Algebraic Identities Research Articles

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

Weighted lattice walks and universality classes

In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.

Generalized Rogers–Ramanujan identities for twisted affine algebras

The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers–Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of the twisted affine Lie algebras. A conjecture for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers–Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, using Lie algebraic theoretic methods. We use Freudenthal’s formula, which we computerized, to verify the identities for all the algebras at low rank and low level. We find complete agreement with the conjecture.

ABC of ladder operators for rationally extended quantum harmonic oscillator systems

The problem of construction of ladder operators for rationally extended quantum harmonic oscillator (REQHO) systems of a general form is investigated in the light of existence of different schemes of the Darboux–Crum–Krein–Adler transformations by which such systems can be generated from the quantum harmonic oscillator. Any REQHO system is characterized by the number of separated states in its spectrum, the number of ‘valence bands’ in which the separated states are organized, and by the total number of the missing energy levels and their position. All these peculiarities of a REQHO system are shown to be detected and reflected by a trinity , , of the basic (primary) lowering and raising ladder operators related between themselves by certain algebraic identities with coefficients polynomially-dependent on the Hamiltonian. We show that all the secondary, higher-order ladder operators are obtainable by a composition of the basic ladder operators of the trinity which form the set of the spectrum-generating operators. Each trinity, in turn, can be constructed from the intertwining operators of the two complementary minimal schemes of the Darboux–Crum–Krein–Adler transformations.

The Binary Nonlinearization of the Super Integrable System and Its Self-Consistent Sources

Abstract The super integrable system and its super Hamiltonian structure are established based on a loop super Lie algebra and super-trace identity in this paper. Then the super integrable system with self-consistent sources and conservation laws of the super integrable system are constructed. Furthermore, an explicit Bargmann symmetry constraint and the binary nonlinearization of Lax pairs for the super integrable system are established. Under the symmetry constraint,the n $n$ -th flow of the super integrable system is decomposed into two super finite-dimensional integrable Hamilton systems over the supersymmetric manifold. The integrals of motion required for Liouville integrability are explicitly given.

On the Steiner–Routh Theorem for Simplices

It is shown in [28] that, using only tools of elementary geometry, the classical Steiner—Routh theorem for triangles can be fully extended to tetrahedra. In this article, we first give another proof of the Steiner—Routh theorem for tetrahedra, where methods of elementary geometry are combined with the inclusion—exclusion principle. Then we generalize this approach to (n — 1)-dimensional simplices. A comparison with the formula obtained using vector analysis yields an interesting algebraic identity.

Generalization of some hypergeometric functions

In this paper, we make use of the Lagrange interpolation to obtain some algebraic identities, involving one or two infinite sets of variables.

Graded identities of simple real graded division algebras

Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.

Derivations with values in ideals of semifinite von Neumann algebras

Let M be a semifinite von Neumann algebra with a faithful semifinite normal trace τ and let A be an arbitrary C⁎-subalgebra of M. Assume that E is a fully symmetric function space on (0,∞) having Fatou property and order continuous norm and E(M,τ) is the corresponding symmetric operator space. We prove that every derivation δ:A→E(M,τ):=E(M,τ)∩M is inner, strengthening earlier results by Kaftal and Weiss [28]. In the case when M is a semifinite non-finite factor, we show that our assumptions on E(0,∞) are sharp.

Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras

Abstract Let K ⁢ 〈 X 〉 = K ⁢ 〈 X 1 , … , X n 〉 {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by X = { X 1 , … , X n } {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted ℕ {\mathbb{N}} -gradation attached to K ⁢ 〈 X 〉 {K\langle X\rangle} , minimal homogeneous generating sets for finitely generated graded two-sided ideals of K ⁢ 〈 X 〉 {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of K ⁢ 〈 X 〉 {K\langle X\rangle} has a finite Gröbner basis 𝒢 {{\mathcal{G}}} with respect to a graded monomial ordering on K ⁢ 〈 X 〉 {K\langle X\rangle} , then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal 〈 𝐋𝐇 ⁢ ( I ) 〉 {\langle\mathbf{LH}(I)\rangle} .

Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $\mathbb{F} S_n$ of the symmetric group $S_n$ over a field $\mathbb{F}$ of characteristic 0 (or $p > n$). The goal is to obtain a constructive version of the isomorphism $\psi\colon \bigoplus_\lambda M_{d_\lambda} (\mathbb{F}) \longrightarrow \mathbb{F} S_n$ where $\lambda$ is a partition of $n$ and $d_\lambda$ counts the standard tableaux of shape $\lambda$. Young showed how to compute $\psi$; to compute its inverse, we use an efficient algorithm for representation matrices discovered by Clifton. In §2, we discuss constructive methods based on §1 which allow us to analyze the polynomial identities satisfied by a specific (non)associative algebra: fill and reduce algorithm, module generators algorithm, Bondari's algorithm for finite dimensional algebras. In §3, we study the multilinear identities satisfied by the octonion algebra $\mathbb{O}$ over a field of characteristic 0. For $n \le 6$ we compare our computational results with earlier work of Racine, Hentzel \&\ Peresi, Shestakov \&\ Zhukavets. Going one step further, we verify computationally that every identity in degree 7 is a consequence of known identities of lower degree; this result is our main original contribution. This gap (no new identities in degree 7) motivates our concluding conjecture: the known identities for $n \le 6$ generate all of the octonion identities in characteristic 0.

Separating Ore sets for prime ideals of quantum algebras

Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus-invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand–Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac–Moody algebras.

Identities for like-powers of Lucas sequences from algebraic identities

Identities for like-powers of Lucas sequences from algebraic identities

Leinert sets and complemented ideals in Fourier algebras

Let $G$ be a locally compact group. We show how complemented ideals in the Fourier algebra $A(G)$ of $G$ arise naturally from a class of thin sets known as Leinert sets. Moreover, we also present an explicit example of a closed ideal in $A(\mathbb{F}_{N})$, the free group on $N \ge 2$ generators, that is complemented in $A(\mathbb{F}_{N})$ but it is not completely complemented. Then by establishing an appropriate extension result for restriction algebras arising from Leinert sets, we show that any almost connected group $G$ for which every complemented ideal in $A(G)$ is also completely complemented must be amenable.

On Base Field of Linear Network Coding

For a (single-source) multicast network, the size of a base field is the most known and studied algebraic identity that is involved in characterizing its linear solvability over the base field. In this paper, we design a new class $\mathcal{N}$ of multicast networks and obtain an explicit formula for the linear solvability of these networks, which involves the associated coset numbers of a multiplicative subgroup in a base field. The concise formula turns out to be the first that matches the topological structure of a multicast network and algebraic identities of a field other than size. It further facilitates us to unveil \emph{infinitely many} new multicast networks linearly solvable over GF($q$) but not over GF($q'$) with $q < q'$, based on a subgroup order criterion. In particular, i) for every $k\geq 2$, an instance in $\mathcal{N}$ can be found linearly solvable over GF($2^{2k}$) but \emph{not} over GF($2^{2k+1}$), and ii) for arbitrary distinct primes $p$ and $p'$, there are infinitely many $k$ and $k'$ such that an instance in $\mathcal{N}$ can be found linearly solvable over GF($p^k$) but \emph{not} over GF($p'^{k'}$) with $p^k < p'^{k'}$. On the other hand, the construction of $\mathcal{N}$ also leads to a new class of multicast networks with $\Theta(q^2)$ nodes and $\Theta(q^2)$ edges, where $q \geq 5$ is the minimum field size for linear solvability of the network.

Fuzzy PMS Ideals in PMS Algebras

Fuzzy PMS Ideals in PMS Algebras

Exact analytical solution for 2-D transient heat conduction in a rectangle with partial heating on one edge

An exact solution is presented for two-dimensional transient heat conduction in a rectangular plate heated at y = 0 from x = 0 to x = L1 and insulated over the other edges. This problem does not have a steady-state solution, but does have a quasi-steady solution. Because of this, Green's functions are used to determine the exact solution. The solution consists of four parts, each of which is determined separately and totalled at the end to achieve the complete solution. A number of algebraic identities are used to reduce some steady- and quasi-steady state series in the solution. Intrinsic verification principles are used to verify the solution. The solution can be used as a building block for a number of related problems.

Commutativity theorems in rings with involution

ABSTRACTIn this paper, we investigate commutativity of ring R with involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

Truncatable bootstrap equations in algebraic form and critical surface exponents

We describe examples of drastic truncations of conformal bootstrap equations encoding much more information than that obtained by a direct numerical approach. A three-term truncation of the four point function of a free scalar in any space dimensions provides algebraic identities among conformal block derivatives which generate the exact spectrum of the infinitely many primary operators contributing to it. In boundary conformal field theories, we point out that the appearance of free parameters in the solutions of bootstrap equations is not an artifact of truncations, rather it reflects a physical property of permeable conformal interfaces which are described by the same equations. Surface transitions correspond to isolated points in the parameter space. We are able to locate them in the case of 3d Ising model, thanks to a useful algebraic form of 3d boundary bootstrap equations. It turns out that the low-lying spectra of the surface operators in the ordinary and the special transitions of 3d Ising model form two different solutions of the same polynomial equation. Their interplay yields an estimate of the surface renormalization group exponents, y h = 0.72558(18) for the ordinary universality class and y h = 1.646(2) for the special universality class, which compare well with the most recent Monte Carlo calculations. Estimates of other surface exponents as well as OPE coefficients are also obtained.

Rota's Classification Problem, rewriting systems and Gröbner–Shirshov bases

In this paper we revisit Rota's Classification Problem on classifying algebraic identities for linear operators. We reformulate Rota's Classification Problem in the contexts of rewriting systems and Gröbner–Shirshov bases, through which Rota's Classification Problem amounts to the classification of operators, given by their defining operator identities, that give convergent rewriting systems or Gröbner–Shirshov bases. Relationship is established between the reformulations in terms of rewriting systems and that of Gröbner–Shirshov bases. We provide an effective condition that gives Gröbner–Shirshov operators and obtain a new class of Gröbner–Shirshov operators.