Odd-quadratic Leibniz superalgebras

An odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form. In this paper, we prove that a left (resp. right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra). Moreover, we show that any non-abelian (left or right) Leibniz superalgebra does not possess simultaneously a quadratic and an odd-quadratic structure. Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras. Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included.

Non-degenerate Invariant Bilinear Forms on Leibniz Algebras

In this paper, we study Leibniz algebras [Formula: see text] with a non-degenerate Leibniz-symmetric [Formula: see text]-invariant bilinear form B, such a pair [Formula: see text] is called a quadratic Leibniz algebra. Our first result generalizes the notion of double extensions to quadratic Leibniz algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Leibniz algebra to be a quadratic Leibniz algebra by double extension.

The q-Analog Klein Bottle Lie Algebra

In this paper, we study an infinite-dimensional Lie algebra ℬq, called the q-analog Klein bottle Lie algebra. We show that ℬq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬq are also determined.

Octo-bialgebras

The notion of octo-algebra was introduced by Leroux as a Loday algebra with 8 operations. In this paper, we introduce a notion of octo-bialgebra as a bialgebra theory of octo-algebras, which is equivalent to a double construction of a quadri-algebra with a nondegenerate 2-cocycle or a double construction of an octo-algebra with a nondegenerate invariant bilinear form. Some properties of octo-bialgebras are given, including the study of the coboundary cases which leads to a construction from an analogue of the classical Yang-Baxter equation in an octo-algebra.

Non-degenerate Invariant Bilinear Forms on Lie Color Algebras

In this paper, we study Lie color algebras 𝔤 with a non-degenerate color-symmetric, 𝔤-invariant bilinear form B, such a (𝔤,B) is called a quadratic Lie color algebra. Our first result generalizes the notion of double extensions to quadratic Lie color algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie color algebra to be a quadratic Lie color algebra by double extension. At last, we generalize the notion of T*-extensions to Lie color algebras.

On the centre of two-parameter quantum groups

We describe Poincaré–Birkhoff–Witt bases for the two-parameter quantum groups U = Ur,s(sln) following Kharchenko and show that the positive part of U has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on U, which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to determine the centre of U.