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Near-derivations and their applications to Lie algebras

Dmitri Panyushev, Oksana Yakimova

Abstract

E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our substantial applications concern the Poisson algebra $(\mathcal S(\mathfrak q),\{\ ,\,\})$ of a Lie algebra $\mathfrak q$. We develop a method for obtaining quasi-derivations via the use of squares of derivations, which allows us to provide quasi-derivations of the simple Lie algebras. It is shown that (1) a near-derivation $D$ of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ yields a pencil of compatible Poisson brackets on $\mathfrak q^*$ and (2) using $D$ one may naturally construct a Poisson-commutative subalgebra of $\mathcal S(\mathfrak q)$. A special attention is given to near-derivations of $(\mathcal S(\mathfrak q),\{\ ,\,\})$ induced from near-derivations of $\mathfrak q$. This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of $\mathfrak q$ and Nijenhuis operators in $\mathfrak{gl}(\mathfrak q)$.

Near-derivations and their applications to Lie algebras

Abstract

E.B. Vinberg's theory of quasi-derivations of algebras is extended to a broader framework of near-derivations. This deepens connections between Poisson geometry and Lie theory. Although basic results apply to arbitrary algebras, our substantial applications concern the Poisson algebra of a Lie algebra . We develop a method for obtaining quasi-derivations via the use of squares of derivations, which allows us to provide quasi-derivations of the simple Lie algebras. It is shown that (1) a near-derivation of yields a pencil of compatible Poisson brackets on and (2) using one may naturally construct a Poisson-commutative subalgebra of . A special attention is given to near-derivations of induced from near-derivations of . This provides some old and new families of compatible Poisson brackets. We also compare properties of near-derivations of and Nijenhuis operators in .

Paper Structure

This paper contains 12 sections, 33 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Quasi-derivations of algebras
  3. Near-derivations of algebras
  4. Near-derivations of Poisson algebras
  5. Near-derivations of Lie algebras: examples
  6. Periodic gradings and near-derivations
  7. Quasi-gradings and near-derivations
  8. Splittings and near-derivations
  9. Special $1$-parameter groups and near-derivations
  10. Quasi-derivations, ${\mathbb Z}$-gradings, and nilpotent elements
  11. Near-derivations versus Nijenhuis operators
  12. Associative algebras and Nijenhuis operators

Key Result

Theorem 2.1

Suppose that $D$ is a quasi-derivation of $({\EuScript V},{\mathsf T})$. If $x,y\in {\EuScript V}$ have the property that $\psi(D^nx,y)=\psi(x,D^ny)=0$ for all $n\in {\mathbb N}$, then $\psi(D^kx,D^ly)=0$ for all $k,l$.

Theorems & Definitions (68)

  • Definition 1: Vinberg
  • Theorem 2.1
  • Theorem 2.2
  • Example 2.3
  • Remark 2.4
  • Proposition 2.5: Vinberg
  • Definition 2
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 58 more