Inner derivations of simple Lie pencils of rank $1$

N. A. Koreshkov

Kazan (Volga Region) Federal University, 18 Kremlyovskaya str., Kazan, 420008 Russia

Abstract: We prove that simple Lie pencils of rank $1$ over algebraically closed field $P$ of characteristic 0, whose operators of left multiplications have the form of sandwich algebra $M_3(U,\mathcal{D}')$, where $U$ is a subspace of all skew-symmetric matrices in $M_3(P)$, $\mathcal{D}'$ is any subspace containing $\langle E\rangle$ in a space of all diagonal matrices $\mathcal{D}$ in $M_3(P)$.

Keywords: Lie pencil, Cartan subalgebra, torus, inner derivation, sandwich algebra.

UDC: 512.554

Received: 29.09.2015

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2017, 61:4, 11–17

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