An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this are motivated by relations between coproducts twisted primitive elements two-fold tensor product quantum $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb C))$. It shown that bivariate $q$-Racah polynomials appear overlap coefficients eigenvectors generators Furthermore, corresponding $q...

The R12 h matrices of the Jordanian Uh(sl(2)) algebra at arbitrary dimensions may be obtained from the corresponding Rj1;j2 q matrices of the standard q-deformed Uq(sl(2)) algebra through a contraction technique. By extending this method, the coloured two-parametric (h, α) Jordanian R1122 h,α matrices of the Uh,α(gl(2)) algebra may be derived from the corresponding coloured R j1,z1;j2,z2 q,λ ma...

In much of the mathematical literature, there is talk about the (note the definite article) orthogonal group O(n, F I ) of degree n ∈ N I over a field F I . This is extremely misleading, because, given a linear space T of dimension n, one can consider the orthogonal group of any non-degenerate quadratic form Q on T . In Chapter 6 of the book Basic Algebra I [J], Jacobson denotes this orthogonal...

We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $X$, $(X, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $X$. We also show that, for a strong residuated lattice $X$, $bigcap {P_{lambda} ,|,P_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong re...

We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras BCr,s(q). The superalgebra BCr,s(q) is a quantum deformation of the walled BrauerClifford superalgebra BCr,s and a super version of the quantum walled Brauer algebra. We prove that BCr,s(q) is the centralizer superalgebra of the action of Uq(q(n)) on the mixed tensor space V q = V ⊗r q ⊗ (V∗ q) when n ≥...

is surjective. If W is a Q-vector space, Q-algebra or Q-Lie algebra then we write WC for the correspondingC-vector space (respectively, C-algebra orC-Lie algebra) W ⊗Q C. Let f(x) ∈ C[x] be a polynomial of degree n ≥ 2 without multiple roots. Suppose that p is a prime that does not divide n and a positive integer q = p is a power of p. As usual, φ(q) = (p − 1)p denotes the Euler function. Let u...

These rings admit numerous algebraic and geometric realizations, but one of the historically first constructions, dating to the work of Steinitz in 1900 completed later by Hall, was given in terms of what is now called the classical Hall algebra H (see [Ma], Chapter II ). This algebra has a basis consisting of isomorphism classes of abelian q-groups, where q is a fixed prime power, and the stru...

Following work of Pa lasińska and Pigozzi on partially ordered varieties and quasi-varieties of universal algebras, the author recently introduced partially ordered systems (posystems) and partially ordered functors (pofunctors) to cover the case of the algebraic systems arising in categorical abstract algebraic logic. Analogs of the ordered homomorphism theorems of universal algebra were shown...