Related papers: The Free Hamilton Algebra
Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of…
This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_{\alpha}\}_{\alpha\in\Lambda}$, with the left regular free semigroup algebra $\mathfrak{L}_{\Lambda}$ and the…
A free differential for an arbitrary associative algebra is defined as a differential with a uniqueness property. The existence problem for such a differential is posed. The notion of optimal calculi for given commutation rules is…
In this article we consider partial abelianization of associative algebra with respect to a subalgebra. This notion is a generalization of usual abelianization of associative algebra and has an application in Quantum Mechanics and Quantum…
We discuss a spectrum generating algebra in the supersymmetric quantum mechanical system which is defined as a series of solutions to a specific differential equation. All Hamiltonians have equally spaced eigenvalues, and we realize both…
Non-alternating Hamiltonian Lie algebras in three variables over a perfect field of characteristic 2 are considered. A classification of non-alternating Hamiltonian forms over an algebra of divided powers in three variables and of the…
A quaternionic field is a pair $p=\{\alpha,u\}$ of function $\alpha$ and vector field $u$ given on a 3d Riemannian maifold $\Omega$ with the boundary. The field is said to be harmonic if $\nabla \alpha={\rm rot\,}u$\, in $\Omega$. The…
Here we extend the algebro-geometric approach to free probability, started in~\cite{FMcK4,F14}, to general (non)-commutative probability theories. We show that any universal convolution product of moments of independent (non)-commutative…
In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic…
In this paper we prove that there exists only one family of classical Hamiltonian systems of two degrees of freedom with invariant plane $\Gamma=\{q_2=p_2=0\}$ whose normal variational equation around integral curves in $\Gamma$ is…
In this paper, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. Initially, we describe the Hopf algebra associated with the…
Let $K$ be a field and let $\sigma$ be an automorphism and let $\delta$ be a $\sigma$-derivation of $K$. Then we show that the multiplicative group of nonzero elements of the division ring $D=K(x;\sigma,\delta)$ contains a free non-cyclic…
I solve a quantum chain whose Hamiltonian is comprised solely of local four-fermi operators by constructing free-fermion raising and lowering operators. The free-fermion operators are both non-local and highly non-linear in the local…
We introduce the notion of $\lambda$-double Lie algebra, which coincides with usual double Lie algebra when $\lambda = 0$. We state that every $\lambda$-double Lie algebra for $\lambda\neq0$ provides the structure of modified double Poisson…
We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by…
We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees,…
In this paper we consider Hamiltonian systems on the quantum plane and we show that the set of Q-meromorphic Hamiltonians is a Virasoro algebra with central charge zero and the set of Hamiltonian derivations of the algebra of $Q$-analytic…
In this paper we develop a theory of free holomorphic functions on noncommutative Reinhardt domains generated by positive regular free holomorphic functions in n noncommuting variables. We show that the free biholomorphic classification of…
A finite hypergraph $H$ consists of a finite set of vertices $V(H)$ and a collection of subsets $E(H) \subseteq 2^{V(H)}$ which we consider as partition of unity relations between projection operators. These partition of unity relations…
One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial…