BAXTER ALGEBRAS AND DIFFERENTIAL ALGEBRAS

Baxter Algebras and Shuffle Products

In this paper, we construct free commutative integro-differential algebras by applying the method of Gröbner–Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota–Baxter (DRB) algebras of order n. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner–Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional monomials to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.

In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the construction of free commutative Rota-Baxter algebras by mixable shuffles. The first is by evaluations. The second is by the method of Gröbner-Shirshov bases.

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove this construction commutes with base change and also agrees with Bhargava's explicit construction over $\mathbb{Z}$.

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products. Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to $K_{4}$-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a P ∞-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an L ∞-algebra which is not a P ∞-algebra. Our technique produces new all orders constructions which are significantly simpler compared to previous approaches, and we illustrate its applicability in several examples of interest in noncommutative field theory and gravity. We further show that our symplectic embeddings naturally define a P ∞-structure on the exterior algebra of differential forms on a generic almost Poisson manifold, which generalizes earlier constructions of differential graded Poisson algebras, and suggests a new approach to defining noncommutative gauge theories beyond the gauge sector and the semi-classical limit based on A ∞-algebras.

The various canonical subcategories of the category HopfR of Hopf algebras over a commutative ring R, like those of (co)commutative Hopf algebras or Hopf algebras whose antipode is bijective or of order 2, are shown to be locally presentable categories and reflective and coreflective in their respective supercategories. The reflectivity results provided only hold for commutative von Neumann regular rings, while most of the coreflectivity results are valid over any ring. As a consequence one gets existence of free commutative Hopf algebras over coalgebras and cofree cocommutative Hopf algebras over algebras.

In [[6], Theorem 2.2] Doi gave a Hopf-algebraic proof of a generalization of Oberst's theorem on affine quotients of affine schemes. He considered a commutative Hopf algebra H over a field, coacting on a commutative H-comodule algebra A. If A coH denotes the subalgebra of coinvariant elements of A and β : A ⊗ A coH A → A ⊗ H the canonical map, he proved that the following are equivalent: a. AcoH ⊆ A is a faithfully flat Hopf Galois extension; b. the functor (−) coH : M H A → A coH -Mod is an equivalence; c. A is coflat as a right H-comodule and β is surjective. Schneider generalized this result in [[14], Theorem 1] to the non-commutative situation imposing as a condition the bijectivity of the antipode of the underlying Hopf algebra. Interpreting the functor of coinvariants as a Hom-functor, Menini and Zuccoli gave in [10] a module-theoretic presentation of parts of the theory. Refining the techniques involved we are able to generalize Schneiders result to H-comodule-algebras A for a Hopf algebra H (with bijective antipode) over a commutative ring R under fairly weak assumptions.

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

Division algebras D with valuation v are studied, where D is lSnite-dimensional and totally ramified over its center F (i.e., the ramification index of v over wlF equals [D: F]). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When vlF is Henselian a full classification of the F-subalgebras of D is given. When F has a Henselian valuation v with separably closed residue field and A is any tame central simple F-algebra, an algorithm is given for computing the underlying division algebra of A from a suitable subgroup of A*/F*. Some examples are constructed using this valuation theory, including the first example of finitbdimensional F-central division algebras D1 and D2 with D1 XF D2 not a division ring, but D1 and D2 having no common subfield K D F. Valuation theory, long a basic tool in commutative algebra, has been relatively neglected in the study of division algebras, until quite recently. Nontheless, valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties, particularly algebras over iterated Laurent power series fields. For example, such division algebras have been key ingredients in Amitsur's noncrossed product construction [Am] and in Platonov's construction [P] of division algebras D with SK1(D) 7& 1. Valuations are not so prevalent on division algebras as on fields. But if a division algebra D does have a valuation, this structure contains a substantial amount of information about D which would scarcely be accessible otherwise. We consider here valued divisioll algebras D for which D is totQlly ramified and tame over its center F, i.e., for which the ramification index IrD rFI equals the dimension [D: F] of D over F and the characteristic char(D) does not divide [D: F]. (Here rD iS the value group of the valuation on D, and D is the residue division algebra. We assume throughout that [D: F] < oo.) Valued division algebras of this type appear, e.g., in Amitsur's noncrossed product paper [Am, §2], in Saltman's work on indecomposable division algebras [Sa], in certain of the MalcevNeumann division algebras considered by the first author and Amitsur [TA2,§4], etc. However, the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before. This may be because most past work on valued division algebras has concentrated on discrete valuations, when rD -Z; for such a valuation D is never totally ramified over its center F unless D = F (cf. (3.2) below). Received by the editors June 26, 1986. 1980 Mathematics Subject Clmsification (1985 Reon). Primary 16A39. 2Supported in part by F.N.R.S. 2Supported in part by the National Science Foundation. (:)1987 American Mathematlcal Society 0002-9947/87 $}.00 + $ 25 per page

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .

Geometric construction of Hopf cyclic characteristic classes