On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

Heisenberg-type groups and intertwining operators

We extend the theorem of Cowling and Haagerup on limits of K-invariant matrix coefficients at approach to the identity, to arbitrary matrix coefficients.     

Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free Rota-Baxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.     

Integral intertwining operators and quantum homogeneous spaces

Integral representations of functions on quantum homogeneous spaces are considered. The Dirichlet problem for the quantum ball is solved and a q-analog of the Cauchy-Szeg formula is derived.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 355–363, December, 1995.     

q-Difference intertwining operators and q-conformal invariant equations

We first recall a canonical procedure for the construction of the invariant differential operators and equations for arbitrary complex or real noncompact semisimple Lie groups. Then we present the application of this procedure to the case of quantum groups. In detail is given the construction of representations of the quantum algebra U _q (sl(n)) labelled by n–1 complex numbers and acting in the spaces of functions of n(n–1)/2 noncommuting variables, which generate a q-deformed SL(4) flag manifold. The conditions for reducibility of these representations and the procedure for the construction of the q-difference intertwining operators are given. Using these results for the case n=4 we propose infinite hierarchies of q-difference equations which are q-conformal invariant. The lowest member of one of these hierarchies are new q-Maxwell equations. We propose also new q-Minkowski spacetime which is part of a q-deformed SU(2,2) flag manifold.     

OnL-functions and intertwining operators for unitary groups

The ingredients of an “L-function machine” for the quasi-split groupU _{n, n} +1 × Res GL _n are treated here, following similar theories of P. Shapiro and S. Gelbart. We start with a known Rankin-Selberg type integral having an Euler product. In section 2 we compute the local integral to get a localL function. This is done by working with an “L group” related to ^L G and the relative root system. All computations are carried out for the split and the non-split case. In section 3 we address the problem of analytic continuation of the Eisenstein series. This involves computation of poles of intertwining operators.     

Singular operators satisfying an intertwining relation

An intertwining relation between the Beltrami-Laplace operator with an added potential and the Beltrami-Laplace operator is considered on a Riemannian manifold. It is shown that the potential singularities of codimension one form completely geodesic hypersurfaces. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 264–270, November, 1999     

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕ _αεG A _α graded by some group G, over a field of characteristic zero, has a nonzero component A ₁ (where 1 stands for the identity element of G), and A ₁ is a semisimple associative algebra. Let B = ⊕ _αεG B _α be a finite-dimensional semisimple Lie algebra over a prime field F _p , and let B be graded by a commutative group G. If B = F _p ? _? A _L , where A _L is the commutator algebra of a ?-algebra A = ⊕ _αεG A _α ; if ? ? _? A is an algebra of associative type, then the 1-component of the algebra K ? _? B, where K stands for the algebraic closure of the field F _p , is the sum of some algebras of the form gl(n _i ,K).     

Novikov-Poisson algebras and associative commutative derivation algebras

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras. Supported by RFBR (grant No. 05-01-00230), by SB RAS (Integration project No. 1.9), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 186–202, March–April, 2008.     

Generalized Toeplitz Kernels and dilations of intertwining operators

It is shown that the basic properties of the so called (operator-valued) generalized Toeplitz Kernels, [5], allow to derive in a unified and natural way different results concerning the dilation of operators, such as the lifting theorem of Nagy-Foias and corresponding unicity conditions.     

Duality and the normalization of standard intertwining operators

Normalized standard intertwining operators associated to an induced representation and its dual (dual in the sense of Aubert) arise in work on a conjecture of Arthur about R-groups. The purpose of this paper is to address the question of relating the normalizing factors used.     

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.     

Permutation orbifolds and associative algebras

Let V be a vertex operator algebra and g =(1 2 ··· k) be a k-cycle which is viewed as an automorphism of the vertex operator algebra V?k. It is proved that Dong-Li-Mason’s associated associative algebra Ag(V?k) is isomorphic to Zhu’s algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g-twisted V?k-modules and irreducible V-modules.     

Logarithmic residues in Banach algebras

Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f ^?1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.     

Power associative composition algebras

Composition algebras of arbitrary dimension over a field and satisfying the identities x ² x=x x ² and (x ²)²=(x ² x)x are shown to be precisely the well-known unital composition algebras, with the exception of three two dimensional algebras over the field of two elements. Received: 1 February 2000