Stefan Berceanu
A realization of coherent state Lie algebras by first-order differential operators with holomorphic polynomial coefficients on Kähler coherent state orbits is presented. Explicit formulas involving the Bernoulli numbers and the structure constants for the semisimple Lie groups are proved.
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We study realizations of Lie algebras by vector fields. A correspondence between classification of transitive local realizations and classification of subalgebras is generalized to the case of regular local realizations. A reasonable classification problem for general realizations is rigorously formulated and an algorithm for construction of such classification is presented.
For every parabolic subgroup $P$ of a Lie supergroup $G$ the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the quesiton whether $\mathfrak{g}=\operatorname{Lie}(G)$ is the maximal symmetry of this supergeometry in the case of exceptional Lie superalgebras $G(3)$ and $F(4)$. Our approach is to consider the negatively graded Lie superalgebras for every choice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, with reduction of the structure group when required (2 resp 3 cases), thus realizing $G(3)$ and $F(4)$ as symmetries of supergeometries. This…
On realizations of the complex Lie groups $ (F_{4,R})^C, (E_{6,R})^C, (E_{7,R})^C ,(E_{8,R})^C$ and those compact real forms $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$
该研究探讨用实数域替换Cayley代数来定义一类复例外李群及其紧实形式，旨在确定这些群的结构，即完成其群实现。
Geometric realizations of the Lie superalgebra D(2,1;a)
本文通过抛物子代数对应的负分次李子代数的 Tanaka–Weisfeiler 延拓，构造了例外李超代数 D(2,1;a) 的 6 个互不等价超几何实现，并显式给出各情形下的超对称。
Convergent realizations of Lie subalgebras
研究有限余维李代数子代数在局部解析向量场意义下收敛实现的刻画，给出性质推广及在控制理论中的应用，恢复并阐明控制仿射系统相关的一些结果。
In order to define the exceptional compact Lie groups $G_2,F_4,E_6,E_7,E_8$, we usually use the Cayley algebra $\mathfrak{C}$ or its complexification $\mathfrak{C}^C$. In the present article, we consider replacing the Cayley algebra $\mathfrak{C}$ with the field of quaternion numbers $\boldmath{H}$ in the definition of the groups above, and these groups are denoted as in title above. Our aim is to determine the structure of these groups. We call realization to determine the structure of the groups.
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez. As homogeneous manifolds, these spaces are of the $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tildeγ$ of oder 4 and $H$ is a fixed points subgroup $G^γ$ of $G$. In the present article, for the exceptional compact Lie group $G=E_8$, we give the explicit form of automorphism of order 4 induced by the $\boldsymbol{R}$-linear transformation $σ'_{4}$ and determine the structure of the group $(E_8)^{σ'_{4}}$. Thereby, we realize the globally exceptional Riemannian $4$-sym…
In this paper, we show that the Lie superalgebra $\mathfrak{spo}(2l+2|n)$ is into the intersection of Lie superalgebra of contact vector fields $\mathcal{K}(2l+1|n)$ and the Lie superalgebra of projective vector fields $\mathfrak{pgl}(2l+2|n)$. We use mainly the embedding used by P. Mathonet and F. Radoux in "\textit{ Projectively equivariant quantizations over superspace $\mathbb{R}^{p|q}$. Lett. Math. Phys, 98: 311-331, 2011}". Explicitly, we use the embedding of a Lie superalgebra constituted of matrices belonging to $\gl(2l+2|n)$ into $\mathrm{Vect}(\R^{2l+1|n})$. We generalize thus in sup…
The Berezin quantization on a simply connected homogeneous Kähler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the Kähler structure. The…
Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case.These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces …
A classification is given of the exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric spaces $G/K$ by A.Kollross, where $G$ is an exceptional compact Lie group or $S\!pin(8)$, and moreover the structure of $K$ is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms (involutions) $\tildeσ, \tildeτ$ of $G$ concretely and determine the structure of group $G^σ \cap G^τ$ corresponding to Lie algebra $\mathfrak{g}^σ\cap \mathfrak{g}^τ$, where $G$ is an exceptional compact Lie group. Thereby, we realize exceptional $\mathbb{Z}_2 \times \mathbb{Z}…
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tildeγ$ of order four on $G$ and $H$ is a fixed points subgroup $G^γ$ of $G$. According to the classification by J.A. Jim{é}nez, there exist seven compact simply connected Riemannian 4-symmetric spaces $ G/H $ in the case where $ G $ is of type $ E_8 $. In the present article, %as Part II continuing from Part I, for the connected c…
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tildeγ$ of order four on $G$ and $H$ is a fixed points subgroup $G^γ$ of $G$. According to the classification by J.A. Jim{é}nez, there exist seven compact simply connected Riemannian 4-symmetric spaces $ G/H $ in the case where $ G $ is of type $ E_8 $. In the present article, %as Part II continuing from Part I, for the connected c…
We introduce a systematic method to solve a type of Cartan's realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We apply our method to the problem of classification of extremal Kähler metrics on surfaces.
Realization of Jordan-Kronecker invariants by Lie algebras
本文刻画 Lie 代数可实现的 Jordan-Kronecker 不变量，完全解决 Jordan 与 Kronecker 情形，并给出一般情形的充分与不可能性判据及非常值特征值相容 Poisson 括号的实现分类。
Relative algebroids and Cartan realization problems
本文提出相对代数胚统一李代数胚与偏微分方程理论，通过延拓与形式可积性构造profinite李代数胚，以解决几何结构的存在与分类问题。
On realizations of the complex Lie groups $ (E_{6,\mathbb{R}})^C, (E_{6,\mathbb{C}})^C, (E_{6,\mathbb{H}})^C $ and those real forms
本文通过将Cayley代数替换为实数、复数、分裂复数、四元数及分裂四元数域，定义了六种类型为 $E_6$ 的Lie群的推广，并研究其结构实现。
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale Lie 2-groupoids with 2-connected source fibres. Finally, we discuss how to lift Lie algebroid morphisms to Lie 2-groupoid morphisms (Lie II Theorem).
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