arXiv subjects
Publications and source records attributed to Hassan Azad.
Brief proofs of classical results of Lie on finite dimensional subalgebras of vector fields in two and three variables are outlined. The results for algebras of maximal rank for vector fields in $\mathbb{C}^N$ -- $N$ arbitrary -- are also given.
We give a complete constructive description of all finite dimensional nilpotent Lie algebras of smooth vector fields in three variables, including intransitive algebras. The description is organized by the rank and dimension of the center, which serves as the key invariant. Since every nonabelian solvable algebra lives in the normalizer of a nilpotent algebra, our normal forms provide the essential building blocks for the study of all solvable algebras of vector fields in three variables.
A brief proof of Lie's classification of finite dimensional subalgebras of vector fields on the complex plane that have a proper Levi decomposition is given. The proof uses basic representation theory of sl(2, C). This, combined with \cite{ABF2} and \cite{ABF3} completes the classification of finite dimensional subalgebras of vector fields on the complex plane.
In this article, we present a constructive procedure for determining all ideals of the Borel subalgebra of a complex semisimple Lie algebra from its root system or, equivalently, its Dynkin diagram. The proposed algorithmic approach has been implemented in Maple. The Maple code has been tested on Borel subalgebras associated with both classical and exceptional Lie algebras. An interactive procedure is also given to determine conjugacy classes of all subalgebras in the derived algebra of the Borel subalgebra.
A local classification of semisimple algebras of vector fields on $\mathbb{C}^{3}$ is given, using the canonical forms of the Heisenberg algebra and of $sl(2,\mathbb{C})\times sl(2,\mathbb{C})$.
A classification of semisimple algebras of vector fields on C^N that have a Cartan subalgebra of dimension N is given. The proof uses basic representation theory and the local canonical form of semisimple Lie algebras of vector fields.
A constructive procedure is given to determine all ideals of a solvable Lie algebra. This is used in determining algorithmically all conjugacy classes of subalgebras of a given solvable Lie algebra.
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea consists of finding common eigenvectors for exterior powers of the matrices concerned. A convenient formulation of the Pl\"ucker relations is then used to ensure that these eigenvectors actually correspond to subspaces or provide the initial constraints for eigenvectors involving parameters. A procedure for computing the divisors of totally decomposable vector is also provided. Several examples are given for which the calculations are too tedious to do by hand and are performed by coding the conditions found into Maple.
A brief proof of Lie's classification of solvable algebras of vector fields on the plane is given. The proof uses basic representation theory and PDEs.
A characterization of the symmetry algebra of the $n$th order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra $\mathfrak{g}$ determines $-$ up to a point transformation $-$ only one linear equation whose symmetry algebra is $\mathfrak{g}$ and an algorithmic procedure is given to find the linearizing coordinates. The procedure is illustrated by several examples from literature.
A constructive method for decomposing finite dimensional representations of semisimple real Lie algebras is developed. The method is illustrated by an example. We also discuss an implementation of the algorithm in the language of the computer algebra system {\sf GAP}4.
Winkelmann considered compact complex manifolds $X$ equipped with a reduced effective normal crossing divisor $D\, \subset\, X$ such that the logarithmic tangent bundle $TX(-\log D)$ is holomorphically trivial. He characterized them as pairs $(X,\, D)$ admitting a holomorphic action of a complex Lie group $\mathbb G$ satisfying certain conditions \cite{Wi1}, \cite{Wi2}; this $\mathbb G$ is the connected component, containing the identity element, of the group of holomorphic automorphisms of $X$ that preserve $D$. We characterize the homogeneous holomorphic principal $H$--bundles over $X$, where $H$ is a connected complex Lie group. Our characterization says that the following three are equivalent: (1)~ $E_H$ is homogeneous. (2)~ $E_H$ admits a logarithmic connection singular over $D$. (3)~ The family of principal $H$--bundles $\{g^*E_H\}_{g\in \mathbb G}$ is infinitesimally rigid at the identity element of the group $\mathbb G$.
Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry.
If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical form of L in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates -- for a suitable choice of coordinates. This result is used to classify semisimple algebras of vector fields on R^3 and to determine all representations of sl(N, R) as vector fields on R^N. These representations are used to find linearizing coordinates for any second order ordinary differential equation that admits sl(3, R) as its symmetry algebra and for a system of two second order ordinary differential equations that admits sl(4, R) as its symmetry algebra.
Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Γ\,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kähler manifold and $ρ\, :\, π_1(X, x_0)\,\longrightarrow\, Γ$ a homomorphism such that the adjoint action of $ρ(π_1(X, x_0))$ on $\text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $ρ$ a harmonic map $X\, \longrightarrow\, K\backslash G/Γ$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.
We classify a class of Bianchi type II spacetimes according to their Noether symmetries. We briefly discuss the conservation laws admitted by these Noether symmetries and classify their possible algebras and determine their structures also.
The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular the following results are proved: a) If L is an algebraic subgroup of a connected real algebraic group G such that the complexification of L contains a maximal torus of the complexification of G, then L contains a Cartan subgroup of G b) Let G be a solvable real algebraic group whose eigenvalues are all real. If the complexification of G operates algebraically on a complex variety V, and some G orbit is compact, then this orbit is a point.
Let $G$ be a semisimple affine algebraic group defined over a field $k$ of characteristic zero. We describe all the maximal connected solvable subgroups of $G$, defined over $k$, up to conjugation by rational points of $G$.