We use the following rule of composition of automorphisms: if φ, ψ ∈ Aut(F2) and x ∈ F2 then φψ(x) = ψ(φ(x)). For x ∈ F2 denote by St(x) the stabilizer of x in Aut(F2). For a subset X of a group denote by 〈X〉 the subgroup generated by X. Denote [x, y] = x−1y−1xy, x = y−1xy. For g ∈ F2 denote by ĝ the automorphism induced by the conjugation by g: ĝ(x) = g−1xg, x ∈ F2. Let − : Aut(F2) → GL2(Z) be...

Let F be a perfect field and M(F ) the nonassociative simple Moufang loop consisting of the units in the (unique) split octonion algebra O(F ) modulo the center. Then Aut(M(F )) is equal to G2(F )o Aut(F ). In particular, every automorphism of M(F ) is induced by a semilinear automorphism of O(F ). The proof combines results and methods from geometrical loop theory, groups of Lie type and compo...

Abstract We formulate an interpretation of the theory of physical superselection sectors in terms of vertex operator algebra language. Using this formulation we give a construction of simple current from a primary semisimple element of weight one. We then prove that if a rational vertex operator algebra V has a simple current M satisfying certain conditions, then V ⊕M has a natural rational ver...

We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complement...

We extend the Larson–Sweedler theorem [10] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplik...

A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic automorphism. Mathematics Subject Cla...

Let F = (F1, . . . , Fn) : Cn → Cn be any polynomial mapping. By multidegree of F, denoted mdegF, we call the sequence of positive integers (deg F1, . . . , degFn). In this paper we addres the following problem: for which sequence (d1, . . . , dn) there is an automorphism or tame automorphism F : Cn → Cn with mdegF = (d1, . . . , dn). We proved, among other things, that there is no tame automor...

‎‎in this paper we determine all finite $2$-groups of‎ ‎class $2$ in which every automorphism of order $2$ leaving the frattini subgroup elementwise fixed is inner‎.

We construct some codes, designs and graphs that have the first or second Janko group, J1 or J2, respectively, acting as an automorphism group. We show computationally that the full automorphism group of the design or graph in each case is J1, J2 or J̄2, the extension of J2 by its outer automorphism, and we show that for some of the codes the same is true.

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism...