Quasidiagonality of nuclear C*-algebras

Peer reviewed

Norms of inner derivations for multiplier algebras of C ∗ -algebras and group C ∗ -algebras, II

Peer reviewed

High-dimensional Z-stable AH algebras

Peer reviewed

K-Theoretic Characterization of C*-Algebras with Approximately Inner Flip

The author is supported by an NSERC PDF.

Norms of inner derivations for multiplier algebras of C ∗ -algebras and group C ∗ -algebras, II

Peer reviewed

High-dimensional Z-stable AH algebras

Peer reviewed

K-Theoretic Characterization of C*-Algebras with Approximately Inner Flip

The author is supported by an NSERC PDF.

The Berkovits Complex and Semi-free Extensions of Koszul Algebras

Peer reviewed

Relative commutants of strongly self-absorbing C∗-algebras

Peer reviewed

Covering Dimension of C*-Algebras and 2-Coloured Classification

Peer reviewed

Localization of quantum biequivariant D-modules and q-W algebras

Peer reviewed

High-dimensional Z-stable AH algebras

Peer reviewed

Spectral isometries into commutative Banach algebras

We determine the structure of spectral isometries between unital Banach algebras under the hypothesis that the codomain is commutative.

Optimal 5-step nilpotent quadratic algebras

By the Golod–Shafarevich theorem, an associative algebra $R$ given by $n$ generators and $<n^2/3$ homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer $n$, there is an algebra $R$ given by $n$ generators and $\lceil n^2/3\rceil$ homogeneous quadratic relations such that $R$ is 5-step nilpotent.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).