Abstract:
The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a bracket-generating distribution of constant type does not have non-zero complex weak characteristics, then its symmetry algebra is necessarily finite-dimensional. The paper also contains a number of illustrative algebraic and geometric examples including the proof that any metabelian Lie algebra with a 2-dimensional center always has an infinite-dimensional Tanaka prolongation.

Abstract:
A new class of infinite dimensional simple Lie algebras over a field with characteristic 0 are constructed. These are examples of non-graded Lie algebras. The isomorphism classes of these Lie algebras are determined. The structure space of these algebras is given explicitly.

Abstract:
In this paper, we construct a new class of infinite rank $\Z$-graded Lie conformal algebra, denoted by $CW(a,c)$. And $CW(a,c)$ contains the loop Virasoro Lie conformal algebra and a Block type Lie conformal algebra. $CW(a,c)$ has a $\C[\partial]$-basis $\{L_{\a}\,|\,{\a}\in\Z\}$ and $\lambda$-brackets $[L_{\a}\, {}_\lambda \, L_{\b}]=((a\a+c)\partial+(a(\a+\b)+2c)\lambda) L_{\a+\b}$, where $\a,\b\in\Z$, $a,c\in\C$. Then the associated Lie algebra $W(a,c)$ is studied, where $W(a,c)$ has a basis $\{L_{\a,i}\,|\,\a,\,\b,i,j\in\Z\}$ over $\C$ and Lie brackets $[L_{\a,i},L_{\b,j}]=(a(\b(i+1)-\a(j+1))+c(i-j))L_{\a+\b,i+j}$, where $\a,\b,i,j\in\Z$, $a,c\in\C$. Clearly, we find that $W(a,c)$ is also a new class of infinite dimensional $\Z$-graded Lie algebras. In particular, the conformal derivations of $CW(a,c)$ are determined. Finally, rank one conformal modules over $CW(a,c)$ are classified

Abstract:
We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.

Abstract:
We describe the isomorphism classes of infinite-dimensional graded Lie algebras of maximal class, generated by elements of weight one, over fields of odd characteristic.

Abstract:
This paper is devoted to deformation theory of graded Lie algebras over $\Z$ or $\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic deformations using this technique.

Abstract:
In this thesis we describe the universal central extension of two important classes of so-called root-graded Lie algebras defined over a commutative associative unital ring $k.$ Root-graded Lie algebras are Lie algebras which are graded by the root lattice of a locally finite root system and contain enough $\mathfrak{sl}_2$-triples to separate the homogeneous spaces of the grading. Examples include the infinite rank analogs of the simple finite-dimensional complex Lie algebras. \\ In the thesis we show that in general the universal central extension of a root-graded Lie algebra $L$ is not root-graded anymore, but that we can measure quite easily how far it is away from being so, using the notion of degenerate sums, introduced by van der Kallen. We then concentrate on root-graded Lie algebras which are graded by the root systems of type $A$ with rank at least 2 and of type $C$. For them one can use the theory of Jordan algebras.

Abstract:
We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie algebras). In particular we describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list. It is proved that a symplectic filiform Lie algebra is a filtered deformation of some N-graded symplectic filiform Lie algebra. But this condition is not sufficient. A spectral sequence is constructed in order to answer the question whether a given deformation of a N-graded symplectic filiform Lie algebra admits a symplectic structure or not. Other applications and examples are discussed.