Automorphisms Of Algebras Research Articles

Cuntz algebra automorphisms: Graphs and stability of permutations

We characterize the permutative automorphisms of the Cuntz algebra O n \mathcal {O}_n (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube [ n ] t [n]^t . As applications we show that in the limit of large t t (resp. n n ) almost all permutations are not stable, thus proving Conj. 12.5 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], characterize (and enumerate) stable quadratic 4 4 and 5 5 -cycles, as well as a notable class of stable quadratic r r -cycles, i.e. those admitting a compatible cyclic factorization by stable transpositions. Some of our results use new combinatorial concepts that may be of independent interest.

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

On multiplicative automorphisms of incidence algebras

Let [Formula: see text] be the incidence algebra of the preordered set [Formula: see text] over the ring [Formula: see text]. In the case of a finite connected partially ordered set [Formula: see text], we prove that the subgroup of inner multiplicative automorphisms is a direct factor of the group of multiplicative automorphisms of the algebra [Formula: see text]. As a consequence, we obtain several matching criteria of the subgroup of inner multiplicative automorphisms with the group of multiplicative automorphisms.

Affine Grassmannians for G2

We study affine Grassmannians for the exceptional group of type G2. This group can be given as automorphisms of octonion algebras (or para-octonion algebras). By using this automorphism group, we consider all maximal parahoric subgroups in G2, and give a description of affine Grassmannians for G2 as functors classifying suitable orders in a fixed space.

Automorphisms of Leavitt path algebras: Zhang twist and irreducible representations

In this article, we construct (graded) automorphisms fixing all vertices of Leavitt path algebras of arbitrary graphs in terms of general linear groups over corners of these algebras. As an application, we study Zhang twist of Leavitt path algebras and describe new classes of irreducible representations of Leavitt path algebras of rose graphs Rn with n petals.

Hopf algebroids and twists for quantum projective spaces

We study the relationship between antipodes on a Hopf algebroid ▪ in the sense of Böhm–Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H-Hopf–Galois extensions B⊆A and related Ehresmann–Schauenburg bialgebroid. In particular, we find that the twists are in one-to-one correspondence with H-comodule algebra automorphism of A. We work out in detail the U(1)-extension O(CPqn−1)⊆O(Sq2n−1) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K-theory of the base algebra O(CPqn−1).

Automorphisms of quantum polynomial rings and Drinfeld Hecke algebras

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.

Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras.

2-term averaging L∞-algebras and non-abelian extensions of averaging Lie algebras

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L∞-algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context.

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 $(A^P)^\times$.

The criteria for automorphisms on finite-dimensional algebras

<p>In this paper, we will establish a criterion for automorphisms of finite-dimensional algebras. As an application, we will describe all automorphisms of the single-parameter generalized quaternion algebra. Additionally, we will obtain all automorphisms of Sweedler's 4-dimensional Hopf algebra.</p>

Автомофизмы нильтреугольных подколец алгебр Шевалле типа $G_2$ над областями целостности. I

Let $N\Phi(K)$ be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring $K$ with the identity associated with a root system $\Phi$. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings $N\Phi(K)$. Automorphisms of the Lie algebra $N\Phi(K)$ under restrictions $K=2K=3K$ on ring $K$ are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type $A_{n}$, a description of automorphisms of Lie rings $N\Phi(K)$ over $K$ was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring $N\Phi(K)$ are described by V.M. Levchuk (1990) for type $D_4$ over $K$, and for other types by A.V. Litavrin (2017), excluding types $G_2$ and $F_4$. In this paper we describe automorphisms of a nil-triangular Lie ring of type $G_2$ over an integrity domain $K$ under the following restrictions on $K$: either $K=2K=3K$, or $3K = 0$. To study automorphisms, the upper and lower central series described in this paper, are essentially used.

Автоморфизмы нильтреугольных подколец алгебр Шевалле типа G2 над полем характеристики 2

Let NΦ(K) be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring K with the identity associated with a root system Φ (The basis of NΦ(K) consists of all elements er ∈ Φ+ of the Chevalley basis). This paper studies the well-known problem of describing automorphisms of Lie algebras and rings NΦ(K). Automorphisms of the Lie algebra NΦ(K) under restrictions K = 2K = 3K on ring K are described by Y. Cao, D. Jiang, J. Wang (Intern. J. Algebra and Computation, 2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type An, a description of automorphisms of Lie rings NΦ(K) over K was obtained by V.M. Levchuk (Siberian Mathematical Journal, 1983). Automorphisms of the Lie ring NΦ(K) are described by V.M. Levchuk (Algebra and Logic, 1990) for type D4 over K, and for other types by A.V. Litavrin (Thesis for: Cand. Sc. (Physics and Mathematics) – 01.01.06. Siberian Federal University, 2017), excluding types G2 and F4. The author (2022) obtained a description of automorphisms of Lie rings NΦ(K) of type G2 when K is an integrity domain and K = 2K = 3K or 3K = 0. In this paper we describe automorphisms of a niltriangular Lie ring of type G2 over a field K under restriction 2K = 0. To study automorphisms, the upper and lower central series described in this work are essentially used. A new non-standard automorphism was found, called an S-automorphism.

Local and 2-local automorphisms of solvable Leibniz algebras with abelian and model nilradicals

In the paper, we describe the automorphisms on the solvable Leibniz algebras with the model or abelian nilradicals, whose complementary spaces are equal to two and one, respectively. We show that any local automorphism on a solvable Leibniz algebra with model nilradical of maximal codimension is an automorphism. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimension complementary space, admit local automorphisms which are not automorphisms. Moreover, similar problems concerning 2-local automorphisms of such algebras are investigated. We give examples of solvable Leibniz algebras in which 2-local automorphisms are automorphisms and examples of such algebras which admit 2-local automorphisms which are not automorphisms.

On exp(Der(E)) of nilpotent evolution algebras of maximal nilindex

In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras of maximal nilindex, allows the set exp ⁡ ( Der ⁡ ( E ) ) to be investigated. Moreover exp ⁡ ( Der ⁡ ( E ) ) is shown to be a normal subgroup of Aut ( E ) , and its corresponding index is calculated.

Local automorphisms of complex solvable Lie algebras of maximal rank

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism.

Perturbations of Nonhyperbolic Algebraic Automorphisms of the 2-Torus

Perturbations of Nonhyperbolic Algebraic Automorphisms of the 2-Torus

Eigenvalue estimates for 3-Sasaki structures

Abstract We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.

Anick type automorphisms and new irreducible representations of Leavitt path algebras

In this article, we give a new class of automorphisms of Leavitt path algebras of arbitrary graphs. Consequently, we obtain Anick type automorphisms of these Leavitt path algebras and new irreducible representations of Leavitt algebras of type (1, n) .

Integrable Generators of Lie Algebras of Vector Fields on textrm{SL}_2(mathbb {C}) and on xy = z^2

For the special linear group textrm{SL}_2(mathbb {C}) and for the singular quadratic Danielewski surface x y = z^2 we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on x y = z^2.