Related papers: Representations via differential algebras and equa…

The Witt algebra W_n is the Lie algebra of all derivations of the n-variable polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A representation of W_n is polynomial if it arises as a subquotient of a sum of tensor…

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all…

In this paper we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations…

Let $k$ be an arbitrary field of characteristic $0$. It is shown that for any $n\geq 1$ the universal enveloping algebras of the Poisson symplectic algebra $P_n(k)$ and the Weyl algebra $A_n(k)$ are isomorphic and the canonical isomorphism…

It is a classical result that the multilinear component of the free Lie algebra is isomorphic (as a representation of the symmetric group) to the top (co)homology of the proper part of the poset of partitions $\Pi_n$ tensored with the sign…

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie…

Let G be a connected reductive group acting on a finite dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring C[V] of polynomial functions becomes a Poisson algebra. The ring C[V]^G of…

Let X be an ordered alphabet. Lie_2(n) (and P_2(n) respectively) are the multilinear parts of the free Lie algebra (and the free Poisson algebra respectively) on X with a pair of compatible Lie brackets. In this paper, we prove the…

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu.…

The linear span P_n of the sums of all permutations in the symmetric group S_n with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D_n; and the…

Let $A_n$ be an $n$-dimensional algebra with zero multiplication over a field $K$ of characteristic $0$. Then its universal (multiplicative) enveloping algebra $U_n$ in the variety of left-symmetric algebras is a homogeneous quadratic…

We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of…

It is well known that the Poisson Lie algebra is isomorphic to the Hamiltonian Lie algebra. We show that the Poisson Lie algebra can be embedded properly in the special type Lie algebra. We also generalize the Hamiltonian Lie algebra using…

In this paper, we develop a construction of Poisson $n$-Lie algebras arising from $n$-Lie algebras of Jacobians and establish conditions under which this construction yields a Poisson $n$-Lie algebra. We also formulate a general conjecture…

We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in…

In this paper we continue with the program to explore the topography of the space of W-type algebras. In the present case, the starting point is the work of Khesin, Lyubashenko and Roger on the algebra of q-deformed pseudodifferential…

For the two Cartan type S subalgebras of the Witt algebra $\W_n$, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra $\h_n$. We also give…

We construct the classical W-algebras for some non-abelian Toda systems associated with the Lie groups GL(2n,R) and Sp(n,R). We start with the set of characteristic integrals and find the Poisson brackets for the corresponding Hamiltonian…

Using a Poisson bracket representation, in 3D, of the Lie algebra $\mathfrak{sl}(2)$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras…

The notions of left-right noncommutative Poisson algebra ($\NP^{lr}$-algebra) and left-right algebra with bracket $\AWB^{lr}$ are introduced. These algebras are special cases of $\NLP$-algebras and algebras with bracket $\AWB$,…