Little and big q-Jacobi polynomials and the Askey–Wilson algebra

The Ramanujan Journal - The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey–Wilson algebra. The operators that these polynomials...     

Interpolating Blaschke products and the ideals of the algebra H∂

To each vector-function f, f H(Z²) one can associate two ideals of the algebra. In the note one characterizes those functions f for which some interpolation Blaschke product is contained in I(f) (or J (f)). One also characterizes those functions u, u H such that if for some f, then u I(f) (in the corona theorem a similar statement is proved for the case u =1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 196–201, 1983.     

On ℤp-graded identities and cocharacters of the Grassmann algebra

Let p be an odd prime number, F a field of characteristic zero, and let Ebe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector space L. We determine the bases of the ?_p-graded identities.Moreover we compute the ?_p-graded codimension and cocharacter sequences for the algebra E endowed with any ?_p-grading such that L is a homogeneous subspace.     

Schur elements for the Ariki–Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki–Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type G(l,p,n) in characteristic 0.     

Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra

We give a proof of a Schur–Weyl duality statement between the Brauer algebra and the ortho-symplectic Lie superalgebra \(\mathfrak {osp}(V)\).     

Blaschke products satisfying the Carleson-Newman conditions and ideals of the algebra H∞

One obtains tests for an ideal of the algebra H to contain a Blaschke product satisfying the Carleson-Newman condition. If the generalized divisor of the ideal is bounded on the space of maximal ideals of H, then the ideal contains a function with bounded divisor.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 93–102, 1986.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

Biderivations of the twisted Heisenberg–Virasoro algebra and their applications

In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg–Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg–Virasoro algebra are given. It is also proved that every biderivation of the graded twisted Heisenberg–Virasoro left-symmetric algebra is trivial.     

Extensions of the disc algebra and of Mergelyanʼs theorem

We investigate the uniform limits of the set of polynomials on the closed unit disc $\overline{D}$ with respect to the chordal metric χ. More generally, we examine analogous questions replacing $\mathbb{C}\cup \{\infty \}$ by other metrizable compactifications of $\mathbb{C}$.     

Nonlinear algebra and Bogoliubov’s recursion

We give many examples of applying Bogoliubov’s forest formula to iterative solutions of various nonlinear equations. The same formula describes an extremely wide class of objects, from an ordinary quadratic equation to renormalization in quantum field theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 316–343, February, 2008.     

Einstein’s equations and Clifford algebra

Einstein’s equations of the general theory of relativity are rewritten within a Clifford algebra. This algebra is otherwise isomorphic to a direct product of two quaternion algebras. A multivector calculus is developed within this Clifford algebra which differs from the corresponding complexified algebra used in the standard spacetime algebra approach.     

Harmonic analysis on the Iwahori–Hecke algebra

These are purely expository notes of Opdam’s analysis [O1] of the trace form τ(f) = f(e) on the Hecke algebra H = C _c (I\G/I) of compactly supported functions f on a connected reductive split p-adic group G which are biinvariant under an Iwahori subgroup I, extending Macdonald’s work. We attempt to give details of the proofs, and choose notations which seem to us more standard. Many objects of harmonic analysis are met: principal series, Macdonald’s spherical forms, trace forms, Bernstein forms. The latter were introduced by Opdam under the name Eisenstein series for H. The idea of the proof is that the last two linear forms are proportional, and the proportionality constant is computed by projection to Macdonald’s spherical forms. Crucial use is made of Bernstein’s presentation of the Iwahori–Hecke algebra by means of generators and relations, as an extension of a finite dimensional algebra by a large commutative subalgebra. We give a complete proof of this using the universal unramified principal series right H-module M = C _c (A(O)N\G/I) to develop a theory of intertwining operators algebraically.     

Antisymmetry and analytic structure in the spectrum of a uniform Fréchet algebra

Let M be a C-smooth submanifold of a domain GCⁿ. Denote by 0(M) the algebra of germs of holomorphic functions on M, that is, each f0(M) is holomorphic in some neighbourhood of M, the neighbourhood dependent of f. Now define A(M) as the closure of 0(M) with respect to the topology of compact convergence on M.If M coincides with A(M), the spectrum of A(M), we give a necessary and sufficient condition for A(M) such that M is a complex manifold.     

Representation theory of the Yokonuma–Hecke algebra

We develop an inductive approach to the representation theory of the Yokonuma–Hecke algebra _Yd,n(q)${\mathrm{Y}}_{d,n}(q)$, based on the study of the spectrum of its Jucys–Murphy elements which are defined here. We give explicit formulas for the irreducible representations of _Yd,n(q)${\mathrm{Y}}_{d,n}(q)$ in terms of standard d  -tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on _Yd,n(q)${\mathrm{Y}}_{d,n}(q)$ and calculate the Schur elements with respect to that form.     

Analogue of the Cayley–Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form

The number of linearly independent homogeneous invariants of degree k for the n-ary form of order d is calculated. Also, a formula for the Poincaré series for the algebra of invariants of the n-ary form is found.     

Remarks on the physical states and the chiral algebra of 2D gravity coupled toC≤1 matter

Some elaboration is givent to the structure of physical states in 2D gravity coupled to C1 matter, and to the chiral algebra (w of C_M=1 theory which has been found recently, in the continuum approach, by Witten and by Klebanov and Polyakov. It is then shown that the chiral algebra is realized as well in the minimal models of gravity (C_M<1), so that it stands as a general symmetry of 2D gravity theories.Dedicated to the memory of the late Professor M. C. PolivanovL. D. Landau Institute of Theortetical Physics, Russian Academy of Sciences. LPTHE, Laboratoire associé No. 280 au CNRS, Universitè Pierre et Marie Curie, Paris VI, France. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 357–373, September, 1992.     

Teaching of real numbers by using the Archimedes–Cantor approach and computer algebra systems

Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes–Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes–Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ?, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes–Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.