Related papers: On the stochastic Lie algebra

We continue our study, initiated in our prior work with Richard Stanley, of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra known as the free Filippov $n$-algebra…

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g} =…

The symplectic group branching algebra, B, is a graded algebra whose components encode the multiplicities of irreducible representations of Sp(2n-2,C) in each irreducible representation of Sp(2n,C). By describing on B an ASL structure, we…

We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of…

In this paper, we explore natural connections among the representations of the extended affine Lie algebra $\widehat{sl_N}(\mathbb{C}_q)$ with $\mathbb{C}_q=\mathbb{C}_q[t_0^{\pm1},t_1^{\pm1}]$ an irrational quantum 2-torus, the simple…

For the semi simple and deployed Lie algebra $\mathfrak g=\mathfrak{sl}(n, \R)$, we give an explicit construction of an overalgebra $\mathfrak g^+=\mathfrak g\rtimes V$ of $\mathfrak g$, where $V$ is a finite dimensional vector space. In…

The main aim of this paper is to classify the distinct multiplicative Lie algebra structures (up to isomorphism) on a given group. We also see that for a given group $G$, every homomorphism from the non-abelian exterior square $G \wedge G$…

We study the abelianization of Kontsevich's Lie algebra associated with the Lie operad and some related problems. Calculating the abelianization is a long-standing unsolved problem, which is important in at least two different contexts:…

We construct generators of the center of the universal enveloping algebra of the complex orthogonal Lie algebra realized as the alternative matrices of size $n$. These elements are constructed in accordance with the Iwasawa decomposition of…

It is well known that a symplectic Lie algebra admit a left symmetric product. In this work, we study the case where this product is Novikov, we show that the left-symmetric product associated to the symplectic Lie algrbra is Novikov if and…

We consider the pair of degenerate compatible antibrackets satisfying a generalization of the axioms imposed in the triplectic quantization of gauge theories. We show that this actually encodes a Lie group structure, with the antibrackets…

We make several observations relating the Lie algebra $\mathfrak{g}_2 \subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other…

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra $\mathfrak{n}_G$ from a simple directed graph $G$ in 2005. There is a natural inner product on $\mathfrak{n}_G$ arising from the construction. We study…

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…

Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{\theta \times \theta}$, where $\mathbf{k}$ is an algebraically closed field of characteristic…

In this paper we investigate how a typical, large-dimensional representation looks for a complex Lie algebra. In particular, we study the family $\mathfrak{sl}_{r+1}(\mathbb{C})$ of Lie algebras for $r \geq 2$ and derive asymptotic…

Let $F$ be a field of characteristic not $2$ . An associative $F$-algebra $R$ gives rise to the commutator Lie algebra $R^{(-)}=(R,[a,b]=ab-ba).$ If the algebra $R$ is equipped with an involution $*:R\rightarrow R$ then the space of the…

We classify simple linearly compact n-Lie superalgebras with n>2 over a field F of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive Z-graded Lie superalgebras…

We set up foundations of representation theory over $S$, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat $S$-Lie algebras and their representations, characters, $gl_n(S)$-Verma modules and…

In this paper, we construct a representation of loop group and derive the formula of the corresponding representation of the affine Kac-Moody algebra with level 1. And we also provide a concrete realization of Whittaker functionals in the…