Finite Order Automorphisms Research Articles

Derivations of Tensor Product Algebras

It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras.

Realization of graded-simple algebras as loop algebras

Multiloop algebras determined by n commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a Zn-graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and sufficient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group. We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields.

Non-finiteness results for Nil-groups

Generalizing an idea of Farrell we prove that for a ring E and a ring automorphism of finite order the groups Nil0.EI/ and all of its p ‐primary subgroups are either trivial or not finitely generated as an abelian group. We also prove that if and are ring automorphisms such that is of finite order then Nil0.EIE ;E / and all of its p ‐primary subgroups are either trivial or not finitely generated as an abelian group. These Nil-groups include the Nil-groups appearing in the decomposition of Ki of virtually cyclic groups for i 1. 18F25; 19B28, 19D35

Chevalleyp–local finite groups

We describe the spaces of homotopy fixed points of unstable Adams operations acting on p–compact groups and also of unstable Adams operations twisted with a finite order automorphism of the p–compact group. We obtain new exotic p–local finite groups.

Symmetric X 9 singularities and complex affine reflection groups

We establish a natural correspondence between the finite order automorphisms of the function singularities X9 and the complex crystallographic groups. A complete list of the related objects is obtained.

Automorphism-invariant representations of real reductive groups

We characterize representations of a connected real reductive group which are invariant under an automorphism of finite order. The intertwining operators constructed in this characterization are then used to prove a Paley-Wiener theorem for a class of real reductive groups twisted by involutions.

Graded Lie algebras with few nontrivial components

We prove that if a (ℤ/nℤ)-graded Lie algebra L = ↕i=0n−1Li has d nontrivial components Li and the null component L0 has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (ℤ/nℤ)-graded Lie rings L = ↕i=0n−1 with few nontrivial components Li if the null component L0 has finite order m. These results generalize Kreknin’s theorem on the solvability of the (ℤ/nℤ)-graded Lie rings L = ↕i=0n−1Li with trivial component L0 and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components Li. The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].

Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori

AbstractWe investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study themin this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

Sylvester-’t Hooft generators and relations between them for $$\mathfrak{s}\mathfrak{l}(n)$$ and $$\mathfrak{g}\mathfrak{l}(n|n)$$

Among the simple finite-dimensional Lie algebras, only \(\mathfrak{s}\mathfrak{l}(n)\) has two finite-order automorphisms that have no common nonzero eigenvector with the eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow generating all of \(\mathfrak{s}\mathfrak{l}(n)\) under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all n×n matrices Mat(n). These generators appear in the description of elliptic solutions of the classical Yang-Baxter equation, the orthogonal decompositions of Lie algebras, ’t Hooft’s work on confinement operators in QCD, and various other instances. Here, we give an algorithm that both generates \(\mathfrak{s}\mathfrak{l}(n)\) and explicitly describes a set of defining relations. For simple (up to the center) Lie superalgebras, analogues of Sylvester generators exist only for \(\mathfrak{g}\mathfrak{l}(n|n)\). We also compute the relations for this case.

ON THE INDUCTIVE LIMIT STRUCTURE OF ORDER FOUR AUTOMORPHISMS OF THE IRRATIONAL ROTATION ALGEBRA

It is shown that there exists an order four automorphism of the irrational rotation algebra Aθ that leaves invariant an increasing sequence of unital C*-subalgebras consisting of two circles and two points and such that its induced map on K1(Aθ) is given by the matrix [Formula: see text]. The resulting fixed point algebra is shown to be approximately finite-dimensional. A brief survey of recent work on the canonical finite order automorphisms of Aθ is given and some open problems are posed.

Generalized Laurent polynomial rings as quantum projective 3-spaces

Given a ring R, we introduce the notion of a generalized Laurent polynomial ring over R. This class includes the generalized Weyl algebras. We show that these rings inherit many properties from the ground ring R. This construction is then used to create two new families of quadratic global dimension four Artin–Schelter regular algebras. We show that in most cases the second family has a finite point scheme and a defining automorphism of finite order. Nonetheless, a generic algebra in this family is not finite over its center.

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.

Some finite quantum [formula omitted]s that are infinite modules over their centers

A result of M. Artin, J. Tate and M. Van den Bergh asserts that a regular algebra of global dimension three is a finite module over its center if and only if the automorphism encoded in the point scheme has finite order. We prove that the analogous result for a regular algebra of global dimension four is false by presenting families of quadratic, noetherian regular algebras A of global dimension four such that (i) A is an infinite module over its center, (ii) A has a finite point scheme, which is the graph of an automorphism of finite order, and (iii) A has a one-parameter family of line modules. Such algebras are candidates for generic regular algebras of global dimension four. The methods used to construct the algebras provide new techniques for creating other potential candidates.

Fixed point subalgebras of extended affine Lie algebras

It is a well-known result that the fixed point subalgebra of a finite dimensional complex simple Lie algebra under a finite order automorphism is a reductive Lie algebra so it is a direct sum of finite dimensional simple Lie subalgebras and an abelian subalgebra. We consider this for the class of extended affine Lie algebras and are able to show that the fixed point subalgebra of an extended affine Lie algebra under a finite order automorphism (which satisfies certain natural properties) is a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace which is contained in the centralizer of the core. Moreover, we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands.

Maximal Subalgebras of Kac–Moody Algebras#

We use the classification of finite order automorphisms by Kac to characterize all maximal subalgebras, regular, semisimple, reductive or not of a simple complex Lie algebra (up to conjugacy) that we can determine from its Dynkin diagram. Using Barnea et al. [Barnea, Y., Shalev, A., Zelmanov, E. I. (1998). Graded subalgebras of affine Kac–Moody algebras. Israel J. Math. 104:321–334] we extend our results to the case of affine Kac–Moody algebras. We also point out some inaccuracies in the Dynkin paper [Dynkin, E. B. (1957a). Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl t. 6:111–244].

On the periods of motives with complex multiplication and a conjecture of Gross–Deligne

We prove that the existence of an automorphism of finite order on a Q-variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Γ-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne [11, p. 205] 1 . Our proof relies on the arithmetic fixed-point

Finite-order automorphisms of a certain torus

A classical result of Higman [H], [S, Exer. 6.3], asserts that the roots of unity in the group ring Z[Γ] of a finite commutative group Γ are the elements ±γ for γ ∈ Γ. One application of this result is the determination of the group of finite-order automorphisms of a torus T = ResS′/S(Gm) for a finite etale Galois covering f : S′ → S with S and S′ connected schemes and abelian Galois group Γ = Gal(S′/S) (e.g., an unramified extension of local fields). Indeed, the torus T has character group Z[Gal(S′/S)] with Gal(S′/S) acting through left translation, and an automorphism of T is “the same” as an automorphism of its character group X(T ) as a Galois module. Thus, to give an automorphism of T is to give a generator of Z[Gal(S′/S)] as a left module over itself. This in turn is just a unit in the group ring. Once we have the list of roots of unity, this gives the list of finite-order automorphisms. It follows that the group of finite-order automorphisms of the torus T is the finite group that is generated (as a direct product) by inversion and the action of Gal(S′/S). Now consider the problem (raised to the author by G. Prasad) of finding all finite-order automorphisms of the norm-1 subtorus T 1 in T when Γ is cyclic (e.g., the Galois group of an unramified extension of local fields). This subtorus is functorially described as the kernel of the determinant map

Almost solubility of Lie algebras with almost regular automorphisms

The well-known theorem of Borel–Mostow–Kreknin on solubility of Lie algebras with regular automorphisms is generalized to the case of almost regular automorphisms. It is proved that if a Lie algebra L admits an automorphism ϕ of finite order n with finite-dimensional fixed-point subalgebra of dimension dimCL(ϕ)=m, then L has a soluble ideal of derived length bounded by a function of n whose codimension is bounded by a function of m and n (Theorem 1). A virtually equivalent formulation is in terms of a (Z/nZ)-graded Lie algebra L whose zero component L0 has finite dimension m. The functions of n and of m and n in Theorem 1 can be given explicit upper estimates. The proof is of combinatorial nature and uses the criterion for solubility of Lie rings with an automorphism obtained in [E.I. Khukhro, Siberian Math. J. 42 (2001) 996–1000]. The method of generalized, or graded, centralizers is developed, which was originally created in [E.I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63] for almost regular automorphisms of prime order. As a corollary we prove a result analogous to Theorem 1 on locally nilpotent torsion-free groups admitting an automorphism of finite order with the fixed points subgroup of finite rank (Theorem 3). We also prove an analogous result for Lie rings with an automorphism of finite order having finitely many fixed points (Theorem 2).

Covering algebras II: Isomorphism of loop algebras

This paper studies the loop algebras that arise from pairs consisting of a symmetrizable Kac-Moody Lie algebra g and a finite order automorphism σ of g. We obtain necessary and sufficient conditions for two such loop algebras to be isomorphic.

Almost solubility of Lie algebras with almost regular automorphisms

The well-known theorem of Borel–Mostow–Kreknin on solubility of Lie algebras with regular automorphisms is generalized to the case of almost regular automorphisms. It is proved that if a Lie algebra L admits an automorphism ϕ of finite order n with finite-dimensional fixed-point subalgebra of dimension dim C L ( ϕ)= m, then L has a soluble ideal of derived length bounded by a function of n whose codimension is bounded by a function of m and n (Theorem 1). A virtually equivalent formulation is in terms of a ( Z/n Z) -graded Lie algebra L whose zero component L 0 has finite dimension m. The functions of n and of m and n in Theorem 1 can be given explicit upper estimates. The proof is of combinatorial nature and uses the criterion for solubility of Lie rings with an automorphism obtained in [E.I. Khukhro, Siberian Math. J. 42 (2001) 996–1000]. The method of generalized, or graded, centralizers is developed, which was originally created in [E.I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63] for almost regular automorphisms of prime order. As a corollary we prove a result analogous to Theorem 1 on locally nilpotent torsion-free groups admitting an automorphism of finite order with the fixed points subgroup of finite rank (Theorem 3). We also prove an analogous result for Lie rings with an automorphism of finite order having finitely many fixed points (Theorem 2).