math.RT — Representation Theory

Class-two Lie algebras and quadratic-dual current constructions

Contributed by Ammar Husain

For a reductive Lie algebra $\s$, a finite-dimensional $\s$-module $R$ and an $\s$-submodule $K\subseteq\bigwedge^2R$, the class-two radical $R\oplus(\bigwedge^2R/K)$ 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 $\sigma X$; we give the structure of the switches $X\to\sigma X$, the obstruction to an involutive switch, and its indecomposable objects in small dimension for semisimple $\s$.

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