All-derivable points in nest algebras

Suppose that $\mathscr{A}$ is an operator algebra on a Hilbert space $H$. An element $V$ in $\mathscr{A}$ is called an all-derivable point of $\mathscr{A}$ for the strong operator topology if every strong operator topology continuous derivable mapping $\phi$ at $V$ is a derivation. Let $\mathscr{N}$ be a complete nest on a complex and separable Hilbert space $H$. Suppose that $M$ belongs to $\mathscr{N}$ with $\{0\}\neq M\neq\ H$ and write $\hat{M}$ for $M$ or $M^{\bot}$. Our main result is: for any $\Omega\in alg\mathscr{N}$ with $\Omega=P(\hat{M})\Omega P(\hat{M})$, if $\Omega |_{\hat{M}}$ is invertible in $alg\mathscr{N}_{\hat{M}}$, then $\Omega$ is an all-derivable point in $alg\mathscr{N}$ for the strong operator topology.Comment: 12 pages, late

Multi-parameter deformed and nonstandard Y(glM)Y(gl_M) Yangian symmetry in integrable variants of Haldane-Shastry spin chain

By using `anyon like' representations of permutation algebra, which pick up nontrivial phase factors while interchanging the spins of two lattice sites, we construct some integrable variants of Haldane-Shastry (HS) spin chain. Lax equations for these spin chains allow us to find out the related conserved quantities. However, it turns out that such spin chains also possess a few additional conserved quantities which are apparently not derivable from the Lax equations. Identifying these additional conserved quantities, and the usual ones related to Lax equations, with different modes of a monodromy matrix, it is shown that the above mentioned HS like spin chains exhibit multi-parameter deformed and `nonstandard' variants of $Y(gl_M)$ Yangian symmetry.Comment: 18 pages, latex, no figure

Characterizing Jordan derivations of matrix rings through zero products

Let \Mn be the ring of all $n \times n$ matrices over a unital ring $\mathcal{R}$, let $\mathcal{M}$ be a 2-torsion free unital \Mn-bimodule and let D:\Mn\rightarrow \mathcal{M} be an additive map. We prove that if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0 whenever \A,\B\in \Mn are such that \A\B=\B\A=0, then D(\A)=\delta(\A)+\A D(\textbf{1}), where \delta:\Mn\rightarrow \mathcal{M} is a derivation and $D(\textbf{1})$ lies in the centre of $\mathcal{M}$. It is also shown that $D$ is a generalized derivation if and only if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A D(\textbf{1})\B-\B D(\textbf{1})\A=0 whenever \A\B=\B\A=0. We apply this results to provide that any (generalized) Jordan derivation from \Mn into a 2-torsion free \Mn-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if \varphi:\Mn\rightarrow \Mn is an additive map satisfying \varphi(\A \B+\B \A)=\A\varphi(\B)+\varphi(\B)\A \quad (\A,\B \in \Mn), then \varphi(\A)=\A\varphi(\textbf{1}) for all \A\in \Mn, where $\varphi(\textbf{1})$ lies in the centre of \Mn. By applying this result we obtain that every Jordan derivation of the trivial extension of \Mn by \Mn is a derivation.Comment: To appear in Mathematica Slovac

Tuplix Calculus

We introduce a calculus for tuplices, which are expressions that generalize matrices and vectors. Tuplices have an underlying data type for quantities that are taken from a zero-totalized field. We start with the core tuplix calculus CTC for entries and tests, which are combined using conjunctive composition. We define a standard model and prove that CTC is relatively complete with respect to it. The core calculus is extended with operators for choice, information hiding, scalar multiplication, clearing and encapsulation. We provide two examples of applications; one on incremental financial budgeting, and one on modular financial budget design.Comment: 22 page