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 $\T/F$.
Let $\kk$ be a \emph{field of characteristic $p>0$, not assumed algebraically closed}, and let $V=\kk[x]/(x^p)$ be the Novikov algebra with product $a\circ b=ab'$. For $\lambda\in\kk$, let $M(\lambda)$ be Xu's module. We compute the second cohomology $\Ht(V,M(\lambda))$ for all $\lambda$ and $p$, and describe the assoc...
Let $\ell$ and $p$ be distinct prime numbers, $K$ a local field with residue characteristic $p$, $C$ a complete algebraically closed extension of $K$, and $X$ a quasi-compact quasi-separated rigid space over $K$. When $\mathop{\mathrm{char}} K = 0$, we prove that the alternating sum of traces of elements in the positiv...
Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat...
A. Giaquinto, J. Irving, A. Lauve et al.· 2 citations
Let $G$ be a finite abelian group of order $N$, $S \subseteq G \setminus \{0\}$ a symmetric connection set, and $K$ a field with $char(K) \nmid N$. The spectral algebra $\mathscr{A_K}(Cay(G,S)) = K[A]$ generated by the adjacency matrix of the Cayley graph is proved to decompose, via the character-orbit decomposition, a...
Deep Bhattacharjee, P. Mandal, Ushashi Bhattacharya· 0 citations
Let $K/\Qp$ be a finite extension, $\bG$ a connected reductive group over $K$, and set $G = \bG(K)$, considered as a $p$-adic Lie group. We show first that the Bernstein center $\cC_G$ of the category of solid locally $\Qp$-analytic representations of $G$ is isomorphic to the center of the locally analytic distribution...
Daniel Le, Matthias Strauch, Zi-Chuan Wang· 0 citations
Let $\mathfrak g$ be a finite-dimensional simple complex Lie algebra, and consider the loop algebra $L\mathfrak g = \mathfrak g[s, s^{-1}]$ and the affine Lie algebra $\widehat{\mathfrak g}$, which is an extension of $L\mathfrak g$ by a central element $c$. It is known that the rings $\operatorname{U}(\widehat{\mathfra...
A. Petukhov, S. Sierra· 0 citations
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