We consider Lie algebras admitting an ad-invariant metric, and we study the problem of uniqueness of the ad-invariant metric up to automorphisms. This is a common feature in low dimensions, as one can observe in the known classification of nilpotent Lie algebras of dimension $$$\le 7$ admitting an ad-invariant metric. We prove that uniqueness of the metric on a complex Lie algebra $$$\mathfrak{g}$ is equivalent to uniqueness of ad-invariant metrics on the cotangent Lie algebra $$$T^{\ast }\mathfrak{g}$; a slightly more complicated equivalence holds over the reals. This motivates us to study the broader class of Lie algebras such that the ad-invariant metric on $$$T^{\ast }\mathfrak{g}$ is unique.

We prove that uniqueness of the metric forces the Lie algebra to be solvable, but the converse does not hold, as we show by constructing solvable Lie algebras with a one-parameter family of inequivalent ad-invariant metrics. We prove sufficient conditions for uniqueness expressed in terms of both the Nikolayevsky derivation and a metric counterpart introduced in this paper.

Moreover, we prove that uniqueness always holds for irreducible Lie algebras which are either solvable of dimension $$$\le 6$ or real nilpotent of dimension $$$\le 10$.

We present several formulas for some specific multiple $$-values of conductor four. This grew out from the study of zeta functions of level four of Arakawa-Kaneko type. Closely related is a new version of multiple poly-Euler numbers and we briefly discuss this too.

In this paper, we consider a translating soliton for the mean curvature flow starting from a graph of a function on a domain in a unit sphere which is constant along each leaf of isoparametric foliation. First, we show that such a function is given as a composition of an isoparametric function on the sphere and a function which is given as a solution of a certain ordinary differential equation. Further, we analyze the shape of the graphs of the solutions of the ordinary differential equation. This analysis leads to the classification of the shape of such translating solitons. Finally, we investigate a domain of the function which is given as a composition of the isoparametric function and the solution of the ordinary differential equation in the case where the number of distinct principal curvatures of the isoparametric hypersurface defined by the regular level set for the isoparametric function is 1, 2, or 3.

A quasi contact metric manifold in some sense is a generalization of contact metric manifold. There is an open question whether quasi contact metric manifolds of dimension greater than 3 are contact metric or not. In this paper, we show that the answer is positive for ones of dimension $$ when the Riemannian curvature tensor in the direction of the characteristic vector field satisfies the nullity condition (*), and then study the Riemannian and Ricci curvature in the 5-dimensional case.

In this paper, we provide a volume presentation of a hyperbolic fibered knot using the Bell polynomials. More precisely, we show that the hyperbolic volume of a fibered knot can be expressed in terms of the traces of powers of a monodromy matrix. We also show that the hyperbolic volume of the figure-eight knot is obtained by the asymptotics of a positive integer sequence consisting of the special values of the twisted Alexander invariants.

In this paper, we study the ramification of extensions of a function field generated by division points of rank 2 Drinfeld modules. Also conductors of certain rank 2 Drinfeld modules are defined as analogues of those for elliptic curves. A calculation of these conductors allows us to show an analogue of Szpiro's conjecture under a certain limited situation.