Automorphism Groups Of Algebras Research Articles

The group of splendid Morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups

Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that \begin{align*} \calT(B)\cong \Out_P(A)\rtimes \Out(P,\calF), \end{align*} where $\Out(P,\calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\calF$ of $G$ on $P$ and $\Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^\times$.

Graded trace generating functions of Aut(ΓKn) acting on A(ΓKn)

In this paper, we describe the algebras [Formula: see text] associated with the Hasse graph of the poset of the faces of the complete graph at n vertices [Formula: see text]. Present the relations that define this algebra, calculate the Hilbert series of quadratic grade algebra [Formula: see text] and the graded trace generating functions of [Formula: see text] acting on [Formula: see text] and show that [Formula: see text] is a Koszul algebra. The methodology adopted in this paper consists of building the algebra [Formula: see text] using [I. Gelfand, V. Retakh, S. Serconek and R. L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. 11(2) (2005) 281], giving a presentation based on what was established by [V. Retakh, S. Serconek and R. L. Wilson, On a class of koszul algebras associated to directed graphs, J. Algebra 304(2) (2006) 1114–1129] in terms of the vertices of the graph [Formula: see text]. After that, we use [V. Retakh, S. Serconek and R. L. Wilson, Hilbert series of algebras associated to directed graphs, J. Algebra 312(1) (2007) 142–151] to determine the Hilbert series of [Formula: see text] and use [9] again to show that this algebra is a Koszul algebra. We use [J. Caldeira, A. De Souza Lima and J. Eder Salvador De Vasconcelos, Representations of automorphism groups of algebras associated to star polygons, J. Algebra Appl. 18(10) (2019) 1950197; C. Duffy, Representations of [Formula: see text] acting on homogeneous components of [Formula: see text] and [Formula: see text], Adv. Appl. Math. 42(1) (2009) 94–122] to determine the generating functions of the graduated trace of [Formula: see text] acting on [Formula: see text].

Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L, we are interested in finding the pairs (ϕ,ψ)∈Aut(V)×Aut(L), which are inducible by an automorphism in Aut(L˜). We connect the inducibility problem to the (2,3)-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut(V)×Aut(L) to be inducible lies in the (2,3)-cohomology group H(2,3)(L,V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2,3)-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2.

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

Description of the automorphism groups of some Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras.

Formal matrix rings: Automorphisms

Automorphism groups of formal matrix algebras with zero trace ideals are studied. Such an algebra is represented as a splitting extension of some ring by some nilpotent ideal. Using this extension, the study of the structure of the automorphism group of an algebra in a certain sense is reduced to the study of the structure of some its subgroups and quotient groups. Then the structure of these subgroups and factor groups is found. The case of formal triangular matrix rings is specially considered. At the end of the paper we study automorphisms of ordinary matrix rings. In this case, a well-known homomorphism from the group of outer automorphisms to the Picard group of a certain ring is used.

Automorphism Groups and Uniqueness of Holomorphic Vertex Operator Algebras of Central Charge 24

We describe the automorphism groups of all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras by using their constructions as simple current extensions. We also confirm a conjecture of G. H\"ohn on the numbers of holomorphic vertex operator algebras of central charge $24$ obtained as inequivalent simple current extensions of certain vertex operator algebras, which gives another proof of the uniqueness of holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras.

The description of the automorphism groups of finite-dimensional cyclic Leibniz algebras

In the study of Leibniz algebras, the information about their automorphisms (as well as about endomorphisms, derivations, etc.) is very useful. We describe the automorphism groups of finite-dimensional cyclic Leibniz algebras. In particular, we consider the natural relationships between Leibniz algebras, groups and modules over associative rings.

Characterization of Automorphisms of (θ,ω)-Twisted Radford’s Hom-Biproduct

In this paper, we study the Hom–Hopf algebra automorphism group of a (θ,ω)-twisted-Radford’s Hom-biproduct, which satisfies certain conditions. First, we study the endomorphism monoid and automorphism group of (θ,ω)-twisted Radford’s Hom-biproducts, and show that the endomorphism has a factorization closely related to the factors (A,α) and (H,β). Then, we consider (θ,ω)-twisted Radford’s Hom-biproduct automorphism group AutHom-Hopf(A×ωθH,p) as a subgroup of some semidirect product U(C,A)op⋊φG(A). Finally, we characterize the automorphisms of a concrete example.

The Conway-Miyamoto correspondences for the Fischer 3-transposition groups

In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups F i 23 \mathrm {Fi}_{23} and F i 22 \mathrm {Fi}_{22} and c = 25 / 28 c=25/28 and c = 11 / 12 c=11/12 Virasoro vectors of subalgebras of the moonshine vertex operator algebra.

Automorphisms of character varieties

We show that the algebraic automorphism group of the SL(2,C) character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminations.

ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY

If X is a smooth manifold and $$ \mathcal{G} $$ is a subgroup of Diff(X) we say that (X, $$ \mathcal{G} $$ ) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ $$ \mathcal{G} $$ there is some x ∈ X whose stabilizer Gx ≤ $$ \mathcal{G} $$ satisfies [G : Gx] ≤ C. We say that X has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if X is compact and possibly with boundary and has no odd cohomology then (X, Diff(X)) has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if Z is a non-necessarily compact smooth real affine variety, and Z has no odd cohomology, then (Z, Aut(Z)) has the almost fixed point property, where Aut(Z) is the group of algebraic automorphisms of Z lifting the identity on Spec ℝ. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.

Fine gradings and automorphism groups on associative algebras

ABSTRACT First, we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an associative algebra such that the algebra is graded-simple and satisfies the DCC on graded left ideals. We give necessary and sufficient conditions for the grading to be fine. Then we assume that one of these necessary conditions to be fine is satisfied, and we compute the automorphism groups of the grading; the results are expressed in terms of the automorphism groups of a graded-division algebra. Finally, we compute the automorphism groups of graded-division algebras in the case in which the ground field is the field of real numbers, and the underlying algebra (disregarding the grading) is simple and of finite dimension.

Automorphism groups of pseudo H-type algebras

In the present paper we determine the group of automorphisms of pseudo H-type Lie algebras, that are two step nilpotent Lie algebras closely related to the Clifford algebras Cl(Rr,s).

On automorphism groups of Hardy algebras

Let $E$ be a $W^{*}$-correspondence and let $H^{\infty}(E)$ be the associated Hardy algebra. The unit disc of intertwiners $\mathbb{D}((E^{\sigma})^{*})$ plays a central role in the study of $H^{\infty}(E)$. We show a number of results related to the automorphism groups of both $H^{\infty}(E)$ and $\mathbb{D}((E^{\sigma})^{*})$. We find a matrix representation for these groups and describe several features of their algebraic structure. Furthermore, we show an application of $Aut(\mathbb{D}({(E^{\sigma}})^*))$ to the study of Morita equivalence of $W^{*}$-correspondences.

Some Properties of Almost Abelian Lie Algebras

We study finite-dimensional Lie algebras which have a codimension one abelian ideal and prove some their properties including those related to the structure of the space of all Lie algebras. We also compute the automorphism groups of such Lie algebras.

Automorphism groups of the holomorphic vertex operator algebras associated with Niemeier lattices and the $-1$-isometries

In this article, we determine the automorphism groups of 14 holomorphic vertex operator algebras of central charge 24 obtained by applying the $\mathbb{Z}_2$-orbifold construction to the Niemeier lattice vertex operator algebras and lifts of the $-1$-isometries.

Characterization of automorphisms of Hom-biproducts

We study certain subgroups of the full group of monoidal Hom-Hopf algebra automorphisms of a Hom-biproduct, which gives a Hom-version of Radford’s results.

Characterization of automorphisms of twisted tensor biproducts

Abstract We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

Jordan property for algebraic groups and automorphism groups of projective varieties in arbitrary characteristic

We show an analogue of Jordan's theorem for algebraic groups defined over a field $\mathbb k$ of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over $\mathbb k$.