Jordan Pairs Research Articles

Inversion of regular analytic matrix functions: Local Smith form and subspace duality

This paper analyzes the relation between the local rank-structure of a regular analytic matrix function and the one of its inverse function. The local rank factorization ( lrf) of a matrix function is introduced, which characterizes extended canonical systems of root functions and the local Smith form. An interpretation of the local rank factorization in terms of Jordan chains and Jordan pairs is provided. Duality results are shown to hold between the subspaces associated with the lrf of the matrix function and the one of its reduced adjoint.

ON SUBGROUPS OF CLIFFORD GROUPS DEFINED BY JORDAN PAIRS OF RECTANGULAR MATRICES

AbstractSome embeddings of general linear groups into hyperbolic Clifford groups are constructed generically by using Jordan pairs of rectangular and alternating matrices over a ring. In low rank cases through exceptional isomorphisms, their direct description and relationships to some automorphisms of Clifford groups are given. Generic norms are calculated in detail, and equivariant embeddings of representation spaces are constructed.

Steinberg groups for Jordan pairs

We announce results on projective elementary groups and on Steinberg groups associated to Jordan pairs V with a grading by a locally finite 3-graded root system Φ: The projective elementary group PE ( V ) of V is a group with Φ-commutator relations with respect to appropriately defined root subgroups. Under some mild additional conditions, the Steinberg group associated to PE ( V ) uniquely covers all central extensions of PE ( V ) and is the universal central extension of PE ( V ) if Φ is irreducible and has infinite rank.

Irreducible Lie–Yamaguti algebras of generic type

Lie–Yamaguti algebras (or generalized Lie triple systems) are binary–ternary algebras intimately related to reductive homogeneous spaces. The Lie–Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterparts of the isotropy irreducible homogeneous spaces. These systems splits into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie–Yamaguti algebras of generic type are classified by relating them to several other nonassociative algebraic systems: Lie and Jordan algebras and triple systems, Jordan pairs or Freudenthal triple systems.

On Polynomial Identities in Associative and Jordan Pairs

We prove that a Jordan system satisfies a polynomial identity if and only if it satisfies a homotope polynomial identity. In the obtention of that result, we also prove an analogue for associative pairs with involution of Amitsur’s theorem on associative algebras satisfying a polynomial identity with involution.

Transitivity of inner automorphisms in infinite dimensional Cartan factors

Let C be a Cartan-factor having arbitrary dimension dimC. It is shown that the group Inn(C) of inner automorphisms of C acts transitively on the manifold \({\mathcal {U}_r(C)}\) of tripotents with finite rank r in C. This extends results by Loos (Bounded Symmetric Domains and Jordan Pairs. Mathematical Lectures. University of California, Irvine, 1977) valid in finite dimensions, and similar findings by Isidro et al. (Math Z 233(4):741–754, 2000; Acta Sci Math (Szeged) 66(3–4), 2000; Expo Math 20(2):97–116, 2002; Q J Math 57(4):505–525, 2006). Hence, the results presented here close a significant gap concerning the transitivity property of the general infinite-dimensional case. The proofs given here are based on new methods, independent of those used for the finite-dimensional cases.

Inner ideals and intrinsic subspaces of linear pair geometries

Abstract We introduce the notion of intrinsic subspaces of linear and affine pair geometries, which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [W. Bertram, K.-H. Neeb, Projective completions of Jordan pairs. I. The generalized projective geometry of a Lie algebra. J. Algebra 277 (2004), 474–519. MR2067615 (2005f:17031) Zbl 02105235], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic subspaces defined by the Peirce-decomposition which is related to 5-gradings of the projective Lie algebra. These examples, as well as the examples of general and Lagrangian flag geometries, lead to the conjecture that geometries of intrinsic subspaces tend to be themselves linear pair geometries.

An Artinian theory for Lie algebras

Complemented Lie algebras are introduced in this paper (a notion similar to that studied by O. Loos and E. Neher in Jordan pairs). We prove that a Lie algebra is complemented if and only if it is a direct sum of simple nondegenerate Artinian Lie algebras. Moreover, we classify simple nondegenerate Artinian Lie algebras over a field of characteristic 0 or greater than 7, and describe the Lie inner ideal structure of simple Lie algebras arising from simple associative algebras with nonzero socle.

A Construction of Gradings of Lie Algebras

In this article we present a method to construct gradings of Lie algebras. It requires the existence of an abelian inner ideal B of the Lie algebra whose subquotient, a Jordan pair, is covered by a finite grid, and it produces a grading of the Lie algebra L by the weight lattice of the root system associated to the covering grid. As a corollary one obtains a finite Z-grading L = L−n ⊕ · · · ⊕ Ln such that B = Ln. In particular, our assumption on B holds for abelian inner ideals of finite length in nondegenerate Lie algebras.

Inverse Spectral Problems for Semisimple Damped Vibrating Systems

Computational schemes are investigated for the solution of inverse spectral problems for $n \times n$ real systems of the form $L(\lambda) = M\lambda^2 + D\lambda + K$. Thus, admissible sets of data concerning systems of eigenvalues and eigenvectors are examined, and procedures for generating associated (isospectral) families of systems are developed. The analysis includes symmetric systems, systems with mixed real/nonreal spectrum, systems with positive definite coefficients, and hyperbolic systems (with real spectrum). A one‐to‐one correspondence between Jordan pairs and structure preserving similarities is clarified. An examination of complex symmetric matrices is included.

Jordan pairs of quadratic forms with values in invertible modules

Jordan pairs of quadratic forms are generalized so that they have forms with values in invertible modules. The role of such pairs turns out to be natural in describing ‘big cells’, a kind of open charts around unit sections, of Clifford and orthogonal groups as group schemes. Group germ structures on big cells are particularly interested in and related also to Cayley-Lipschitz transforms.

An elemental characterization of strong primeness in Lie algebras

In this paper we prove that a Lie algebra L is strongly prime if and only if [ x , [ y , L ] ] ≠ 0 for every nonzero elements x , y ∈ L . As a consequence, we give an elementary proof, without the classification theorem of strongly prime Jordan algebras, of the fact that a linear Jordan algebra or Jordan pair T is strongly prime if and only if { x , T , y } ≠ 0 for every x , y ∈ T . Moreover, we prove that the Jordan algebras at nonzero Jordan elements of strongly prime Lie algebras are strongly prime.

Homotope polynomial identities in prime Jordan systems

We prove an analogue of the Posner–Rowen theorem for strongly prime Jordan pairs and triple systems: the central closure of a strongly prime Jordan system satisfying a homotope polynomial identity is simple with finite capacity. We also prove that if a Jordan system satisfies a homotope polynomial identity it also satisfies a strict homotope polynomial identity.

Projective Completions of Jordan Pairs, Part II: Manifold Structures and Symmetric Spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields $$\mathbb{K}$$ , and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ‘compact-like’ duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ‘standard models’ – they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.

3-Graded Lie Algebras with Jordan Finiteness Conditions#

A notion of socle is introduced for 3-graded Lie algebras (over a ring of scalars Φ containing ) whose associated Jordan pairs are non-degenerate. The socle turns out to be a 3-graded ideal and is the sum of minimal 3-graded inner ideals each of which is a central extension of the TKK-algebra of a division Jordan pair. Non-degenerate 3-graded Lie algebras having a large socle are essentially determined by TKK-algebras of simple Jordan pairs with minimal inner ideals and their derivation algebras, which are also 3-graded. Classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space provide a source of examples of infinite dimensional strongly prime 3-graded Lie algebras with non-zero socle. Other examples can be found within the class of finitary simple Lie algebras

Hopf duals, algebraic groups, and Jordan pairs

The Hom functor is used to construct various algebras and coalgebras, including the continuous dual of a Hopf algebra with a residually finitely generated projective linear topology. The dual is used to construct a k-group scheme. The results are applied to the study of algebraic groups associated with Jordan pairs.

Projective completions of Jordan pairs: Part I. The generalized projective geometry of a Lie algebra

We prove that the projective completion ( X +, X −) of the Jordan pair ( g 1, g −1) corresponding to a 3-graded Lie algebra g= g 1⊕ g 0⊕ g −1 can be realized inside the space F of inner 3-filtrations of g in such a way that the remoteness relation on X +× X − corresponds to transversality of flags. This realization is used to give geometric proofs of structure results which will be used in Part II of this work in order to define on X + and X − the structure of a smooth manifold (in arbitrary dimension and over general base fields or -rings).

Tits–Kantor–Koecher algebras of strongly prime hermitian Jordan pairs

In this paper we describe the Tits–Kantor–Koecher algebras of strongly prime hermitian Jordan pairs in terms of ∗-prime associative pairs and their Martindale pairs of symmetric quotients.

Strongly prime Jordan pairs with nonzero socle

A characteristic-free description of strongly prime Jordan pairs with minimal inner ideals is provided in this paper.

Spin Groups Over a Commutative Ring and the Associated Root Data

Usual constructions of spin and Clifford groups, restricted to the case where the basic quadratic spaces is hyperbolic, are examined as those of group schemes over a commutative ring. Not surprisingly they yield indeed smooth and reductive group schemes, but the proof involves a lot of calculation and uses a connection with Jordan pairs. Maximal tori and their associated root data are also constructed explicitly.