Shuffle Algebra Research Articles

The q-AGT–W Relations Via Shuffle Algebras

We construct the action of the q-deformed W-algebra on its level r representation geometrically, using the moduli space of U(r) instantons on the plane and the double shuffle algebra. We give an explicit LDU decomposition for the action of W-algebra currents in the fixed point basis of the level r representation, and prove a relation between the Carlsson–Okounkov Ext operator and intertwiners for the deformed W-algebra. We interpret this result as a q-deformed version of the AGT–W relations.

Monotone, free, and boolean cumulants: A shuffle algebra approach

The theory of cumulants is revisited in the “Rota way”, that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.

The cohomological Hall algebra of a preprojective algebra

We introduce for each quiver $Q$ and each algebraic oriented cohomology theory $A$, the cohomological Hall algebra (CoHA) of $Q$, as the $A$-homology of the moduli of representations of the preprojective algebra of $Q$. This generalizes the $K$-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When $A$ is the Morava $K$-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the $A$-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when $A$ is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of $A$. As applications, we obtain a shuffle description of the Yangian.

Interlacing adjacent levels of $$\beta $$–Jacobi corners processes

We study the asymptotics of the global fluctuations for the difference between two adjacent levels in the $$\beta $$ –Jacobi corners process (multilevel and general $$\beta $$ extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the 2d Gaussian free field. Our main tools are integral forms for the (Macdonald-type) difference operators originating from the shuffle algebra.

The Moments of Lévy's Area Using a Sticky Shuffle Hopf Algebra

The Moments of Lévy's Area Using a Sticky Shuffle Hopf Algebra

Analysis of periodically forced bioreactors using nonlinear transfer functions

Nonlinearity is an underlying property of process systems that allows for time-average performance enhancement with periodic forcing of external parameters. One can use the well-known π-criterion to obtain a sufficient condition for optimal periodic operation, but it is valid only for weakly nonlinear systems with infinitesimal forcing amplitudes. Among such process systems, bioreactors often exhibit highly nonlinear dynamics, thus posing difficulties for a systematic analysis of their periodic operation. It becomes desirable to understand how inherent nonlinearities would contribute to performance improvement if periodic forcing is applied to such processes. This paper explores how nonlinear characteristics affect periodic process operation in the context of bioreactor systems. By taking advantage of iterated integrals and shuffle algebra, an analytical solution of a nonlinear bioprocess is approximated with the functional expansion method. The resulting Laplace-Borel transform of the nonlinear system facilitates the expression of the solution in the frequency domain based on the nonlinear transfer function. The method enables the separation of the transient dynamics and the stationary periodic behavior, facilitating the analysis of periodic operations and leading to a generalizable platform. Specifically, the optimal periodic operation problem is solved by finding the proper forcing amplitude and frequency to maximize the offset. Compared with the π-criterion, our method is proven to be globally valid for any forcing frequency and amplitude, so long as the process is globally stable.

Antipodes and involutions

If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmetric functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation-free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto–Reutenauer and Poirier–Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research.

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

We give a natural and complete description of Ecalle's mould-comould formalism within a Hopf-algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes-Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogeneous coarborification, using the explicit duality between decorated Connes-Kreimer and Grossman-Larson algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.

Integrable Systems, Shuffle Algebras, and Bethe Equations

Integrable Systems, Shuffle Algebras, and Bethe Equations

Structure of polyzetas and explicit representation on transcendence bases of shuffle and stuffle algebras

Polyzetas, indexed by words, satisfy shuffle and quasi-shuffle identities. In this respect, one can explore the multiplicative and algorithmic (locally finite) properties of their generating series. In this paper, we construct pairs of bases in duality on which polyzetas are established in order to compute local coordinates in the infinite dimensional Lie groups where their non-commutative generating series live. We also propose new algorithms leading to the ideal of polynomial relations, homogeneous in weight, among polyzetas (the graded kernel) and their explicit representation (as data structures) in terms of irreducible elements.

Center Problem, Abel Equation and the Faà di Bruno Hopf Algebra for Output Feedback

A combinatorial interpretation is given of Devlin's word problem underlying the classical center-focus problem of Poincare for non-autonomous differential equations. It turns out that the canonical polynomials of Devlin are from the point of view of connected graded Hopf algebras intimately related to the graded components of a Hopf algebra antipode applied to the formal power series of Ferfera. The link is made by passing through control theory since the Abel equation, which describes a center, is equivalent to an output feedback equation, and the Hopf algebra of output feedback is derived from the composition of iterated integrals rather than just the products of iterated integrals, which yields the shuffle algebra. This means that the primary algebraic structure at play in Devlin's approach is actually not the shuffle algebra, but a Faa di Bruno type Hopf algebra, which is defined in terms of the shuffle product but is a distinct algebraic structure.

The Hall algebra of a curve

Let X be a smooth projective curve over a finite field. We describe H, the full Hall algebra of vector bundles on X, as a Feigin–Odesskii shuffle algebra. This shuffle algebra corresponds to the scheme S of all cusp eigenforms and to the rational function of two variables on S coming from the Rankin–Selberg L-functions. This means that the zeroes of these L-functions control all the relations in H. The scheme S is a disjoint union of countably many \({\mathbb G}_m\)-orbits. In the case when X has a theta-characteristic defined over the base field, we embed H into the space of regular functions on the symmetric powers of S.

Bethe subalgebras of $$U_q(\widehat{\mathfrak {gl}}_n)$$ U q ( gl ^ n ) via shuffle algebras

In this article, we construct certain commutative subalgebras of the big shuffle algebra of cyclic type. This can be considered as a generalization of the similar construction for the small shuffle algebra, obtained by Feigin-Hashizume-Hoshino-Shiraishi-Yanagida. We present a Bethe algebra realization of these subalgebras. The latter identifies them with the Bethe subalgebras of quantum affine gl(n).

On the homology and homotopy of commutative shuffle algebras

For commutative algebras there are three important homology theories, Harrison homology, Andre-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quillen equivalent to the model category of pointed simplicial commutative monoids in symmetric sequences.

On the homology and homotopy of commutative shuffle algebras

For commutative algebras there are three important homology theories, Harrison homology, Andre-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quillen equivalent to the model category of pointed simplicial commutative monoids in symmetric sequences.

The Feigin Tetrahedron

The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra. In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined. To relate these constructions, we establish a homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel algebra to the quantum shuffle algebra which relates the generalized Feigin homomorphisms. These constructions can be compactly described by a commuting tetrahedron of maps beginning with the quantum group and terminating in a quantum polynomial algebra. The second goal in this project is to better understand the dual canonical basis conjecture for skew-symmetrizable quantum cluster algebras. In the symmetrizable types it is known that dual canonical basis elements need not have positive multiplicative structure constants, while this is still suspected to hold for skew-symmetrizable quantum cluster algebras. We propose an alternate conjecture for the symmetrizable types: the cluster monomials should correspond to irreducible characters of a KLR algebra. Indeed, the main conjecture of this note would establish this "KLR conjecture" for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture that the images of rigid representations under the quantum shuffle character give irreducible characters for KLR algebras.

Deformations of shuffles and quasi-shuffles

We investigate deformations of the shuffle Hopf algebra structure Sh(A) which can be defined on the tensor algebra over a commutative algebra A. Such deformations, leading for example to the quasi-shuffle algebra QSh(A), can be interpreted as natural transformations of the functor Sh, regarded as a functor from commutative nonunital algebras to coalgebras. We prove that the monoid of natural endomophisms of the functor Sh is isomorphic to the monoid of formal power series in one variable without constant term under composition, so that in particular, its natural automorphisms are in bijection with formal diffeomorphisms of the line. These transformations can be interpreted as elements of the Hopf algebra of word quasi-symmetric functions WQSym, and in turn define deformations of its structure. This leads to a new embedding of free quasi-symmetric functions into WQSym, whose relevance is illustrated by a simple and transparent proof of Goldberg's formula for the coefficients of the Hausdorff series.

Shuffle algebras, homology, and consecutive pattern avoidance

Shuffle algebras are monoids for an unconvential monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations.

The Shuffle Algebra Revisited

In this paper, we introduce certain new features of the shuffle algebra of [3] that will allow us to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra of [1, 7]. These results are necessary for our work in [4, 6], where they will be applied to the study of the Hilbert scheme and to computing knot invariants.

NATURAL ENDOMORPHISMS OF SHUFFLE ALGEBRAS

We show that there exist two natural endomorphism algebras for shuffle bialgebras such as Sh (X), where X is a graded set. One of these endomorphism algebras is a natural extension of the Malvenuto–Reutenauer Hopf algebra and is defined using graded permutations. The other one, the dendriform descent algebra, is a subalgebra of the first defined by mimicking the definition of the descent algebras by convolution from the graded projections in the tensor algebra. We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.