Automorphisms and twisted forms of the \(N = 1, 2, 3\) Lie conformal superalgebras
The authors have a twofold purpose with the article. Firstly, to give the physicists new ideas, that is methods coming from differential non-abelian cohomology. These methods are applied on reductive group schemes, torsors, and descent, and can be adapted to the study of Lie conformal superalgebras by replacing the base scheme \(\text{Spec}(\mathbb C[t^{\pm 1}]\) in this case, by a differential scheme.NEWLINENEWLINEThe second purpose is to complete the classification of the \(N=1,2,3,4\) Lie superalgebras, referred to as superconformal Lie algebras by the physicists. The authors give a uniform proof of the cases \(N=1,2,3\), by using the fact that the relevant Lie conformal superalgebras used as base objects for the twisted loop construction can be described in terms of exterior algebras. Also, a precise argument that describes the passage from Lie conformal superalgebras to their corresponding Lie superalgebras is given.NEWLINENEWLINEThe \(N=1,2,3,4\) Lie superalgebras are closely related to the twisted loop Lie conformal superalgebra based on a complex Lie conformal superalgebra. They were realized as differential Lie conformal algebras. A twisted loop Lie conformal superalgebra has both a complex conformal superalgebra structure and a \(\mathcal R\)-Lie conformal superalgebra structure, where \(\mathcal R=(\mathbb C[t^{\pm 1}],\frac{d}{dt}).\) The \(\mathcal R\)-structure makes it possible to introduce cohomological methods.NEWLINENEWLINENEWLINEKac-Moody Lie algebras are defined a priori by generators and relations, and the construction of the twisted loop Lie conformal superalgebras is built on the same principles. The classification of these has been established by non-abelian étale cohomology. The loop algebras based on a finite-dimensional simple Lie algebra \(\mathfrak g\) are parameterized by the conjugacy classes of the finite group of symmetries of the corresponding Coxeter-Dynkin diagram. If the automorphism group of \(\mathfrak g\) is connected, then all loop algebras based on \(\mathfrak g\) are trivial, i.e., isomorphic to \(\mathfrak g\otimes_{\mathbb C}\mathbb C[t^{\pm 1} ].\)NEWLINENEWLINERealizing the \(N\)-superconformal Lie algebras by loops shows that there exists only two isomorphism classes. This agrees with the cohomological point of view because in the case \(N=2\) the automorphism group \(\mathbf{O}_2\) of the corresponding Lie conformal superalgebra has two connected components. In contrast, there is an infinite family if non-isomorphic superconformal Lie algebras in the \(N=4\)-case, even if the automorphism group in this case is connected. The main reason for this to happen is that the base ring \(\mathbb C[t^{\pm1}]\) doesn't contain enough information to geometrically measure superconformal Lie algebras. This is solved by replacing it with the \textit{complex differential ring} \((\mathbb C[t^{\pm 1}],\frac{d}{dt})\). The case \(\delta=\frac{d}{dt}=0\) gives the classification of current algebras.NEWLINENEWLINEForms of a given \(\mathcal R\)-Lie conformal superalgebra \((\mathcal A,\partial)\) are classified in terms of the non-abelian coholomology pointed set \(H^1(\mathcal R,\Aut(\mathcal A)).\) The authors focus on the \(N=1,2,3\) Lie conformal superalgebras. The classification of their twisted loop Lie conformal superalgebras is carried out by explicitly computing their automorphism group functors functors and the corresponding non-abelian cohomology sets.NEWLINENEWLINEThe article only considers Lie conformal superalgebras over a complex differential ring \(\mathcal D\). Thus \(\mathcal D=(D,\delta)\) consists of a commutative \(\mathbb C\)-algebra \(D\) together with a derivation \(\delta:D\rightarrow D\). A \(\mathcal D\)-Lie conformal superalgebra is a triple \((\mathcal A,\partial_{\mathcal A},(-_(n)-)_{n\in\mathbb N})\) of (i) a \(\mathbb Z/2\mathbb Z\)-graded \(D\)-module \(\mathcal A=\mathcal A_0\oplus\mathcal A_1\), (ii) \(\partial_A\in\text{End}_{\mathbb C}(\mathcal A)\) stabilizing the grading of \(\mathcal A\), and (iii) a \(\mathbb C\)-bilinear product \((a,b)\mapsto a_{(n)}b\), for wach \(n\in\mathbb N\), all three satisfying explicit conditions.NEWLINENEWLINEFor an extension \(\mathcal D\rightarrow\mathcal D^\prime\) of differential rings, a \(\mathcal D^\prime/\mathcal D\)-form of \(\mathcal A\) is a \(\mathcal D\)-Lie conformal superalgebra \(\mathcal L\) such that \(\mathcal L\otimes_{\mathcal D}\mathcal D^\prime\cong\mathcal A\otimes_{\mathcal D}\mathcal D^\prime\). When the ring extension \(D\rightarrow D^\prime\) is faithfully flat, the set of isomorphism classes of \(\mathcal D^\prime/\mathcal D\)-forms is identified with the non-abelian Čech cohomology point set \(H^1(\mathcal D^\prime/\mathcal D,\Aut(\mathcal A))\), where \(\Aut(\mathcal A)\) is the group functor from the category of differential extensions of \(\mathcal D\) to the category of groups which assigns to an extension \(\mathcal D^\prime\) of \(\mathcal D\) the group \(\Aut(\mathcal A)(\mathcal D^\prime)\), explicitly defined.NEWLINENEWLINEThe essential Lie conformal superalgebra of this text is \(\mathcal K_N=\mathbb C[\delta]\otimes_C\Lambda(N)\). The first main result, building the rest, is that for an arbitrary complex differential ring \(\mathcal D\), for \(N=1,2,3\), there is a functorial isomorphism of groups NEWLINE\[NEWLINE\iota_{\mathcal D}:\mathbf{O}_N(D)\overset\sim\longrightarrow\text{GrAut}(\mathcal K_N)(\mathcal D).NEWLINE\]NEWLINENEWLINENEWLINENEWLINEBased on \(\mathcal K_N\), the authors classify the twisted loop Lie conformal superalgebras. The classification is completed in two steps: Classify \(\hat{\mathcal S}/\mathcal R\)-forms of \(\mathcal K_N\otimes_{\mathbb C}\mathcal R\), and then look at the passage from isomorphism classes of the \(\mathcal R\)-Lie conformal superalgebras to isomorphism classes of the complex lie conformal superalgebras. The result is that there are exactly two \(\hat{\mathcal S}/\mathcal R\)-forms of \(\mathcal K_N\otimes_{\mathbb C}\mathcal R\), and these are explicitly given.NEWLINENEWLINENEWLINEThe article is very explicit, and the classification is given in a good way.
scientific article; zbMATH DE number 6092505