Automorphism Of Order Research Articles

ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

AbstractWe consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebras

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus LT which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite-dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial LT-action and where the centre of the underlying Lie algebra acts trivially.

Erratum to the paper “New Uniform Subregular Parallelisms of PG (3, 4) Invariant under an Automorphism of Order 2” (Published in Vol. 20, 2020, No 6, pp. 18-27, DOI: 10.2478/cait-2020-0057)

Erratum to the paper “New Uniform Subregular Parallelisms of PG (3, 4) Invariant under an Automorphism of Order 2” (Published in Vol. 20, 2020, No 6, pp. 18-27, DOI: 10.2478/cait-2020-0057)

Representations of twisted toroidal Lie algebras from twisted modules over vertex algebras

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.

Levi Factors and Admissible Automorphisms

Let mathfrak {g} be a complex simple Lie algebra. We consider subalgebras mathfrak {m} which are Levi factors of parabolic subalgebras of mathfrak {g}, or equivalently mathfrak {m} is the centralizer of its center. We introduced the notion of admissible systems on finite order mathfrak {g}-automorphisms \U0001d703, and show that \U0001d703 has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize such automorphisms, and look for automorphisms of minimal order.

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 1)

Let VL be the vertex algebra associated to a non-degenerate even lattice L, θ the automorphism of VL induced from the −1-isometry of L, and VL+ the fixed point subalgebra of VL under the action of θ. In this series of papers we classify the irreducible weak VL+-modules and show that any irreducible weak VL+-module is isomorphic to a weak submodule of some irreducible weak VL-module or to a submodule of some irreducible θ-twisted VL-module.In this paper (Part 1), we show that when the rank of L is 1, every non-zero weak VL+-module contains a non-zero M(1)+-module, where M(1)+ is the fixed point subalgebra of the Heisenberg vertex operator algebra M(1) under the action of θ.

On the Solvability of ℤ3-Graded Novikov Algebras

Symmetries of algebraic systems are called automorphisms. An algebra admits an automorphism of finite order n if and only if it admits a Zn-grading. Let N=N0⊕N1⊕N2 be a Z3-graded Novikov algebra. The main goal of the paper is to prove that over a field of characteristic not equal to 3, the algebra N is solvable if N0 is solvable. We also show that a Z2-graded Novikov algebra N=N0⊕N1 over a field of characteristic not equal to 2 is solvable if N0 is solvable. This implies that for every n of the form n=2k3l, any Zn-graded Novikov algebra N over a field of characteristic not equal to 2,3 is solvable if N0 is solvable.

KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and$U(2)\oplus D_4^{\oplus 2}$lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of$\mathbb {P}^{1}\times \mathbb {P}^{1}$branched along a specific$(4,\,4)$curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient$(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$with the symmetric linearization.

On the influence of the fixed points of an automorphism to the structure of a group

Let α be a coprime automorphism of a group G of prime order and let P be an α-invariant Sylow p-subgroup of G. Assume that p∉π(CG(α)). Firstly, we prove that G is p-nilpotent if and only if CNG(P)(α) centralizes P. In the case that G is Sz(2r) and PSL(2,2r)-free where r=|α|, we show that G is p-closed if and only if CG(α) normalizes P. As a consequence of these two results, we obtain that G≅P×H for a group H if and only if CG(α) centralizes P. We also prove a generalization of the Frobenius p-nilpotency theorem for groups admitting a group of automorphisms of coprime order.

Exponent of a finite group admitting a coprime automorphism of prime order

Abstract Let 𝐺 be a finite group admitting an automorphism 𝜙 of prime order 𝑝 such that ( | G | , p ) = 1 (\lvert G\rvert,p)=1 . It is shown that if the fixed-point subgroup for 𝜙 has rank 𝑟 and ( x - 1 ⁢ x ϕ ) e = 1 (x^{-1}x^{\phi})^{e}=\nobreak 1 for each x ∈ G x\in G , then the exponent of [ G , ϕ ] [G,\phi] is ( e , p , r ) (e,p,r) -bounded.

New Uniform Subregular Parallelisms of PG(3, 4) Invariant under an Automorphism of Order 2

Abstract A spread in PG(n, q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A parallelism is uniform if all its spreads are isomorphic. Up to isomorphism, there are three spreads of PG(3, 4) – regular, subregular and aregular. Therefore, three types of uniform parallelisms are possible. In this work, we consider uniform parallelisms of PG(3, 4) which possess an automorphism of order 2. We establish that there are no regular parallelisms, and that there are 8253 nonisomorphic subregular parallelisms. Together with the parallelisms known before this work, this yields a total of 8623 known subregular parallelisms of PG(3, 4).

The problem of Euler/Tarry revisited

The problem of Euler/Tarry concerning 36 officers can be formulated in mathematical terms: Can a latin square of order 6 have an orthogonal square, or equivalently, are there 6 pairwise disjoint transversals? This was first answered (in the negative) by Tarry (1900/01). We prove the following Theorem: If a latin square of order 6 admits a reflection, i. e. an automorphism of order two which fixes the main diagonal elementwise, then it has no orthogonal square. We list the 12 isomorphism types of latin squares of order 6 and see: they all admit such a reflection. So we get a solution of the Euler problem without the tedious task of tracing the transversals.

The smallest symmetric cubic graphs with given type

It is known that arc-transitive group actions on finite cubic (3-valent) graphs fall into seven classes, denoted by 1, 21, 22, 3, 41, 42 and 5, where k, k1 or k2 indicates that the action is k-arc-regular, with k2 indicating that there is no arc-reversing automorphism of order 2 (for k=2 or 4). These classes can be further subdivided into 17 sub-classes, according to the types of arc-transitive subgroups of the full automorphism group of the graph, sometimes called the ‘action type’ of the graph. In this paper, we complete the determination of the smallest graphs in each of these 17 classes (begun by Conder and Nedela in (2009) [7]).

Fgc extended affine Lie algebras as fixed point subalgebras

It is known that the centreless cores of fgc extended affine Lie algebras can be viewed as fixed point subalgebras. In this paper, we study fixed point subalgebras of fgc extended affine Lie algebras. Our main result is that the fixed point subalgebras of untwisted extended affine Lie algebras under a finite abelian group (which is determined by some commuting finite order automorphisms) are fgc extended affine Lie algebras.

On central extensions and simply laced Lie algebras

Let Λ be a simply laced root lattice and w an elliptic automorphism of Λ of order d. This paper gives a construction that begins with a central extension of the group of coinvariants Λw and produces a semisimple Lie algebra of Dykin type Λ with an automorphism of order d lifting w. The input for this construction naturally arises when considering certain families of algebraic curves. The construction generalizes one used by Thorne to study plane quartics and one used by the author and Thorne to study a family of genus-2 curves.

Recursion operators and hierarchies of $$\text{mKdV}$$ equations related to the Kac–Moody algebras $$D_4^{(1)}$$, $$D_4^{(2)}$$, and $$D_4^{(3)}$$

We constructed the three nonequivalent gradings in the algebra $D_4 \simeq so(8)$. The first one is the standard one obtained with the Coxeter automorphism $C_1=S_{\alpha_2} S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}$ using its dihedral realization. In the second one we use $C_2 = C_1R$ where $R$ is the mirror automorphism. The third one is $C_3 = S_{\alpha_2}S_{\alpha_1}T$ where $T$ is the external automorphism of order 3. For each of these gradings we constructed the basis in the corresponding linear subspaces $\mathfrak{g}^{(k)}$, the orbits of the Coxeter automorphisms and the related Lax pairs generating the corresponding mKdV hierarchies. We found compact expressions for each of the hierarchies in terms of the recursion operators. At the end we wrote explicitly the first nontrivial mKdV equations and their Hamiltonians. For $D_4^{(1)}$ these are in fact two mKdV systems, due to the fact that in this case the exponent $3$ has multiplicity 2. Each of these mKdV systems consist of 4 equations of third order with respect to $\partial_x$. For $D_4^{(2)}$ this is a system of three equations of third order with respect to $\partial_x$. Finally, for $D_4^{(3)}$ this is a system of two equations of fifth order with respect to $\partial_x$.

Planar Whitehead graphs with cyclic symmetry arising from the study of Dunwoody manifolds

A fundamental theorem in the study of Dunwoody manifolds is a classification of finite graphs on 2n vertices that satisfy seven conditions (concerning planarity, regularity, and a cyclic automorphism of order n). Its significance is that if the presentation complex of a cyclic presentation is a spine of a 3-manifold then its Whitehead graph satisfies the first five conditions (the remaining conditions do not necessarily hold). In this paper we observe that this classification relies implicitly on an unstated 8th condition and that this condition is not necessary for such a presentation complex to be the spine of a 3-manifold. We expand the scope of Dunwoody’s classification by classifying all graphs that satisfy the first five conditions.

Plethysm and cohomology representations of external and symmetric products

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules on spaces such as (possibly singular) complex quasi-projective varieties. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant Künneth formula to a more general generating series identity for abstract characters of tensor powers V⊗n of an element V in a suitable symmetric monoidal category A. This abstract approach applies directly also in the equivariant context for spaces with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms), as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in A.

Fixed Points on Asymmetric Riemann Surfaces

We study actions of cyclic groups on asymmetric Riemann surfaces for which all conformal automorphisms of prime orders have the same number of fixed points. We find the sharp upper bound on the number of points fixed by the square of an anticonformal automorphism of an asymmetric surface. Moreover, we determine the minimal genus of an asymmetric Riemann surface on which a given finite group G acts without fixed points.

Star order automorphisms on the poset of type 1 operators

Let H be a complex infinite dimensional Hilbert space and B(H) the algebra of all bounded linear operators on H. The star partial order is defined by A≤⁎B if and only if A⁎A=A⁎B and AA⁎=AB⁎ for any A and B in B(H). We give a type decomposition of operators with respect to star order. For any A∈B(H), there are unique type 1 operator A1 which is 0 or the supremum of those rank 1 operators less than A1 and type 2 operator A2 which is not greater than any rank 1 operator under star order such that Ai≤⁎A(i=1,2) and A=A1+A2. Moreover, we determine all automorphisms on the poset of type 1 operators. As a consequence, we characterize continuous automorphisms on B(H).