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A Lie algebra L is said to be (Θn,sln)-graded if it contains a simple subalgebra g isomorphic to sln such that L is generated by g as an ideal and the g-module L decomposes into copies of the adjoint module, the trivial module, the natural module V, its symmetric and exterior squares S2V and , and their duals. In Yaseen (Generalized Root Graded Lie Algebras 2018), Müller et al. (Int Math Res Not IMRN 2010:783–823, 2010), we classified (Θn,sln)-graded Lie algebras for n>4. In this paper we describe the multiplicative structures and the coordinate algebras of (Θn,sln)-graded Lie algebras for n=3,4.

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In the present paper we start to build a bridge from the algebraic theory of root graded Lie algebras to the global Lie theory of infinite-dimensional Lie groups by showing how root graded Lie algebras can be defined and analyzed in the context of locally convex Lie algebras. Our main results concern the description of locally convex root graded Lie algebras in terms of a locally convex coordinate algebra and its universal covering algebra, which has to be defined appropriately in the topological context. Although the structure of the isogeny classes is much more complicated in the topological context, we give an explicit description of the universal covering Lie algebra which implies in particular that in most cases (called regular) it depends only on the root system and the coordinate algebra. Not every root graded locally convex Lie algebra is integrable in the sense that it is the Lie algebra of a Lie group. In a forthcoming paper we will discuss criteria for the integrability of root graded Lie algebras.

This paper is about toroidal Lie algebras, certain intersection matrix Lie algebras defined by Slodowy, and their relationship to one another and to certain Lie algebra analogues of Steinberg groups. The main result of the paper is the identification of the intersection matrix algebras arising from multiply-affinized Cartan matrices of types A, D and E with certain Steinberg Lie algebras and toroidal Lie algebras (Propositions 5.9 and 5.10). A major part of the paper studies and classifies Lie algebras graded by finite root systems. These become the princi- pal tool in our analysis of intersection matrix algebras. Each Lie algebra graded by a simply-laced finite root system of rank > 2 has attached to it an algebra which, according to the type and rank, is either commutative and associative, only associative, or alternative. All these possibilities occur in our description of inter- section matrix algebras. Let R be any associative algebra with identity, not necessarily finite dimen- sional, over a field k of characteristic 0. For each positive integer n the associative algebra M,(R) of n “ n matrices with coefficients in R forms a Lie algebra over k under the commutator product. We denote this Lie algebra by ol,(R). Let Eis be the (i, j) matrix unit of M.(R) and assume that n > 2. The subalgebra e.(R) of OI.(R) generated by the elements rEis, r ~ R, i 4= j, is an ideal of 91,(R) and is perfect, i.e. it is its own derived algebra. Now any perfect Lie algebra O has a universal central extension, also perfect, called a universal covering algebra (u.c.a.) of O [Ga], so in particular, e.(R) has a u.c.a, that we will denote by ~t,(R). We define 112,.(R) by the exact sequence (0.1) 0 ~ f2,,(R) ~ ~t.(R) ~ e,(R) ~ 0 * Dedicated to our teacher Maria J. Wonenburger ** Both authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada

In this work, we consider Lie algebras L containing a subalgebra isomorphic
to sl3 and such that L decomposes as a module for that sl3 subalgebra into
copies of the adjoint module, the natural 3-dimensional module and its dual,
and the trivial one-dimensional module. We determine the multiplication in L
and establish connections with structurable algebras by exploiting symmetry
relative to the symmetric group S4.

We prove that any simple Lie subalgebra of a locally finite associative algebra is either finite-dimensional or isomorphic to the commutator algebra of the Lie algebra of skew symmetric elements of some involution simple locally finite associative algebra. The ground field is assumed to be algebraically closed of characteristic 0. This result can be viewed as a classification theorem for simple Lie algebras that can be embedded in locally finite associative algebras. We also establish a link between this class of Lie algebras and that of Lie algebras graded by finite root systems.

In this article, we introduce (weakly) root graded Banach–Lie algebras and corresponding Lie groups as natural generalizations
of group like
for a Banach algebra A or groups like C(X,K) of continuous maps of a compact space X into a complex semisimple Lie group K. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups P to representations of G on holomorphic sections of homogeneous vector bundles over G/ P. One of our main results is an algebraic characterization of the space of sections which is used to show that this space
actually carries a natural Banach structure, a result generalizing the finite dimensionality of spaces of sections of holomorphic
bundles over compact complex manifolds. We also give a geometric realization of any irreducible holomorphic representation
of a (weakly) root graded Banach–Lie group G and show that all holomorphic functions on the spaces G/ P are constant.

A Lie algebra L is said to be (Θn,sln)-graded if it contains a simple subalgebra g isomorphic to sln such that the g-module L decomposes into copies of the adjoint module, the trivial module, the natural module V, its symmetric and exterior squares S2V and ∧2V and their duals. We describe the multiplicative structures and the coordinate algebras of (Θn,sln)-graded Lie algebras for n≥5, classify these Lie algebras and determine their central extensions.

Let g be a non-zero finite-dimensional split semisimple Lie algebra with root system
∆. Let Γ be a finite set of integral weights of g containing ∆ and {0}. We say that a Lie
algebra L over F is generalized root graded, or more exactly (Γ,g)-graded, if L contains
a semisimple subalgebra isomorphic to g, the g-module L is the direct sum of its weight
subspaces Lα (α ∈ Γ) and L is generated by all Lα with α ̸= 0 as a Lie algebra. If g is the
split simple Lie algebra and Γ = ∆∪ {0} then L is said to be root-graded. Let g ∼= sln and
Θn = {0,±εi ±ε j,±εi ,±2εi| 1 ≤ i ̸= j ≤ n} where {ε1,..., εn} is the set of weights of the natural sln-module. Then a Lie algebra L is (Θn,g)-graded if and only if L is generated by g as an ideal and the g-module L decomposes into copies of the adjoint module, the natural module V, its symmetric and exterior squares S 2V and ∧ 2V, their duals and the one dimensional trivial g-module.
In this thesis we study properties of generalized root graded Lie algebras and focus
our attention on (Θn,sln)-graded Lie algebras. We describe the multiplicative structures
and the coordinate algebras of (Θn,sln)-graded Lie algebras, classify these Lie algebras
and determine their central extensions.e.

We classify the Lie algebras of characteristic zero graded by the finite nonreduced root systems BCr for r ≥ 2 and determine their derivations, central extensions, and invariant forms.

Lie Groups and Algebraic Groups.- Structure of Classical Groups.- Highest-Weight Theory.- Algebras and Representations.- Classical Invariant Theory.- Spinors.- Character Formulas.- Branching Laws.- Tensor Representations of GL(V).- Tensor Representations of O(V) and Sp(V).- Algebraic Groups and Homogeneous Spaces.- Representations on Spaces of Regular Functions.

We classify the Lie algebras that are graded by the nonre-duced root system BC 1 and determine their central extensions, derivations, and invariant forms.

Some locally finite simple Lie algebras are graded by finite (pos-sibly nonreduced) root systems. Many more algebras are sufficiently close to being root graded that they still can be handled by the techniques from that area. In this paper we single out such Lie algebras, describe them, and suggest some applications of such descriptions.

This paper classifies the Lie algebras graded by doubly-laced finite root systems and applies this classification to identify
the intersection matrix algebras arising from multiply affinized Cartan matrices of types B,C,F, and G. This completes the determination of the Lie algebras graded by finite root systems initiated by Berman and Moody who studied
the simply-laced finite root systems of rank ≧2.