Two-sided ideals in q-deformed Heisenberg algebras

In this article, the structure of two-sided ideals in the $q$-deformed Heisenberg algebras defined by the $q$-deformed Heisenberg canonical commutation relation$\mathit{AB}-\mathit{qBA}=I$is investigated. We show that these algebras are simple if and only if $q=1$. For $q\ne 1,0$ we present an infinite descending chain of non-trivial two-sided ideals, thus deducing by explicit construction that the $q$-deformed Heisenberg algebras are not just non-simple but also non-artinian for $q\ne 1,0$. We establish a connection between the quotients of the $q$-deformed Heisenberg algebras by these ideals and the quotients of the quantum plane. We also present a number of reordering formulae in $q$-deformed Heisenberg algebras, investigate properties of deformed commutator mappings, show their fundamental importance for investigation of ideals in $q$-deformed Heisenberg algebras, and demonstrate how to apply these results to the investigation of faithfulness of representations of $q$-deformed Heisenberg algebras.     

Structures and lower bounds for binary covering arrays

A $q$-ary $t$-covering array is an $m\times n$ matrix with entries from $\{0,1,\dots ,q?1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for given $t$ and $n$, or maximize $n$ for given $t$ and $m$. For $t=2$ and $q=2$, it is completely solved by Rényi, Katona, and Kleitman and Spencer. They also show that maximal binary 2-covering arrays are uniquely determined. Roux found a lower bound of $m$ for a general $t,n$, and $q$. In this article, we show that $m\times n$ binary 2-covering arrays under some constraints on $m$ and $n$ come from the maximal covering arrays. We also improve the lower bound of Roux for $t=3$ and $q=2$, and show that some binary 3 or 4-covering arrays are uniquely determined.     

Mappings of Butson-type Hadamard matrices

A $\mathrm{BH}(q,n)$ Butson-type Hadamard matrix $H$ is an $n\times n$ matrix over the complex $q$th roots of unity that fulfils $H{H}^1=n{I}_n$. It is well known that a $\mathrm{BH}(4,n)$ matrix can be used to construct a $\mathrm{BH}(2,2n)$ matrix, that is, a real Hadamard matrix. This method is here generalised to construct a $\mathrm{BH}(q,pn)$ matrix from a $\mathrm{BH}(pq,n)$ matrix, where $q$ has at most two distinct prime divisors, one of them being $p$. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case $p=q=2$ is developed and used to classify the $\mathrm{BH}(4,16)$ matrices from a classification of the $\mathrm{BH}(2,32)$ matrices.     

La forme seconde trace d'une algèbre simple centrale de degré 4 de caractéristique 2

Let A be a central simple algebra of degree 4 over a field k of characteristic 2 and let $q_A$ be the quadratic form on A given by the second coefficient of the reduced characteristic polynomial. We show that A uniquely determines a 2-fold Pfister form $q_2$ and a 4-fold Pfister form $q_4$ such that $q_A=[1,1]+q_2+q_4$ in the Witt group of k, where $[1,1]$ is the form $x^2+xy+y^2$. The form $q_2$ is the norm form of the quaternion algebra Brauer-equivalent to $A{?}_{k}A$, and $q_4$ is hyperbolic if and only if A is cyclic. To cite this article: J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006).