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Varieties of affine Kac-Moody algebras. (English. Russian original) Zbl 0922.17005
Math. Notes 62, No. 1, 80-86 (1997); translation from Mat. Zametki 62, No. 1, 95-102 (1997).
Let $$G$$ be a finite dimensional complex simple Lie algebra. The affine Kac-Moody algebras are defined as $$\widetilde G=G\otimes {\mathbb{C}}[t^{-1},t]\oplus{\mathbb{C}}\theta$$, with multiplication $$(a\otimes f)(b\otimes g)=ab\otimes fg+ (a,b)\text{Res}(g{df\over dt})\theta$$, where $$a,b\in G$$, $$(a,b)$$ is the value of the Killing form in $$G$$, $$f,g\in{\mathbb{C}}[t^{-1},t]$$ are Laurent polynomials, $$\text{Res}(u)$$ is the residue of $$u\in{\mathbb{C}}[t^{-1},t]$$ and $$\theta$$ is a central element of $$\widetilde G$$. The algebra $$\widehat G=\widetilde G+{\mathbb{C}}d$$ is obtained using a concrete derivation $$d$$ of $$\widetilde G$$. The algebras $$\widetilde G$$ and $$\widehat G$$ are called nontwisted. There are also twisted versions $$\widetilde G(\sigma,m)$$ and $$\widehat G(\sigma,m)$$ constructed using an automorphism $$\sigma$$ of $$G$$ of order $$m$$.
In the paper under review the author studies the polynomial identities of Kac-Moody algebras. First he shows that the varieties of Lie algebras $$\text{var }\widetilde G$$ and $$\text{var }\widetilde G(\sigma,m)$$ are special, i.e., are generated by Lie subalgebras of associative PI-algebras. On the other hand, the varieties $$\text{var }\widehat G$$ and $$\text{var }\widehat G(\sigma,m)$$ are not special and even do not belong to the class of the so called varieties of associative type. Then the author proves that the algebras $$\widetilde G$$ and $$\widetilde G(\sigma,m)$$, respectively $$\widehat G$$ and $$\widehat G(\sigma,m)$$, have the same polynomial identities and two algebras $$\widehat G$$ and $$\widehat H$$ generate the same variety if and only if $$\widehat G$$ and $$\widehat H$$ are isomorphic which is an analogue of a previous result of the same author [M. V. Zaitsev, Mosc. Univ. Math. Bull. 51, No. 2, 29-31 (1996); translation from Vestn. Mosk. Univ., Ser. I 1996, No. 2, 33-36 (1996; Zbl 0881.17021)] on the identities of $$\widetilde G$$ and $$\widetilde H$$.
Another important result in the paper is that for any subfield $$\Phi$$ of $$\mathbb{C}$$ and any finite dimensional Lie $$\Phi$$-algebra $$G$$ the relatively free algebras of the variety $${\mathfrak V}=\text{var} G$$ are subalgebras of the algebra $$W_s$$ of derivations of $$\Phi[x_1,\ldots,x_s]$$ and the variety $${\mathfrak V}{\mathfrak A}$$ is a subvariety of $$\text{var} W_s$$ for some $$s$$. In particular this result implies that the variety $$\text{var }\widehat G$$ and any subvariety of $${\mathfrak A}^3$$ generated by a finitely generated algebra have exponential growth of the codimension sequence and this property is typical for associative PI-algebras.

##### MSC:
 17B01 Identities, free Lie (super)algebras 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B30 Solvable, nilpotent (super)algebras 17B66 Lie algebras of vector fields and related (super) algebras
Zbl 0881.17021
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##### References:
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