A category of Brauer diagrams, analogous to Turaev’s tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O$\left(V\right)$ or the symplectic group Sp$\left(V\right)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain presentations...

Let $\left(𝕃\right(V),d)$ be a free graded connected differential Lie algebra over the field $\mathbb{Q}$ of rational numbers. An ideal $I$ in the Lie algebra $H\left(𝕃\right(V),d)$ is called nice if, for every cycle $\alpha \in 𝕃\left(V\right)$ such that $\left[\alpha \right]$ belongs to $I$, the kernel of the map $H\left(𝕃\right(V),d)\to H\left(𝕃\right(V\oplus \mathbb{Q}x),d)$, $d\left(x\right)=\alpha$, is contained in $I$. We show that the center of $H\left(𝕃\right(V),d)$ is a nice ideal and we give in that case some informations on the structure of the Lie algebra $H\left(𝕃\right(V\oplus \mathbb{Q}x),d)$. We apply this computation for the determination of the rational homotopy Lie algebra $L_X={\pi }_{*}\left(\Omega X\right)\otimes \mathbb{Q}$ of a simply connected space $X$. We deduce that the...

Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building ${𝔅}^1\left(H\right)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building ${𝔅}^1\left(G\right)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{\prime }$ be maps from ${𝔅}^1\left(H\right)$ to ${𝔅}^1\left(G\right)$ which preserve the Moy–Prasad filtrations. We prove that...

The natural filtration of the infinite-dimensional simple modular Lie superalgebra M over a field of characteristic p > 2 is proved to be invariant under automorphisms by discussing ad-nilpotent elements. Moreover, an intrinsic property is obtained and all the infinite-dimensional simple modular Lie superalgebras M are classified up to isomorphisms. As an application, a property of automorphisms of M is given.

A Lie algebra $𝔤$ is called two step nilpotent if $𝔤$ is not abelian and $[𝔤,𝔤]$ lies in the center of $𝔤$. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension $8$ over the...

Let $${𝒪}_1$$ and $${𝒪}_2$$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $${𝒪}_1$$ ≥ $${𝒪}_2$$ if $${𝒪}_2$$ is contained in the closure of $${𝒪}_1$$ . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E 8. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible...