Differential calculi on commutative algebras

The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy nontrivial commutation relations. For a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice. These differential calculi generate a lattice on a space continuum. The whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus. In this framework one just has to express the Lagrangian for the continuum theory in terms of differential forms. This expression then also makes sense for the deformed differential calculus. There is a natural integral associated with the latter. Integration of the Lagrangian over a space continuum then produces the correct lattice action for a large class of theories. This is explicitly shown for the scalar field action and the action for SU(m) gauge theory.

3 - Fiber Bundles

In a previous paper all bicovariant differential calculi on the two-parameter quantum group GLp,q(2) were determined. In this work the authors elaborate some of their properties and discuss relations with work by other authors. Furthermore, they show that there are two different ways in which bicovariant differential calculi on GLq(2) (where p=q) induce corresponding calculi on the quantum subgroup SLq(2). It is shown that on SLq(2) there are only two different bicovariant differential calculi. From these one obtains the 4D+or- calculi on SUq(2). The classical limit of the two differential calculi on SLq(2) is investigated revealing a relation with recent work on deformed differential calculi on commutative (function) algebras.

In this note we highlight a common origin for many ubiquitous geometric structures, as well as several new ones by using only the functors of differential calculus in A.M Vinogradov’s original sense, adapted to special classes of (graded) commutative algebras. Special attention is given to the particularly simple cases of diole and triole algebras and we show the latter environment is the appropriate one to describe calculus in vector bundles in the presence of a vector-valued fiber metric.

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra, the orbits of the action Φ ofongiven byare pseudo‐Hessian manifolds, where. We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called $\varphi$% -monogenic. A relation between the concepts of $\varphi$-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras.

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac–Moody algebras.

Shirshov published six papers on Lie algebras in which he found the following results (in order of publication. 1953–1962):

In this article, we generalize a construction of graded q-differential algebra with ternary differential satisfying the property d 3 = 0 and q-Leibniz rule on the non-coordinate case, that is on the case where the differentials of generators of underlying algebra do not coincide with generators of bimodule of first order forms. Our starting point is the coordinate first order differential calculus on associative complex algebra with n generators and the bimodule of second order differentials.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

Let F be a field, and let $q(\alpha ) = q_1^{d_1 } (\alpha ) \cdots q_1^{d_k } (\alpha ) \in F[\alpha ]$ be a polynomial of degree n, where $q_1 (\alpha ) \cdots q_k (\alpha )$ are distinct irreducible polynomials. Let $y(\alpha ),y_1 (\alpha ), \ldots ,y_r (\alpha ),x_1 (\alpha ), \ldots ,x_s (\alpha )$ be $(n - 1)$-degree polynomials with distinct nonscalar coefficients. The authors show the following: the number of nonscalar multiplications/divisions required to compute the coefficients of $x_1 (\alpha ),y(\alpha )\bmod q(\alpha )$ for $i = 1, \ldots ,s$ by straight line algorithms is $s(2n - k)$. If H is a $s \times r$- matrix with entries from F, then the number of nonscalar multiplications/ divisions required to compute the coefficients of $(x_1 (\alpha ), \ldots ,x_s (\alpha ))H(y_1 (\alpha ), \ldots ,y_r (\alpha ))^T \bmod q(\alpha )$ by straight line algorithms is equal to $(2n - k)$ rank $(H)$. All the above systems satisfy the direct sum conjecture strongly. The above results also hold for some other algebras that are direct sums of local algebras, such as commutative algebras and division algebras.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

We start off with the idea of a chart. This is sometimes called a system of coordinates, but we feel, as does Lang (see [32]), that this both obscures the basic idea and impairs the recognition of an immediate generalization to Banach manifolds by introducing scads of unnecessary notation. The idea that Banach spaces provide an appropriate scenario for doing differential calculus goes back at least to the treatise [8] of Dieudonne. An extremely well-written text on smooth manifolds is Lee’s book [33].

We introduce a unital associative algebra A associated with degenerate CP1. We show that A is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby, we can regard A as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov–Pandharipande [e-print arXiv:math-ph/0411210].

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the commutative algebra are replaced by $${\mathbb{D}{\rm er}(A)}$$ , the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on $${\mathcal{F}(A)}$$ , a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra $${\mathcal{D}(\mathcal{F}(A))}$$ of differential operators is filtered and gr $${\mathcal{D}(\mathcal{F}(A))}$$ , the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr $${\mathcal{D}(\mathcal{F}(A))}$$ is closely related to the double Poisson structure on $${T_{A} \mathbb{D}{\rm er}(A)}$$ introduced by Van den Bergh. Specifically, we prove that gr $${\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),}$$ provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space $${\mathcal{F}(A)}$$ is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras. In this language, $${\mathcal{D}(\mathcal{F}(A))}$$ becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper, we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on $${\mathbb{D}{\rm er}(A)}$$ gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator, that makes $${\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))}$$ a BV wheelgebra. In the final section, we explain how the wheeled differential operators $${\mathcal{D}(\mathcal{F}(A))}$$ produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1.