On the Hopf Algebraic Structure of Lie Group Integrators

A commutative but not cocommutative graded Hopf algebra H_N, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H_C, developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H_N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H_N and four other Hopf algebras is discussed. The dual of H_N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H_C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H_N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H_Sh is obtained from H_N by a quotient construction. The Hopf algebra H_P of ordered trees by Foissy differs from H_N in the definition of the product (noncommutative concatenation for H_P and shuffle for H_N) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.     

Lie algebras of infinitesimal norm isometries

Given a norm on a finite dimensional vector space V, we may consider the group of all linear automorphisms which preserve it. The Lie algebra of this group is a Lie subalgebra of the endomorphism algebra of V having two properties: (1) it is the Lie algebra of a compact subgroup, and (2) it is “saturated” in a sence made precise below. We show that any Lie subalgebra satisfying these conditions is the Lie algebra of the group of linear automorphisms preserving some norm. There is an appendix on elementary Lie group theory.     

Group Algebras of Finite Groups as Lie Algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G.     

On Post-Lie Algebras, Lie–Butcher Series and Moving Frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on Euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with Euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan’s method of moving frames. Lie–Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie–Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie–Butcher series are related to invariants of curves described by moving frames.     

High-Order Symplectic Partitioned Lie Group Methods

In this article, a unified approach to obtain symplectic integrators on \(T^{*}G\) from Lie group integrators on a Lie group \(G\) is presented. The approach is worked out in detail for symplectic integrators based on Runge–Kutta–Munthe-Kaas methods and Crouch–Grossman methods. These methods can be interpreted as symplectic partitioned Runge–Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.     

ABELIAN BALANCED HERMITIAN STRUCTURES ON UNIMODULAR LIE ALGEBRAS

Let g be a 2n-dimensional unimodular Lie algebra equipped with a Hermitian structure (J; F) such that the complex structure J is abelian and the fundamental form F is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of SU(n – k), being 2k the dimension of the center of g. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent.     

On the compact real forms of the lie algebras of type E 6 and F 4

We give a construction of the compact real form of the Lie algebra of type E ₆, using the finite irreducible subgroup of shape 3³⁺³: SL₃(3), which is isomorphic to a maximal subgroup of the orthogonal group Ω₇(3). In particular we show that the algebra is uniquely determined by this subgroup. Conversely, we prove from first principles that the algebra satisfies the Jacobi identity, and thus give an elementary proof of existence of a Lie algebra of type E ₆. The compact real form of F ₄ is exhibited as a subalgebra.     

Multiplicative Lie algebras and Schur multiplier

In this paper, we attempt to study the structure of multiplicative Lie algebras, the theory of extensions, the second cohomology groups of multiplicative Lie algebras, and in turn the Schur multipliers. The Schur–Hopf formula is established for multiplicative Lie algebras. We also introduce the group of nontrivial relations satisfied by the Lie product in a multiplicative Lie algebra, and study it as a functor arising from the presentations of multiplicative Lie algebras. Some applications in K-theory are also discussed.     

The Relation Between Automorphism Group and Isometry Group of Randers Lie Groups

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invariant vector field, in a special case, the Lie group admits a left invariant Randers metric such that this intersection is a maximal compact subgroup of the group of automorphisms with respect to which the considered vector field is invariant.     

Group algebras with almost maximal lie nilpotency index

LetK G be a non-commutative Lie nilpotent group algebra of a groupG over a fieldK. It is known that the Lie nilpotency index ofKG is at most |G′|+1, where |G′| is the order of the commutator subgroup ofG. In [4] the groupsG for which this index is maximal were determined. Here we list theG’s for which it assumes the next highest possible value. The present paper is a part of the PhD dissertation of the author.     

Complexity Theory for Lie-Group Solvers

Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge–Kutta–Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by derivation of the computational cost of Fer and Magnus expansions, whose conclusion is that for order four, six, and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Lie-group method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g., Krylov-subspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods.     

Backward Error Analysis for Lie-Group Methods

Backward error analysis has proven to be very useful in stability analysis of numerical methods for ordinary differential equations. However the analysis has so far been undertaken in the Euclidean space or closed subsets thereof. In this paper we study differential equations on manifolds. We prove a backward error analysis result for intrinsic numerical methods. Especially we are interested in Lie-group methods. If the Lie algebra is nilpotent a global stability analysis can be done in the Lie algebra. In the general case we must work on the nonlinear Lie group. In order to show that there is a perturbed differential equation on the Lie group with a solution that is exponentially close to the numerical integrator after several steps, we prove a generalised version of Alekseev-Gr: obner's theorem. A major motivation for this result is that it implies many stability properties of Lie-group methods.     

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.     

Lie Derived Lengths of Group Algebras of Groups with Cyclic Derived Subgroup

In this article the Lie derived length and the strong Lie derived length of group algebras are determined in the case when the derived subgroup of the basic group is cyclic of odd order. As a consequence, we have the characterization of the group algebras of minimal strong Lie derived length.     

Algebras of Quotients of Jordan–Lie Algebras

In this article, we introduce the notion of algebra of quotients of a Jordan–Lie algebra. Properties such as semiprimeness or primeness can be lifted from a Jordan–Lie algebra to its algebras of quotients. Finally, we construct a maximal algebra of quotients for every semiprime Jordan–Lie algebra.     

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820-839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.     

广义Weyl型李代数Wgl_n(F)上的2-上同调群

设F是一特征为零的域,W是F上的广义Weyl代数,gl_n(F)为F上的一般线性李代教,则结合代数Wgl_n(F)上具有一个诱导的李代数结构,本文讨论了李代数Wgl_n(F)的2-上同调群的结构.     

Universal deformation formulas and breaking symmetry

We show that an algebra with a non-nilpotent Lie group of automorphisms or “symmetries” (e.g., smooth functions on a manifold with such a group of diffeomorphisms) may generally be deformed (in the function case, “quantized”) in such a way that only a proper subgroup of the original group acts. This symmetry breaking is a consequence of the existence of certain “universal deformation formulas” which are elements, independent of the original algebra, in the tensor algebra of the enveloping algebra of the Lie algebra of the group.     

Weyl Groups are Finite — and Other Finiteness Properties of Cartan Subalgebras

For each pair (??,??) consisting of a real Lie algebra ?? and a subalgebra a of some Cartan subalgebra ?? of ?? such that [??, ??]∪ [??, ??] we define a Weyl group W(??, ??) and show that it is finite. In particular, W(??, ??,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra ??, the normalizer N(??, G) acts on the finite set Λ of roots of the complexification ??c with respect to hc, giving a representation π : N(??, G)→ S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C(??) of G with respect to h; the image is isomorphic to W(??, ??), and C(??)= {g G : Ad(g)(h)— h ε [h,h] for all h ε h }. All concepts introduced and discussed reduce in special situations to the familiar ones. The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ ?? the set ??? ?(b) remains finite as ? ranges through the full group of inner automorphisms of ??.