Lie Ideals and Lie Derivations of the Algebraic Toeplitz Algebra

Abstract

Let $K$ be a field and let $\T=L_K(E_T)$ be the algebraic Toeplitz algebra. We classify the Lie ideals of $\T^-$. If $w$ is the sink of the Toeplitz graph and $F=I(w)$, then $F\cong M_\infty(K)$, $\T/F\cong K[t,t^{-1}]$, and $[\T,\T]=F$. We prove that every noncentral Lie ideal contains $\mathfrak s=[F,F]$, the finitary trace-zero Lie algebra. Thus the classification reduces to the Heisenberg-type quotient $\T/\mathfrak s$. The Lie ideals are precisely $0$, $K1$, the spaces $\mathfrak s+U$, where $U\leq P_K\oplus Kw$ and $w\notin U$, and the inverse images $\rho^{-1}(W)$, where $W\leq K[t,t^{-1}]$. Here $P_K=K1$ in characteristic $0$, and $P_K=\operatorname{span}_K\{1,e^{m\ell},(e^*)^{m\ell}:m\geq1\}$ if $\operatorname{char}K=\ell>0$. We also describe the Lie derivations of $\T^-$ as sums of associative derivations of $\T$ and central Lie derivations factoring through $\T/F$.