Abstract:
Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article is written based on the author's seminar talks on nongraded infinite-dimensional simple Lie algebras. The key constructional ingredients of our Lie algebras are locally-finite derivations. The structure spaces of some families of these simple Lie algebras can be viewed as analogues of vector bundles with Lie algebras as fibers. We also believe that some of our simple Lie algebras could be related to noncommutative geometry.

Abstract:
In this article we review the main results of the earlier papers [I. Penkov, K. Styrkas, Tensor representations of infinite-dimensional root-reductive Lie algebras, in Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, Birkh\"auser, 2011, pp. 127-150], [I. Penkov, V. Serganova, Categories of integrable $\mathfrak{sl}(\infty)$-, $\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, in "Representation Theory and Mathematical Physics", Contemporary Mathematics 557 (2011), pp. 335-357] and [E. Dan-Cohen, I. Penkov, V. Serganova, A Koszul category of representations of finitary Lie algebras, preprint 2011, arXiv:1105.3407], and establish related new results in considerably greater generality. We introduce a class of infinite-dimensional Lie algebras $\mathfrak{g}^{M}$, which we call Mackey Lie algebras, and define monoidal categories $\mathbb{T}_{\mathfrak{g}^M}$ of tensor $\mathfrak{g}^M-$modules. We also consider dense subalgebras $\mathfrak{a} \subset \mathfrak{g}^M$ and corresponding categories $\mathbb{T}_\mathfrak{a}$. The locally finite Lie algebras $\mathfrak{sl}(V,W), \mathfrak{o}(V), \mathfrak{sp}(V)$ are dense subalgebras of respective Mackey Lie algebras. Our main result is that if $\mathfrak{g}^M$ is a Mackey Lie algebra and $\mathfrak{a} \subset \mathfrak{g}^M$ is a dense subalgebra, then the monoidal category $\mathbb{T}_\mathfrak{a}$ is equivalent to $\mathbb{T}_{\mathfrak{sl}(\infty)}$ or $\mathbb{T}_{\mathfrak{o}(\infty)}$; the latter monoidal categories have been studied in detail in [E. Dan-Cohen, I. Penkov, V. Serganova, A Koszul category of representations of finitary Lie algebras, preprint 2011, arXiv:1105.3407]. A possible choice of $\mathfrak{a}$ is the well-known Lie algebra of generalized Jacobi matrices.

Abstract:
The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.

Abstract:
In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

Abstract:
In this paper we study the complete reducibility of representations of infinite-dimensional Lie algebras from the perspective of the representation theory of vertex algebras.

Abstract:
We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank affine Lie algebra, the $\winf$ algebra or its certain Lie subalgebras. We give a formulation in the framework of vertex algebras. We also formulate several conjectures and open problems.

Abstract:
We consider a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. We identify two subclasses of Nottingham Lie algebras as loop algebras of finite-dimensional simple Lie algebras of Hamiltonian Cartan type. A property of Laguerre polynomials of derivations, which is related to toral switching, plays a crucial role in our constructions.

Abstract:
In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation $\D$ of any Lie algebra $\g$. Here it is shown how infinite dimensional Lie algebras appear naturally within the framework of fractional supersymmetry. Using a differential realization of $\g$ this infinite dimensional Lie algebra, containing the Lie algebra $\g$ as a sub-algebra, is explicitly constructed.

Abstract:
We study in detail the semi-infinite or BRST cohomology of general affine Lie algebras. This cohomology is relevant in the BRST approach to gauged WZNW models. We prove the existence of an infinite sequence of elements in the cohomology for non-zero ghost numbers. This will imply that the BRST approach to topological WZNW model admits many more states than a conventional coset construction. This conclusion also applies to some non-topological models. Our work will also contain results on the structure of Verma modules over affine Lie algebras. In particular, we generalize the results of Verma and Bernstein-Gel'fand-Gel'fand,for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules.