Showing papers on "Quasitriangular Hopf algebra published in 1988"

Hopf Algebras for Physics at the Planck Scale

Abstract: Applies ideas of non-commutative geometry to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime. The reformulation maintains an interesting symmetry between observables and states in the form of a Hopf algebra structure. In the simplest example both the dynamics and quantum mechanics are completely determined by the Hopf algebra consideration. The simplest example is a two-parameter algebra-co-algebra KAB which is the unique Hopf algebra extension, such as is possible, of the self-dual Hopf algebra of functions on flat phase space C(R*R). In the limit (A=0,B) the author recovers functions on a curved phase space with curvature proportional to B2 and in another limit (A= infinity , B= infinity ), A/B=h(cross), the author recovers quantum mechanics on R>or=0 with an absorbing wall at the origin. The algebra in this way corresponds to a toy model of quantum mechanics of a particle in one space dimension combined with gravity-like forces. It has an interesting Z2 symmetry interchanging A to or from B, and thereby, in some sense, the quantum element with the geometric element. The compatibility conditions that are solved are a generalisation of the classical Yang-Baxter equations.

Tame objects for finite commutative Hopf algebras

Abstract: Let S be an A-module algebra for a commutative Hopf algebra A, both projective of the same rank over a commutative ring. Let I be the space of integrals in A. Then S is an invertible A-module iff it is a faithful rnodule which satisfies the "trace surjectivity" condition that 1 is in IS. Let A be a commutative Hopf algebra, projective of rank n over a commutative ring R. Let S be a commutative algebra, also projective of rank n over R, and suppose we have made S an A-module algebra in the sense of Sweedler [4, p. 153]. That is, we have extended the f-module structure to an A-module structure that satisfies b 1 = (b)l and b . st = Z(bi . s)(ci t) for the counit E and the comnultiplication A (b) = E bi 0ci in A. If A is also cocommutative, this means that we have an action of the finite commutative group scheme Spec(A*) on Spec(S). Let I denote the set of integrals in A, so I = {c E Albc = e(b)c for all b E A}; this set is known to be a projective R-module of rank one [3, p. 592]. In a recent paper, Childs and Hurley [2] introduced some conditions on S weaker than those making it a Galois object for A; they call it a tame object if (in addition to haviing the same rank as A) it satisfies (i) S is a faithful A-module, (ii) R = {s E Slb s = E(b)s for all b E A}, and (iii) I S =R. When R is an integral domain and certain supplementary conditions are satisfied, Childs and Hurley showed that any tame object is actually an invertible A-module. (Taking their terminology from the case where A is a commutative group algebra, they describe this by saying that S has "local normal bases", reflecting the fact that when R is local the ring S will then-like A*--be free over A.) In this note, I shall show that the conclusion actually holds in full generality for arbitrary R. Furthermore, condition (ii) can be dropped, and (i) can be weakened; the "trace surjectivity" condition (iii) is the crucial property. THEOREM. Let R be a commutative ring, A a commutative Hopf algebra projective of rank n over R. Let I be the integrals in A, and let S be an A-module algebra also projective of rank n over R. The following are equivalent: (1) S is tame in the sense of Childs and Hurley. (2) SQ :A 0 for all Q E Spec(A), and 1 E IS. (3) S is an invertible A-module. Received by the editors March 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 14L20; Secondary 13B05. This work was supported in part by the National Science Foundation, Grant MCS 8400649. (D1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Most quasidiagonal operators are not block-diagonal

Abstract: The set of all block-diagonal operators is a dense first category subset of the class (QD) of all quasidiagonal operators. On the other hand, the subset of all irreducible quasidiagonal operators with thin spectra, that are similar to block-diagonal ones, includes a G6-dense subset of (QD). C.-K. Fong proved that most normal operators are diagonal, in the sense that the class of all normal operators includes a G3-dense subset of diagonal ones [1]. By using his results in [6], the author proved that Fong's argument can be modified to show that most quasitriangular operators are triangular and most biquasitriangular operators are bitriangular ("most" in the same sense as above [7]). Here operator means a bounded linear mapping from a complex, separable, infinite dimensional Hilbert space A' into itself; T E ](Z) (:= the algebra of all operators acting on A') is called triangular if it admits a representation as an upper triangular matrix with respect to a suitable orthonormal basis of A. T is bitriangular if both T and its adjoint T* are triangular (not necessarily with respect to the same basis, of course). Recall that T E Y(') is quasidiagonal if there exists an increasing sequence {Pn}?O'1 of finite rank orthogonal projections such that Pn -1 (strongly) and IITPn-PnTJJ-*O (n-*oo). An operator B is block-diagonal if there exists {Pn}J01 as above such that BPn = PnB for all n = 1,2,.... (and therefore B = eE)L1 Bn where Bn = (Pn -Pn-l )B I ran(Pn -Pn), n = 1, 2, ... .; Po = ?). The results of [1 and 7] might suggest that most quasidiagonal operators (class (QD)) are block-diagonals (class (BD)). But this is definitely false. Reason. If X(Z) denotes the ideal of compact operators, then (QD) + X(d) = (BD) + X(X?) = (BD)= (QD) [3]. (As usual, the upper bar denotes norm-closure.) P. R. Halmos has shown that the irreducible operators form a G3-dense subset of i2 (A) [2] (see also [10]; A E YG(A) is irreducible if there is no nontrivial subspace 4 of A such that AXf C 4 and A(A E X4) c A e. X4). Furthermore, Halmos actually proved that given T in Y (A) and E > 0, there exists K e E (AP) with I 1K 1 < E, such that T K. is irreducible. Since (QD) is a closed subset of Y(i'), invariant under compact perturbations, it is not difficult to check that Halmos's argument also proves that the irreducible quasidiagonal operators form Received by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985 Revtsion). Primary 47A66.