Celebratio Mathematica

G. Benkart and P. Terwilliger: “Irreducible modules for the quantum affine algebra \( U_q(\widehat{\mathfrak{sl}}_2) \) and its Borel subalgebra,” J. Algebra 282 : 1 (2004), pp. 172–194. MR 2095578 Zbl 1106.17014 ArXiv math/0311152 article

Let \( U_q(\mathfrak{b}) \) denote the standard Borel subalgebra of the quantum affine algebra \( U_q(\widehat{\mathfrak{sl}}_2) \). We show that the following hold for any choice of scalars \( \varepsilon_0 \), \( \varepsilon_1 \) from the set \( \{1,-1\} \):

Let \( V \) be a finite-dimensional irreducible \( U_q(\mathfrak{b}) \)-module of type \( (\varepsilon_0,\varepsilon_1) \). Then the action of \( U_q(\mathfrak{b}) \) on \( V \) extends uniquely to an action of \( U_q(\widehat{\mathfrak{sl}}_2) \) on \( V \). The resulting \( U_q(\widehat{\mathfrak{sl}}_2) \)-module structure on \( V \) is irreducible and of type \( (\varepsilon_0,\varepsilon_1) \).
Let \( V \) be a finite-dimensional irreducible \( U_q(\widehat{\mathfrak{sl}}_2) \)-module of type \( (\varepsilon_0,\varepsilon_1) \). When the \( U_q(\widehat{\mathfrak{sl}}_2) \)-action is restricted to \( U_q(\mathfrak{b}) \), the resulting \( U_q(\mathfrak{b}) \)-module structure on \( V \) is irreducible and of type \( (\varepsilon_0,\varepsilon_1) \).

G. Benkart and P. Terwilliger: “The universal central extension of the three-point \( \mathfrak{sl}_2 \) loop algebra,” Proc. Am. Math. Soc. 135 : 6 (2007), pp. 1659–1668. MR 2286073 Zbl 1153.17008 ArXiv math/0512422 article

G. Benkart and P. Terwilliger: “The equitable basis for \( \mathfrak{sl}_2 \),” Math. Z. 268 : 1–2 (June 2011), pp. 535–557. MR 2805446 Zbl 1277.17013 ArXiv 0810.2066 article

This article contains an investigation of the equitable basis for the Lie algebra \( \mathfrak{sl}_2 \). Denoting this basis by \( \{x,y,z\} \), we have \[ [x,y] = 2x + 2y, \qquad [y,z] = 2y + 2z, \qquad [z,x] = 2z + 2x. \] We determine the group of automorphisms \( G \) generated by \[ \operatorname{exp}(\operatorname{ad} x^*), \quad\operatorname{exp}(\operatorname{ad} y^*), \quad\operatorname{exp}(\operatorname{ad} z^*) ,\] where \( \{x^*, y^*, z^*\} \) is the basis for \( \mathfrak{sl}_2 \) dual to \( \{x, y, z\} \) with respect to the trace form \( (u,v) = \operatorname{tr}(uv) \) and study the relationship of \( G \) to the isometries of the lattices \[ L = \mathbb{Z}x \oplus \mathbb{Z}y \oplus \mathbb{Z}z \quad\text{and}\quad L^* = \mathbb{Z}x^* \oplus \mathbb{Z}y^* \oplus \mathbb{Z}z^* .\] The matrix of the trace form is a Cartan matrix of hyperbolic type, and we identify the equitable basis with a set of simple roots of the corresponding Kac–Moody Lie algebra \( \mathfrak{g} \), so that \( L \) is the root lattice and \( \frac{1}{2}L^* \) is the weight lattice of \( \mathfrak{g} \). The orbit \( G(x) \) of \( x \) coincides with the set of real roots of \( \mathfrak{g} \). We determine the isotropic roots of \( \mathfrak{g} \) and show that each isotropic root has multiplicity 1. We describe the finite-dimensional \( \mathfrak{sl}_2 \)-modules from the point of view of the equitable basis. In the final section, we establish a connection between the Weyl group orbit of the fundamental weights of \( \mathfrak{g} \) and Pythagorean triples.

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