We show that many noetherian Hopf algebras A have a rigid dualising complex R with R = A[d]. Here, d is the injective dimension of the algebra and ν is a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call ν the Nakayama automorphism of A. We prove that ν = Sξ, where S is the antipode of A and ξ ...

We use a simple description of the outer automorphism of S6 to cleanly describe the invariant theory of six points in P, P, and P. In §1.1–1.2, we give two short descriptions of the outer automorphism of S6, complete with a proof that they indeed describe an outer automorphism. (Our goal is to have a construction that the reader can fully understand and readily verify.) In §1.3 we give another ...

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group SL2(F41)⊗ S̃3 ⊂ GL80(Z) is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the Θ-series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite...

We cryptanalyze a recently proposed matrix-based MOR cryptosystem. The security of the system depends on the difficulty of solving the following discrete logarithm problem: given an inner automorphism φ of SL(d,Fq) and φa (each given in terms of their images on generators of SL(d,Fq)), find a. We show that this problem can be reduced to a small number of similar problems in quotients of polynom...

We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as Iw ◦ σ1 ◦ σ2 ◦ · · · ◦ σk, where Iw is the inner automorphism associated with w, and σj ∈ S for 1 ≤ j ≤ k. As a consequence, M is the matrix of an invertible substitution if and only if it is a f...

Generalized conjugation is the action of a group on its underlying set given by (g, x) 7→ φ(g)xg, where φ : G → G is some fixed endomorphism. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if φ has a nontrivial kernel or if it is an inner automorphism, and we give a construction that allows a part...

We show that the " hard " part of Whitehead's algorithm for solving the automorphism problem in a fixed free group F k terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly linear-time generic-case complexity. We also prove that the stabilizers of generic elements of F k in Aut(F k) are cyclic groups gen...

Given a variety V of universal algebras. A new approach is suggested to characterize algebraically automorphisms of the category of free V-algebras. It gives in many cases an answer to the problem set by the first of authors, if automorphisms of such a category are inner or not. This question is important for universal algebraic geometry [5, 9]. Most of results will actually be proved to hold f...

Assuming the continuum hypothesis, we show that the Calkin algebra has 2 ℵ 1 outer automorphisms. Let H be a separable infinite dimensional Hilbert space, let L(H) be the algebra of bounded operators on H, let K(H) be the algebra of compact operators on H, and let Q = L(H)/K(H) be the Calkin algebra. A long-standing problem asks whether every automorphism of Q is inner, that is, of the form x →...

Let S be a ring with 1, C the center of S, G a finite automorphism group of S of order n invertible in S, and SG the subnng of elements of S fixed under each element in G. It is shown that the skew group ring S,G is a G’-Galois extension of (S,G)G’ that is a projective separable CG-algebra where G’ is the inner automorphism group of S,G induced by G if and only if S is a G-Galois extension of S...