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Lie algebras generated by indecomposables. (English) Zbl 0841.16018

This article introduces a new connection between the representation theory of finite dimensional \(\mathbb{C}\)-algebras of finite representation type and certain Lie algebras and their roots. It is related, in modern times, to the article of C. M. Ringel [in “Topics in Algebra”, Banach Center Publ. 26, 433-447 (1990; Zbl 0778.16004)], to the article of G. Lusztig [in ICM Kyoto 1990, Vol. 1, 155-174 (1991; Zbl 0749.14010)] and to some unpublished work of A. Schofield. What is not clear is if this interesting research will contribute to clarify the context or merely to make it still more rich and complex.
Quoting from the introduction: “Let \(\Lambda\) be a finite dimensional \(\mathbb{C}\)-algebra with unit element which has only finitely many indecomposable (left) modules up to isomorphism. We define a \(\mathbb{Z}\)-Lie algebra structure on the \(\mathbb{Z}\)-module \(L(\Lambda)\) freely generated by the isomorphism classes of indecomposable \(\Lambda\)-modules.” If \(\Lambda\) is hereditary, \(L(\Lambda)\) “is isomorphic to the positive part of a \(\mathbb{Z}\)-form” of the simple Lie algebra corresponding to the Dynkin quiver that defines it.
In the above mentioned paper, Ringel shows that, inside his Hall algebra and when \(\Lambda\) is also directed, the indecomposables form a Lie algebra. The author says that, in this case of a directed algebra, Ringel’s Lie algebra probably coincides with \(L(\Lambda)\), but there is no proof available for this.
For the definition of \(L(\Lambda)\), she uses the Euler-Poincaré characteristic of the corresponding variety of the modules. And, she informs: “The first to work with the Euler-Poincaré characteristic as we do here were Schofield and Lusztig. Schofield generalized Ringel’s results in a different direction: the dimension vectors of the indecomposable representations of any quiver \(Q\) are just the positive roots of the Kac-Moody Lie algebra associated with the underlying graph of \(Q\). (…) The second motivation for this article is then to prove that his Lie algebra is isomorphic to \(\mathbb{Q} \otimes_\mathbb{Z} L(\Lambda)\) if \(\Lambda\) is directed and has only finitely many indecomposables”.

MSC:

16G30 Representations of orders, lattices, algebras over commutative rings
17B20 Simple, semisimple, reductive (super)algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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