Tangent Algebra Research Articles

TANGENT FIELDS ON DEFORMATIONS OF COMPLEX SPACES

Properties of sheaves of graded Lie algebras associated with a flat mapping of complex spaces are established. In particular, for a minimal versal deformation the tangent algebra of a fiber defines a linearization of the algebra of liftable fields on the base, which in turn enables one to find the discriminant of the deformation and its modular subspace. A criterion is obtained for the nilpotency of the tangent algebra of the germ of a hypersurface with a unique singular point. It is proved that in the algebra of liftable fields on the base of a minimal versal deformation of such a germ there always exists a basis with symmetric coefficient matrix.

An algebraic interpretation of quantized gauge interactions and BRS transformations

In an algebraic approach, nonlinear gauge interactions and Becchi-Rouet-Stora transformations arise as two isomorphic aspects of a gauge algebra—consisting of the homogeneous and inhomogeneous structures in the tangent algebra of a Lie group. The quantized representatives are formulated in the associated quantum algebra which has to be of Bose and Fermi type for the gauge vectors and the Faddeev-Popov vectors, respectively. The implementation of the time reflection (hermiticity) properties requires a doubling for the Faddeev-Popov sector of the theory or—equivalently—a 2nd-order time derivative formalism.

On Sophus Lie's fundamental theorem

Sophus Lie's third theorem asserts that for every finite dimensional Lie algebra there is a local Lie group with the given algebra as tangent algebra at the identity. In particular, each subalgebra of the Lie algebra of a local Lie group determines a local subgroup whose Lie algebra is the given one. Recent interest in subsemigroups and local subsemigroups of Lie groups motivate the investigation to which extent local semigroups in Lie groups are determined by infinitesimal generators. It is known that the set L( S) of tangent vectors at the identity of a local semigroup S with identity in a (local) Lie group G is an additively closed convex subset of the Lie algebra L( G) which is topologically closed and contains 0; its largest vector subspace H is a subalgebra and all automorphisms of L( G) of the form e ad x with x ϵ H leave L( S) invariant as a whole (with (ad x)( y) = [ x, y]). We call an additively and topologically closed convex subset W containing 0 in a Lie algebra L a Lie wedge if it satisfies e ad x W = W for all x ϵ W ∩ t- W. In this paper we show that Lie wedges are precisely the infinitesimal generators of local semigroups in Lie groups. More precisely, we show that in a Banach Lie algebra L for every Lie wedge W whose edge W ∩ − W has a closed vector space complement in L, there is an open ball B around 0 in L such that the Cambell Hausdorff series x ∗ y = x + y + 1 2 [x, y] + ··· converges absolutely for x, y ϵ B and there is a subset S ⊂ B with (i) 0 ϵ S, (ii) (S ∗ S) ∩ B ⊂S, and L(S) = W .

Extensions of Lie-graded algebras

For Lie-graded algebras which are generalizations of Lie algebras with respect to a graduation, used recently in physics for the classification of elementary particles, and extension G of F by T, i.e., a short exact sequence T≳→G→≳F can be described by a Lie-graded composition on T⊕F, which is formulated in terms of a pair of mappings ∂:F→derT and Δ:F×F→T. The congruence of two extensions of F by T, i. e., the equivalence of the corresponding short exact sequences, is related to an equivalence relation on the set Z2(F,T) of such 2-cocycles (∂,Δ) such that there exists a bijection between the set of congruence classes of extensions of F by T and the set H2(F,T) of classes in Z2(F,T). This generalizes Lie algebraical results which are also known for groups. Examples for two special cases are given: the semidirect sums with Δ trivial and the almost direct sums with ∂ trivial. Both generalize the concepts of tangent and cotangent algebras of Lie algebras and their central extensions with R, the latter being used in the Bargmann theory of ray representations of these semidirect sums.

On Anti-Commutative Algebras and Analytic Loops

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties(1.1) H contains a two-sided identity element e;(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e;