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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.

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
Full Text: DOI
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[2] V. G. Kac,Infinite Dimensional Lie Algebras. An Introduction, Birkhäuser, Boston-Basel-Stuttgart (1983).
[3] Yu. V. Billig, ”On the homomorphic image of a special Lie algebra,”Mat. Sb. [Math. USSR-Sb.],136, No. 7, 320–323 (1988). · Zbl 0655.17006
[4] S. P. Mishchenko, ”The standard Lie identity in solvable varieties of associative type,”Vestnik Moskov. Univ. Ser. I. Mat. Mekh. [Moscow Univ. Math. Bull.], No. 4, 30–36 (1995). · Zbl 0873.17006
[5] S. P. Mishchenko, ”Growth of varieties of Lie algebras,”Uspekhi Mat. Nauk [Russian Math. Surveys],45, No. 6, 25–45 (1990). · Zbl 0718.17004
[6] M. V. Zaitsev, ”Identities of affine Kac-Moody algebras,”Vestnik Moskov. Univ. Ser. I. Mat. Mekh. [Moscow Univ. Math. Bull.], No. 2, 33–36 (1996). · Zbl 0881.17021
[7] A. Kh. Kushkulei and Yu. P. Razmyslov, ”Varieties generated by irreducible representations of Lie algebras,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 5, 4–7 (1983). · Zbl 0516.17007
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