Related papers: Exponential Formulas and Lie Algebra Type Star Pro…

Given a finite-dimensional Lie algebra, and a representation by derivations on the completed symmetric algebra of its dual, a number of interesting twisted constructions appear: certain twisted Weyl algebras, deformed Leibniz rules,…

Let $\mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$. Let $(u_1, \dots, u_n)$ be a sequence of $n$ algebraically independent elements in $\mathbb{K}[x_1, \dots, x_n]$. Given a polynomial $f$ in…

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary…

Let $H_k = 1 + 1/2 + 1/3 + \cdots + 1/k$ denote the $k$th harmonic number. We present an easy-to-implement algorithm for the computation of explicit closed-form evaluations, in terms of the digamma and polygamma functions, for Euler sums of…

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind and the confluent hypergeometric function under which these special functions become exponential convex…

Motivated by the combinatorial properties of products in Lie algebras, we investigate the subset of permutations that naturally appears when we write the long commutator $[x_1, x_2, ..., x_m]$ as a sum of associative monomials. We…

Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…

We obtain some basic partial differential operators connected with nonholomorphic automorphic forms on $\Gamma \backslash U(2, 1)/K$. We give the corresponding Eisenstein series of weight $k$ and automorphic Green functions of weight $k$.…

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most…

After a brief description of the $\mathbb{Z}$-graded differential Poisson algebra, we introduce a covariant star product for exterior differential forms and give an explicit expression for it up to second order in the deformation parameter…

A generating function is given for the number, $E(l,k)$, of irreducible $k$-fold Euler sums, with all possible alternations of sign, and exponents summing to $l$. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}\mu(d)…

We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$. We prove an existence and uniqueness theorem for a linear differential equation of infinite…

Families of vector-like deformed relativistic quantum phase spaces and corresponding realizations are analyzed. Method for general construction of star product is presented. Corresponding twist, expressed in terms of phase space…

We present a closed formula for a family of star-products by replacing the partial derivatives in the Moyal-Weyl formula with commuting vector fields. We show how to reproduce algebra relations on commutative spaces with these star-products…

We use the notation EX(S>M), EXF(S>M) and DL(S>M), where M is a smooth manifold and S is a geometric structure. EX(S>M) is the question whether S exists in M. EXF(S>M) is the question whether M admits S-foliations. DL(S>M) is the search of…

Given a polynomial P of partial derivatives of the Kahler metric, expressed as a linear combination of directed multigraphs, we prove a simple criterion in terms of the coefficients for $P$ to be an invariant polynomial, i.e. invariant…

Generalizing the notion of a Koszul algebra, a graded k-algebra A is K2 if its Yoneda algebra is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that…

We derive a compact determinant formula for calculating and factorizing the hypersum polynomials S^{(L)}_k(N) \equiv \sum_{n_1=1}^N ...\sum_{n_{L+1}=1}^{n_{L}}(n_{L+1})^k expressed in the variable N(N+L+1)

We express the Hamiltonian of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra $D_4$ in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra.…

We present the star-product algebra of the kappa-deformation of Minkowski space and the formulation of Poincare covariant differential calculus. We use these tools to construct a twisted K-cycle over the algebra and a twisted cyclic…