REALIZATION OF SIMPLE LIE ALGEBRAS VIA HALL ALGEBRAS

HIDETO ASASHIBA

Let A be a finite-dimensional hereditary algebra over a finite field k, and consider the free abelian group H(A) with basis the isoclasses of finite A-modules. In [4] Ringel introduced a multiplication on H(A) to make it an associative ring with identity. The structure constants were given by the numbers of filtrations of modules with factors isomorphic to modules that are multiplied. The ring H(A) obtained in this way is called the Hall algebra of A. Assume in addition that A is representation-finite, i.e., that the underlying graph Δ of the quiver of A is a Dynkin graph. Then it was shown that the structure constants are given by polynomials in the cardinality q of k, called the Hall polynomials. This makes it possible to define a generic Hall algebra H(A, ℤ[T]), which is a free ℤ[T]-module with the same basis as H(A) and the multiplication is given by the Hall polynomials. The specialization H(A)_{1} of H(A, ℤ[T]) for T = 1, called the degenerate Hall algebra of A, yields a Lie subalgebra \overline{L}(A)_{1}, which is the free abelian subgroup of the degenerate Hall algebra with basis the set of isoclasses of indecomposable A-modules. In [5] he has shown that \overline{L}(A)_{1}^{ℂ} := \overline{L}(A)_{1} ⊗_{ℤ} ℂ and the positive part 𝔫_{+}(Δ) of the simple complex Lie algebra 𝔤(Δ) of type Δ are isomorphic as Lie algebras. Namely, the positive part 𝔫_{+}(Δ) of 𝔤(Δ) for a Dynkin graph Δ was recovered by the representation theory of A. After that, Peng and Xiao [2] constructed the simple Lie algebra 𝔤(Δ) of type Δ itself for each Dynkin graph Δ, mainly but partly using a Hall algebra. In their construction, the positive part and the negative part were given by the root category R of the path-algebra of a quiver with the underlying graph Δ (R can be regarded as a double of the category of finite-dimensional Amodules), and the Cartan subalgebra was given by the Grothendieck group of R. The Hall multiplication was used to define the Lie bracket only inside R, and when the bracket should not be closed in R the definition was changed. Their realization of simple Lie algebras is useful, but the definition of the bracket is a patchwork of several multiplication tables and seems to be rather artificial.

In the lecture, for each integer n ≥ 1 we first give simple and explicit realization of the simple Lie algebra 𝔤(A_{n}) of type A_{n}, i.e., the special linear algebra 𝔰𝔩_{n+1}(ℂ) and of the general linear algebra 𝔤𝔩_{n+1}(ℂ) only by the Hall multiplication as a quotient of the degenerate composition Lie algebra (resp. of the degenerate Lie algebra) with coefficients in ℂ of the category of nilpotent representations of the cyclic quiver with n + 1 vertices. Note that until now this categoy was used to realize the positive part of the affine Lie algebra of type Ã_{n}(= A_{n}^{(1)}) as in [7] or [1]. The realization of these immediately yields realization of their universal enveloping algebras via Hall algebras. Next, we will give a general way to realize all types of simple Lie algebras via Hall algebras of finite-dimensional tame hereditary algebras (in contrast with the first part, in which we use an infinite-dimensional algebra by the speciality of a cyclic orientation on Ã_{n}), which were well studied by Ringel [6] and Peng and Xiao [3], and give explicit realization for the types A_{n} (a different presentation) and D_{n}(n ≥ 4).

References

[1] Peng, L. and Xiao, J.: A realization of affine Lie algebras of type Ã_{n−1} via the derived categories of cyclic quivers, Can. Math. Soc. Conference Proc. 18, 539–554, 1996.

[2] Peng, L. and Xiao, J.: Root categories and simple Lie algebras, J. Alg. 198, 19–56, 1997.

[3] Peng, L. and Xiao, J.: Triangulated categories and Kac-Moody algebras, Invent. Math. 140, 563–603, 2000.

[4] Ringel, C.M.: Hall algebras, in “Topics in Algebra,” Banach Center Publ. 26(1), Warszawa, 433–447, 1990.

[5] Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras, Adv. Math. 84 (2), 137–178, 1990.

[6] Ringel, C.M.: From representations of quivers via Hall and Loewy algebras to quantum groups, In: Proc. Novosibirsk Conf. Algebra 1989, Contemporary Math. 131, 381–401, 1992.

[7] Ringel, C.M.: The composition algebra of a cyclic quiver, towards an explicit description of the quantum group of type Ã_{n}, Proc. London Math. Soc. 66 (3), 507–537, 1993.

Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshiku, Osaka, 558, Japan

 E-mail address: asashiba@sci.osaka-cu.ac.jp