# Unitary Lie Algebras and Lie Tori of Type BC r, r geq 3 - Mathematics > Rings and Algebras

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Abstract: A Lie G-torus of type X r is a Lie algebra with two gradings - one by anabelian group G and the other by the root lattice of a finite irreducible rootsystem of type X r. In this paper we construct a centreless Lie G-torus of typeBC r, which we call a unitary Lie G-torus, as it is a special unitary Liealgebra of a nondegenerate G-graded hermitian form of Witt index r over anassociative torus with involution. We prove a structure theorem for centrelessLie G-tori of type BC r, r \geq 3, that states that any such Lie torus isbi-isomorphic to a unitary Lie G-torus, and we determine necessary andsufficient conditions for two unitary Lie G-tori to be bi-isomorphic. Themotivation to investigate Lie G-tori came from the theory of extended affineLie algebras, which are natural generalizations of the affine and toroidal Liealgebras. Every extended affine Lie algebra possesses an ideal which is a Lien-torus of type X r for some irreducible root system X r, where by an n-toruswe mean that the group G is a free abelian group of rank n for some n \geq 0.The structure theorem above enables us to classify centreless Lie n-tori oftype BC r, r \geq 3. We show that they are determined by pairs consisting of aquadratic form K on an n-dimensional Z 2-vector space and of an orbit of theorthogonal group of K. We use that result to construct extended affine Liealgebras of type BC r, r \geq 3. Our article completes a large projectinvolving many earlier papers and many authors to determine the centreless Lien-tori of all types.

Author: ** Bruce Allison, Georgia Benkart**

Source: https://arxiv.org/