), 37--72. doi:10.1090/conm/332/05929 (https://doi.org/10.1090/conm/332/05929).DOI
Show all 19 referencesShow fewer references
  1. F. Grunewald and J. O'Halloran, ``Varieties of nilpotent Lie algebras of dimension less than six,'' Journal of Algebra 112 (1988), 315--325. doi:10.1016/0021-8693(88)90093-2 (https://doi.org/10.1016/0021-8693(88)90093-2).DOI
  2. Jean-Louis Loday and Bruno Vallette, Algebraic Operads , Grundlehren der mathematischen Wissenschaften 346 , Springer, Heidelberg, 2012. PDF (https://www.math.univ-paris13.fr/~vallette/Operads.pdf).link
  3. Vladimir Dotsenko and Anton Khoroshkin, ``Character formulas for the operad of two compatible brackets and for the bihamiltonian operad,'' Functional Analysis and Its Applications 41 (2007), no. 1, 1--17. doi:10.1007/s10688-007-0001-0 (https://doi.org/10.1007/s10688-007-0001-0).DOI
  4. V. Ginzburg and M. Kapranov, ``Koszul duality for operads,'' Duke Mathematical Journal 76 (1994), no. 1, 203--272. arXiv:0709.1228 (https://arxiv.org/abs/0709.1228).arXiv
  5. M. S. Jakobsen, ``On a (no longer) new Segal algebra: a review of the Feichtinger algebra,'' Journal of Fourier Analysis and Applications 24 (2018), 1579--1660. doi:10.1007/s00041-018-9596-4 (https://doi.org/10.1007/s00041-018-9596-4).DOI
  6. I. Z. Golubchik and V. V. Sokolov, ``Compatible Lie brackets and the Yang--Baxter equation,'' Theoretical and Mathematical Physics 146 (2006), no. 2, 159--169.
  7. R. C. Thompson, ``Pencils of complex and real symmetric and skew matrices,'' Linear Algebra and its Applications 147 (1991), 323--371.
  8. M. A. Semenov-Tian-Shansky, ``What is a classical rr-matrix?,'' Funktsional'nyi Analiz i ego Prilozheniya 17 (1983), no. 4, 17--33; English translation: Functional Analysis and Its Applications 17 (1983), no. 4, 259--272. doi:10.1007/BF01076717 (https://doi.org/10.1007/BF01076717).DOI
  9. N. Bourbaki, Lie Groups and Lie Algebras, Chapters 1--3 , Elements of Mathematics, Springer, Berlin, 1989.

Version history

  1. v1 · 51 pagesSubmitted by Ammar HusainThe choices of structures are on the author. So them being maximally unhelpful is due to them. The structures can be too weak or too strong to be of any use and that is retained as the author's responsibility.CurrentNov 12, 2026