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We prove that for matrix algebras $M_n$ there exists a monomorphism $(prod_n M_n/oplus_n M_n)otimes C(S^1) to {cal Q} $ into the Calkin algebra which induces an isomorphism of the $K_1$-groups. As a consequence we show that every vector bundle over a classifying space $Bpi$ which can be obtained from an asymptotic representation of a discrete group $pi$ can be obtained also from a representation of the group $pitimes Z$ into the Calkin algebra. We give also a generalization of the notion of Fredholm representation and show that asymptotic representations can be viewed as asymptotic Fredholm representations.
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Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of Grassmann algebra. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the Grassmann algebra over the base field. We also consider the infinitesimal analog of the Boolean Bercovici--Pata map.
In this paper, we derive, from a general Simonenkos local principle, Fredholm criteria for restriction to isotypical components. More precisely, we gave a full proof, of the equivariant local principle for restriction to isotypical components of invariant pseudodifferential operators announced in cite{BCLN2}. Furthermore, we extend this result by relaxing the hypothesis made in the preceding quoted paper.
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $mathrm{GL}(N,mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painleve VI equation.
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $mathcal{C}$ is graded over a group $Gamma$, a torus-valued 3-cocycle on $Gamma$ can be used to deform the associator of $mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Gamma$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
We consider the group (G,*) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where * denotes the convolution operation. We introduce a larger group (Gtilde,*) of unitized functions from the same incidence algebra, which satisfy a weaker condition of being semi-multiplicative. The natural action of Gtilde on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of Gtilde in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bozejko and Wysoczanski. It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that Gtilde can also be identified as group of characters of a Hopf algebra T, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion of G in Gtilde turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.
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