Formal sums over a Lie algebra

Formal series over an (un)twisted affine Lie algebra

Let be quadratic Lie algebra, be an involutive isometry, and and be the associated extended untwisted affine Lie algebra and extended twisted affine Lie algebra respectively. Then to each , we associate the following formal series in and respectively: #m/def/lie

and we will drop the subscripts when it is clear. In the latter

denotes the appropriate projection into the -eigenspace decomposition .¹

Commutation relations

For , we may recast the commutation relations of using the Formal delta and operator as

and in as

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