The Existence of Affine Structures on the Borel Subalgebra of Dimension 6

The notion of affine structures arises in many fields of mathematics, including convex homogeneous cones, vertex algebras, and affine manifolds. On the other hand, it is well known that Frobenius Lie algebras correspond to the research of homogeneous domains. Moreover, there are 16 isomorphism classes of 6-dimensional Frobenius Lie algebras over an algebraically closed field. The research studied the affine structures for the 6-dimensional Borel subalgebra of a simple Lie algebra. The Borel subalgebra was isomorphic to the first class of Csikós and Verhóczki’s classification of the Frobenius Lie algebras of dimension 6 over an algebraically closed field. The main purpose was to prove that the Borel subalgebra of dimension 6 was equipped with incomplete affine structures. To achieve the purpose, the axiomatic method was considered by studying some important notions corresponding to affine structures and their completeness, Borel subalgebras, and Frobenius Lie algebras. A chosen Frobenius functional of the Borel subalgebra helped to determine the affine structure formulas well. The result shows that the Borel subalgebra of dimension 6 has affine structures which are not complete. Furthermore, the research also gives explicit formulas of affine structures. For future research, another isomorphism class of 6-dimensional Frobenius Lie algebra still needs to be investigated whether it has complete affine structures or not.

Kuperberg invariants for balanced sutured 3-manifolds

We construct quantum invariants of balanced sutured 3-manifolds with a ${\text {Spin}^c}$ structure out of an involutive (possibly nonunimodular) Hopf superalgebra H. If H is the Borel subalgebra of ${U_q(\mathfrak {gl}(1|1))}$ , we show that our invariant is computed via Fox calculus, and it is a normalization of Reidemeister torsion. The invariant is defined via a modification of a construction of Kuperberg, where we use the ${\text {Spin}^c}$ structure to take care of the nonunimodularity of H or $H^{*}$ .

Abelian ideals and amazing roots

Let [Formula: see text] be a simple Lie algebra with a Borel subalgebra [Formula: see text]. To any long positive root [Formula: see text], one associates two ideals of [Formula: see text]: the abelian ideal [Formula: see text] and not necessarily abelian ideal [Formula: see text]. It is known that [Formula: see text], and [Formula: see text] is said to be amazing if the equality holds. The set of amazing roots, [Formula: see text], is closed under the operation “∨” in [Formula: see text], and [Formula: see text] is said to be primitive, if it cannot be written as [Formula: see text] with incomparable amazing roots [Formula: see text]. We classify the amazing roots and notice that the number of primitive roots equals [Formula: see text]. Moreover, if [Formula: see text] (respectively, [Formula: see text]) is the set of simple (respectively, primitive) roots, then there is a natural bijection [Formula: see text]. We also study the subset [Formula: see text] of [Formula: see text].

ABELIAN IDEALS IN A COMPLEX SIMPLE LIE ALGEBRA

In this note, we give a new simple construction of all maximal abelian ideals in a Borel subalgebra of a complex simple Lie algebra. We also derive formulas for dimensions of certain maximal abelian ideals in terms of the theory of Borel de Siebenthal.