Teleparallelism in the algebraic approach to extended geometry

Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered $L_\infty$ structure of the local symmetries (the diffeomorphisms and their reducibility) to incorporate physical fields, field strengths and Bianchi identities, and identify these as elements of the tensor hierarchy algebra. The field strengths arise as generalised torsion, so the naturally occurring complex in the $L_\infty$ algebra is $\ldots\leftarrow$ torsion BI's $\leftarrow$ torsion $\leftarrow$ vielbein $\leftarrow$ diffeomorphism parameters $\leftarrow\ldots$ In order to obtain equations of motion, which are not in this complex, (pseudo-)actions, quadratic in torsion, are given for a large class of models. This requires considering the dual complex. We show how local invariance under the compact subgroup locally defined by a generalised metric arises as a "dual gauge symmetry" associated with a certain torsion Bianchi identity, generalising Lorentz invariance in the teleparallel formulation of gravity. The analysis is performed for a large class of finite-dimensional structure groups, with $E_5$ as a detailed example. The continuation to infinite-dimensional cases is discussed.