Related papers: Categorified Symplectic Geometry and the Classical…
We propose a symplectic structure for the phase space of a generic Lagrangian field theory expressed in the framework of $L_\infty$ algebras. The symplectic structure does not require explicit knowledge of the derivative content of the…
Symplectic (respectively orthogonal) triple systems provide constructions of Lie algebras (resp. superalgebras). However, in characteristic 3, it is shown that this role can be interchanged and that Lie superalgebras (resp. algebras) can be…
We classify finite-dimensional nilpotent Lie algebras with $2$-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to $SO_2(\mathbb R)$. This enables one to enlarge the class of nilpotent Lie algebras of…
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe…
In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of…
Based on the Euler-Lagrange cohomology groups $H_{EL}^{(2k-1)}({\cal M}^{2n}) (1 \leqslant k\leqslant n)$ on symplectic manifold $({\cal M}^{2n}, \omega)$, their properties and a kind of classification of vector fields on the manifold, we…
Recent work applying higher gauge theory to the superstring has indicated the presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie 2-superalgebra' extending the Poincare superalgebra in precisely the dimensions where…
Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify…
Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an algebraically closed field $F$ of characteristic zero. Under the Poisson bracket, $A$ is equipped with a natural Lie algebra structure. It is proven that the maximal…
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold…
We establish a link between the multisymplectic and the covariant phase space approach to geometric field theory by showing how to derive the symplectic form on the latter, as introduced by Crnkovic-Witten and Zuckerman, from the…
We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing…
A symplectic structure on the space of nondegenerate and nonparametrized curves in a locally affine manifold is defined. We also consider several interesting spaces of nondegenerate projective curves endowed with Poisson structures. This…
Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which…
The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we…
We describe the `Lie algebra of classical mechanics', modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. It is a polynomially graded Lie…
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used.…
In this paper, we introduce a new kind of Siegel upper half space and consider the symplectic geometry on it explicitly under the action of the group of all holomorphic transformations of it. The results and methods will form a basis for…
The Z_2^n gradings of the classical Lie algebras are described. To elucidate the grading, the classical Lie algebras are expressed in terms of matrix algebras over one of eight fields or Clifford algebras which carry gradings ranging from…
We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively.…