Automorphisms Of Algebras Research Articles

Kakutani's automorphism theorem for vector-valued bounded holomorphic functions

We give descriptions for algebra homomorphisms and automorphisms of vector-valued bounded holomorphic functions defined on the open unit disc. The characterizations derived allow us to establish the algebraic reflexivity properties of the group of automorphisms on these spaces.

A topological characterisation of the Kashiwara–Vergne groups

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in R 4 \mathbb {R}^4 , which can be finitely presented algebraically as a circuit algebra, or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups K V \mathsf {KV} and K R V \mathsf {KRV} —the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group G R T 1 \mathsf {GRT}_1 as automorphisms of arrow diagrams.

On a Classification of Periodic Maps on the 2-Torus

In this paper, following J. Nielsen, we introduce a complete characteristic of orientation-preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A. Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

О группе автоморфизмов алгебры плюральных чисел

The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element. In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he established that higher-order tangent bundles over smooth manifolds have the structure of a smooth manifold over algebras of plural numbers. This allowed him in the 70s of the twentieth century to construct a theory of lifts of tensor fields and linear connections from a smooth manifold to its tangent bundles of arbitrary order. In this paper, we study automorphisms of the algebra of plural numbers. It is proved that the set of all automorphisms of the algebra of plural numbers forms a group. The structure of this group is described. The groups of automorphisms of the algebra of plural numbers with small dimension are indicated as examples.

Автоморфные алгебры динамических систем и обобщенные контракции Иненю-Вигнера

Lie algebras a with a complex underlying vector space V are
 studied that are automorphic with respect to a given linear
 dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂
 Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a
 Lie algebraic structure to the vector space of trajectories of
 the group Gt. The basics of the general structure of automorphic
 algebras a are described in terms of the eigenspace
 decomposition of the operatorM ∈ der(a) that determines
 the dynamics. Symmetries encoded by the presence of nonabelian
 automorphic algebras are pointed out connected to
 conservation laws, spectral relations and root systems. It is
 shown that, for a given dynamics Gt, automorphic algebras
 can be found via a limit transition in the space of Lie algebras
 on V along the trajectories of the group Gt itself. This procedure
 generalises the well-known Inönü-Wigner contraction
 and links adjoint representations of automorphic algebras to
 isospectral Lax representations on gl(V ). These results can
 be applied to physically important symmetry groups and their
 representations, including classical and relativistic mechanics,
 open quantum dynamics and nonlinear evolution equations.
 Simple examples are given.

Local Derivations and Automorphisms of Direct Sum Null-Filiform Leibniz Algebras

Local Derivations and Automorphisms of Direct Sum Null-Filiform Leibniz Algebras

Local derivations and automorphisms of Cayley algebras

The present paper is devoted to the description of local and 2-local derivations and automorphisms on Cayley algebras over an arbitrary field F. Given a Cayley algebra C with norm n, let C0 be its subspace of trace 0 elements. We prove that the space of all local derivations of C coincides with the Lie algebra {d∈so(C,n)|d(1)=0} which is isomorphic to the orthogonal Lie algebra so(C0,n). Surprisingly, the behavior of 2-local derivations depends on the Cayley algebra being split or division. Every 2-local derivation on the split Cayley algebra is a derivation, so they are isomorphic to the exceptional Lie algebra g2(F) if charF≠2,3. On the other hand, on division Cayley algebras over a field F, the sets of 2-local derivations and local derivations coincide. As a corollary we obtain descriptions of local and 2-local derivations of the seven-dimensional simple non-Lie Malcev algebras over fields of characteristic ≠2,3. Further, we prove that the group of all local automorphisms of C coincides with the group {φ∈O(C,n)|φ(1)=1}. As in the case of 2-local derivations, the behavior of 2-local automorphisms depends on the Cayley algebra being split or division. Every 2-local automorphism on the split Cayley algebra is an automorphism, so they form the exceptional Lie group G2(F) if charF≠2,3. On the other hand, on division Cayley algebras over a field F, the groups of 2-local automorphisms and local automorphisms coincide.

Duality defects in E8

We classify all non-invertible Kramers-Wannier duality defects in the E8 lattice Vertex Operator Algebra (i.e. the chiral (E8)1 WZW model) coming from ℤm symmetries. We illustrate how these defects are systematically obtainable as ℤ2 twists of invariant sub-VOAs, compute defect partition functions for small m, and verify our results against other techniques. Throughout, we focus on taking a physical perspective and highlight the important moving pieces involved in the calculations. Kac’s theorem for finite automorphisms of Lie algebras and contemporary results on holomorphic VOAs play a role. We also provide a perspective from the point of view of (2+1)d Topological Field Theory and provide a rigorous proof that all corresponding Tambara-Yamagami actions on holomorphic VOAs can be obtained in this manner. We include a list of directions for future studies.

Permutative automorphisms of the Cuntz algebras: Quadratic cycles, an involution and a box product

Permutative automorphisms of the Cuntz algebra On are in bijection with a class of permutations of nk elements, that are called stable, and are further partitioned by rank. In this work we mainly focus on stable cycles in the quadratic case (i.e., k=2). More precisely, in such a quadratic case we provide a characterization of the stable cycles of rank one (so proving Conjecture 12.1 in [3]), exhibit a closed formula for the number of stable r-cycles of rank one (valid for all n and r), and characterize and enumerate the stable 3-cycles of any given rank. We also show that the set of stable permutations is equipped with a natural involution that preserves the cycle-type and the rank, and that there is a map that associates to two stable permutations of nk and mk elements, respectively, a stable permutation of (nm)k elements.

Diagram automorphisms and canonical bases for quantum affine algebras, II

Let Uq− be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on Uq− induced from a diagram automorphism. Let U_q− be the quantum algebra obtained from σ, and B˜ (resp. B_˜) the canonical signed basis of Uq− (resp. U_q−). Assume that Uq− is simply-laced of finite or affine type. In our previous papers [10], [11], we have proved by an elementary method, that there exists a natural bijection B˜σ≃B_˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases.

POINTWISE INNER AUTOMORPHISMS OF RELATIVELY FREE LIE ALGEBRAS

Let $K$ be a field of characteristic zero, and $L_{m,c}$ be the free metabelian nilpotent Lie algebra of class $c$ of rank $m$ over $K$. We call an automorphism $\phi$ pointwise inner, if there exists an inner automorphism $\xi_i$ for each generator $x_i$, $i=1,\ldots,m$, such that $\phi(x_i)=\xi_i(x_i)$. In this study, we exemine the group $PI(L_{m,c})$ of pintwise inner automorphisms of the Lie algebra $L_{m,c}$, and we provide a set of generators for this group.

On perturbations of algebraic periodic automorphisms of a two-dimensional torus

According to the results of V. Z. Grines and A. N. Bezdenezhnykh, for each gradient-like diffeomorphism of a closed orientable surface M2 there exist a gradient-like flow and a periodic diffeomorphism of this surface such that the original diffeomorphism is a superposition of a diffeomorphism that is a shift per unit time of the flow and the periodic diffeomorphism. In the case when M2 is a two-dimensional torus, there is a topological classification of periodic maps. Moreover, it is known that there is only a finite number of topological conjugacy classes of periodic diffeomorphisms that are not homotopic to identity one. Each such class contains a representative that is a periodic algebraic automorphism of a two-dimensional torus. Periodic automorphisms of a two-dimensional torus are not structurally stable maps, and, in general, it is impossible to predict the dynamics of their arbitrarily small perturbations. However, in the case when a periodic diffeomorphism is algebraic, we constructed a one-parameter family of maps consisting of the initial periodic algebraic automorphism at zero parameter value and gradient-like diffeomorphisms of a twodimensional torus for all non-zero parameter values. Each diffeomorphism of the constructed one-parameter families inherits, in a certain sense, the dynamics of a periodic algebraic automorphism being perturbed.

Affine Grassmannians for triality groups

We study affine Grassmannians for ramified triality groups. These groups are of type D43, so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebras. Using these composition algebras, we give descriptions of the affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space.

Local and 2-local automorphisms of some solvable Leibniz algebras

In this paper we prove that any local automorphism on the solvable Leibniz algebras with null-filiform and naturally graded non-Lie filiform nilradicals, whose dimension of complementary space is maximal is an automorphism. Furthermore, the same problem concerning 2-local automorphisms of such algebras is investigated and we obtain the analogously results for 2-local automorphisms.

Twisted conjugacy in linear algebraic groups II

Let G be a linear algebraic group over an algebraically closed field k and Autalg(G) the group of all algebraic group automorphisms of G. For every φ∈Autalg(G) let R(φ) denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by (g,x)↦gxφ(g−1), for all g,x∈G). We say that G has the algebraic R∞-property if R(φ) is infinite for every φ∈Autalg(G). In [1] we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group G has the algebraic R∞-property, then Gφ (the fixed-point subgroup of G under φ) is infinite for all φ∈Autalg(G). In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic R∞-property and identify certain classes of solvable algebraic groups for which the property fails.

Differential operators on G/U and the Gelfand-Graev action

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on T⁎(G/U), the cotangent bundle. A long time ago, S. Gelfand and M. Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in an essential way. We give a new algebraic construction of the Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hamiltonian reduction and involves the ring of Whittaker differential operators on G/U, a twisted analogue of D(G/U).Our main result has an interpretation, via geometric Satake, in terms of spherical perverse sheaves on the affine Grassmannian for the Langlands dual group.

CR-manifolds of infinite type in the sense of Bloom and Graham

An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.

On the extension of unitary group automorphisms of Cuntz algebras

On the extension of unitary group automorphisms of Cuntz algebras

Proper Lie automorphisms of incidence algebras

AbstractLet X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.

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.