Related papers: Invariant tensors for simple groups

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…

For any L-infinity algebra L, we construct an A-infinity structure on the space of symmetric tensors Sym*(L), which generalizes the classical universal enveloping for Lie algebras. Our construction is based on an invariant homotopy on a…

We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schr\"odinger algebra, these equations…

First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…

If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go…

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…

Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, >..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/<p_1,…

Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of…

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We…

We introduce the concept of a non-associative (i.e. non-necessarily associtive) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a non-associative algebra, whereas the…

An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of…

We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator…

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed…

This is a survey on the geometric classification of different varieties of algebras (nilpotent, nil-, associative, commutative associative, cyclic associative, Jordan, Kokoris, standard, noncommutative Jordan, commutative power-associative,…

Examples of operator algebras with involution include the operator $*$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix…

Leibniz algebras are non skew-symmetric generalization of Lie algebras. In this paper we introduce the notion of anti-Leibniz algebras as a "non commutative version" of mock-Lie algebras. Low dimensional classification of such algebras is…

This paper is to study vertex operator superalgebras which are strongly generated by their weight-$2$ and weight-$\frac{3}{2}$ homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra $V$ is…

In this paper, we establish a converse to Schur's theorem for Lie superalgebras \( L \), focusing on cases where the minimal generator number pairs \((p \vert q)\) of \( L/Z(L) \) are considered, and where the superdimension \(…

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie…

We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these…