[Paper Review] Lie algebra extensions related with linear bundles of Lie brackets

A.B. Yanovski|ArXiv.org|Aug 20, 2001

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper introduces a natural construction of Lie algebra extensions—specifically Leibniz extensions—using linear bundles of Lie brackets, showing that these extensions arise from compatible Lie brackets defined via graded structures and tensor deformations. The key contribution is a systematic method to generate universal solvable and semisimple extensions over ${\cal G}^n$ using a parameterized bracket that generalizes known Poisson-Lie structures in hydrodynamic models.

ABSTRACT

We consider some special type extensions of an arbitrary Lie algebra ${\cal G}$, arising in the theory of Lie-Poisson structures over $({\cal G}^*)^n$, where ${\cal G}^*$ is the dual of ${\cal G}$. We show that some classes of these extensions can be constructed in a natural way using some linear bundles of Lie algebras.

Motivation & Objective

• To understand the structure of Lie algebra extensions over ${{\cal G}}^n$ arising from Lie-Poisson structures on $({\cal G}^*)^n$.
• To identify conditions under which tensor-defined brackets $[{\bf x},{\bf y}]_W$ yield valid Lie algebra structures.
• To construct a new class of universal extensions—Leibniz extensions—using graded Lie algebra decompositions and deformation techniques.
• To show that these extensions naturally emerge from linear bundles of Lie algebras, particularly through compatible bracket deformations.
• To clarify the correspondence between solvable and semisimple extensions via a 'finding the semisimple part' operation.

Proposed method

• Define a Lie bracket on ${\cal G}^n$ via a tensor $W^{ij}_s$ satisfying symmetry and Jacobi-type conditions: $W^{ij}_s = W^{ji}_s$ and $\sum_k (W^s_{ik}W^p_{kj} - W^q_{ik}W^p_{kj}) = 0$.
• Reinterpret the bracket using $n+1$ matrices $W^{(k)}$ with components $(W^{(k)})_i^j = W^{kj}_i$, showing that the Jacobi condition is equivalent to mutual commutativity of these matrices.
• Use simultaneous block-diagonalization to reduce the problem to irreducible components, focusing on nilpotent matrices for $s > 0$ and analyzing generalized eigenvalues.
• Construct a new class of extensions via a $\mathbb{Z}_n$-graded Lie algebra $\cal H = \bigoplus_{i=0}^{n-1} \cal H^{(i)}$, defining a bracket $[x,y]^\lambda$ that combines standard and shifted components.
• Introduce a commutative algebra $\cal B$ generated by $g$ with $g^n = 0$, and realize the new bracket as the pullback of the Lie bracket on ${\cal H} \otimes_{\bf K} \cal B$ via an embedding $h(x) = \sum x_i \otimes g^i$.
• Define the Leibniz extension bracket as $[x,y]^\lambda = [x,y]^0 + \lambda [x,y]^a$, where $[x,y]^a$ captures terms with $s+l \geq n$, and prove that this defines a Lie bracket for all $\lambda$.

Experimental results

Research questions

• RQ1 Can universal Lie algebra extensions over ${{\cal G}}^n$ be systematically constructed using linear bundles of Lie algebras?
• RQ2 How do the solvable and semisimple cases of Lie algebra extensions relate through a canonical 'semisimple part' operation?
• RQ3 What conditions on the tensor $W^{ij}_s$ ensure that the induced bracket $[{\bf x},{\bf y}]_W$ satisfies the Jacobi identity?
• RQ4 Can the Leibniz extension bracket be derived naturally from a graded Lie algebra structure and a nilpotent algebra deformation?
• RQ5 Is every universal Lie algebra extension realizable as a deformation within the framework of linear bundles of Lie algebras?

Key findings

• The tensor $W^{ij}_s = \delta^{i+j}_s$ defines a Lie bracket on ${\cal G}^n$ that induces a graded structure where $[\cal F_n^{(i)}, \cal F_n^{(j)}]_L \subset \cal F_n^{(i+j)}$ for $i+j \leq n-1$, and zero otherwise.
• The bracket $[x,y]^\lambda = \sum_{s+l=i} [x_s,y_l] + \lambda \sum_{s+l \geq n} [x_s,y_l]$ defines a Lie algebra structure for all $\lambda \in \bf K$, generalizing known compatible Poisson brackets.
• The case $\lambda = 0$ corresponds to the standard graded Lie bracket, while $\lambda = 1$ gives the Leibniz extension, both of which are compatible and yield a new Lie algebra structure on ${\cal G}_L^n$.
• The solvable Leibniz extension is recovered as the 'semisimple part' of the full extension by restricting to indices $1 \leq i,j,k \leq n-1$, with $\bar{W}_k^{ij} = \delta^{i+j}_k$.
• The construction via the algebra $\cal B$ with $g^n = 0$ and the embedding $h(x) = \sum x_i \otimes g^i$ provides a natural realization of the $\lambda$-deformed bracket as a pullback from a tensor product Lie algebra.
• The method establishes that Leibniz extensions are the only regular class of universal extensions, appearing for every $n$, and are naturally embedded in the theory of linear bundles of Lie algebras.