Quantum algebraic tori (Q5951085)

By a quantum torus the author understands an algebra generated by \(x\) and \(y\) over a field \(k\) with defining relation \(xy=qyx\), where \(q\in k^*\) (this arises from the Heisenberg commutation relation \(uv-vu=h/2\pi i\) by setting \(x=e^u\), \(y=e^v\), \(q=e^{h/2\pi i}\)). More generally, an algebra of twisted Laurent polynomials is the algebra \(A\) generated by \(x_1,\dots,x_n\) and their inverses subject to \(x_ix_j=q_{ij}x_jx_i\), where \(q_{ij}q_{ji}=q_{ii}=1\). The group \(D_n\) of diagonal matrices acts by the automorphisms \(x_i\mapsto t_ix_i\) on \(A\), and the latter is also called the quantum form of the diagonal torus \(D_n\). More generally, a quantum algebraic torus is an algebra which takes on the above form after making a Galois extension of \(k\) to \(L\). If \(T\) is an algebraic \(k\)-torus split by \(L/k\), then there is a coaction mapping \(R\colon A\to k[T]\otimes A\) of the Hopf algebra \(k[T]\) on \(A\) and the lifting \(R_L\) taking \(A_L=A\otimes L\) to \(L[T]\otimes A_L\) coincides with the coaction of the diagonal torus \(R_L\colon x_i\to t_i\otimes x_i\). The category of quantum \(k\)-forms of the torus \(T\) split over \(L\) is denoted by \(\text{Quan}_L(T)\) and the category of central extensions \(E\) of \(\Gamma\)-groups (\(\Gamma=\text{Gal}(L/k)\)) \(L^*\) by \(M\) is denoted by \(\text{Ext}^c_T(M,L^*)\). The author proves that the categories \(\text{Quan}_L(T)\) and \(\text{Ext}^c_L(M,L^*)\) are isomorphic. Here the extension \(E\) is the group of units in \(A\otimes L\), while \(M\) is the group of characters of \(T\) defined over an algebraic closure of \(k\). For any such extension \(E\) a matrix \(Q=(q_{ij})\) can be defined over \(L^*\) satisfying a compatibility condition with \(\Gamma\) and the author asks whether conversely, from a matrix satisfying these conditions a central extension of \(L^*\) by \(M\) can be defined. He gives a positive answer when \(M\) is a permutation module or a module over a cyclic group \(\Gamma\) of order \(2\). He next determines the centre of \(A_L\) and obtains an expression for it in the form \(k[T_c]\) for a \(k\)-torus \(T_c\). In the special case where the \(q_{ij}\) are the \(l\)-th roots of 1 he is able to determine the structure of \(A_L\): modulo a certain kernel it is a tensor product of cyclic algebras. He also provides a more detailed description when \(n=[L:k]=2\).