Tensor Hierarchy Algebras and Restricted Associativity

Abstract We study local algebras, which are structures similar to $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -graded algebras concentrated in degrees $$-1,0,1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , but without a product defined for pairs of elements at the same degree $$\pm 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . To any triple consisting of a Kac–Moody algebra $${\mathfrak g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> with an invertible and symmetrisable Cartan matrix, a dominant integral weight of $${\mathfrak g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> and an invariant symmetric bilinear form on $${\mathfrak g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> , we associate a local algebra satisfying a restricted version of associativity. From it, we derive a local Lie superalgebra by a commutator construction. Under certain conditions, we identify generators which we show satisfy the relations of the tensor hierarchy algebra W previously defined from the same data. The result suggests that an underlying structure satisfying such a restricted associativity may be useful in applications of tensor hierarchy algebras to extended geometry.