Non-Commutative Algebras and Quantum Structures

We present a survey on pseudo-effect algebras and pseudo MV-algebras, which generalize effect algebras and MV-algebras by dropping the assumption on commutativity. A non-commutative logic is nowadays used even in programming languages. We show when a pseudo-effect algebra E is an interval in a unital po-group. This is possible, e.g. if E satisfies a Riesz-type decomposition property, i.e. another kind of distributivity with respect to addition. Every pseudo MV-algebra is an interval in a unital ℓ-group. We study a case when compatibility can be expressed by a pseudo MV-structure, i.e. when E can be covered by blocks being pseudo MV-algebras. Finally, we study the state space of such structures.     

Atomic Effect Algebras with the Riesz Decomposition Property

We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.     

Pseudo-Effect Algebras and Pseudo-Difference Posets

In this paper, we introduce two different operations in pseudo-effect algebras and also introduce the pseudo-difference posets. We prove that the pseudo-effect algebras and the pseudo-difference posets are the same thing.     

Ideals and Filters in Pseudo-Effect Algebras

In this paper, we show that the filters and local filters are equivalent in pseudo-effect algebras. Ideals and local ideals and generalized ideals are equivalent in the pseudo-effect algebras, too.     

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.     

Quantum and Classical Implication Algebras with Primitive Implications

Join in an orthomodular lattice is obtained inthe same form for all five quantum implications. Theform holds for the classical implication in adistributive lattice as well. Even more, the definition added to an ortholattice makes it orthomodularfor quantum implications and distributive for theclassical one. Based on this result a quantumimplication algebra with a single primitive — andin this sense unique — implication is formulated. Acorresponding classical implication algebra is alsoformulated. The algebras are shown to be special casesof a universal implication algebra.     

Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs

To directed graphs with unique sink and source we associate a noncommutative associative algebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper. To the memory of Felix Alexandrovich Berezin. Vladimir Retakh was partially supported by NSA     

Pre-Poisson Algebras

A definition of pre-Poisson algebras is proposed, combining structures of pre-Lie and zinbiel algebra on the same vector space. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra by passing to the corresponding Lie and commutative products. Analogs of basic constructions of Poisson algebras (through deformations of commutative algebras, or from filtered algebras whose associated graded algebra is commutative) are shown to hold for pre-Poisson algebras. The Koszul dual of pre-Poisson algebras is described. It is explained how one may associate a pre-Poisson algebra to any Poison algebra equipped with a Baxter operator, and a dual pre-Poisson algebra to any Poisson algebra equipped with an averaging operator. Examples of this construction are given. It is shown that the free zinbiel algebra (the shuffle algebra) on a pre-Lie algebra is a pre-Poisson algebra. A connection between the graded version of this result and the classical Yang–Baxter equation is discussed.     

Uniqueness and Order in Sequential Effect Algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product is defined. We present examples of effect algebras that admit a unique, many and no sequential product. Some general theorems concerning unique sequential products are proved. We discuss sequentially ordered SEAs in which the order is completely determined by the sequential product. It is demonstrated that intervals in a sequential ordered SEA admit a sequential product.     

Second Quantized Frobenius Algebras

We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n^th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n^th tensor powers and the natural permutation action. We also show that naturally graded symmetric group twisted Frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super–grading. Furthermore we discuss several notions of discrete torsion and show that indeed a non–trivial discrete torsion leads to a non–trivial super structure on the second quantization.This work was partially supported by NSF grant #0070681.     

Generalized EMV-Effect Algebras

Recently in Dvure?enskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.     

Inhomogeneous Yang-Mills Algebras

We determine all inhomogeneous Yang–Mills algebras and super Yang–Mills algebras which are Koszul. Following a recent proposal, a non-homogeneous algebra is said to be Koszul if the homogeneous part is Koszul and if the PBW property holds. In this letter, the homogeneous parts are the Yang–Mills algebra and the super Yang–Mills algebra.     

Proper Effect Algebras Admitting No States

We show that there is even a finite proper effect algebra admitting no states. Further, every lattice effect algebra with an ordering set of valuations is an MV effect algebra (consequently it can be organized into an MV algebra). An example of a regular effect algebra admitting no ordering set of states is given. We prove that an Archimedean atomic lattice effect algebra is an MV effect algebra iff it admits an ordering set of valuations. Finally we show that every nonmodular complete effect algebra with trivial center admits no order-continuous valuations.     

Orthomodular Implication Algebras

In Boolean algebras the properties of the implication operation can be modeled by a so-called implication algebra that itself can be considered as a join-semilattice with 1 whose principal filters are Boolean algebras. This situation is generalized from Boolean algebras to orthomodular lattices.     

The Associative Algebras of Conformal Field Theory

Modulo the ideal generated by the derivative fields, the normal ordered product of holomorphic fields in two-dimensional conformal field theory yields a commutative and associative algebra. The zero mode algebra can be regarded as a deformation of the latter. Alternatively, it can be described as an associative quotient of the algebra given by a modified normal ordered product. We clarify the relation of these structures to Zhu's product and Zhu's algebra of the mathematical literature.     

Matrix Algebras in Non-Hermitian Quantum Mechanics

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators.
These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.     

Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy. J. Chuang is supported by an EPSRC advanced research fellowship. A. Lazarev is partially supported by an EPSRC research grant.     

The Lexicographic Product of Po-groups and n-Perfect Pseudo Effect Algebras

We study the existence of different types of the Riesz Decomposition Property for the lexicographic product of two partially ordered groups. Special attention is paid to the lexicographic product of the group of integers with an arbitrary po-group. Then we apply these results to the study of n-perfect pseudo effect algebras. We show that the category of strong n-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups with RDP₁.     

S-Dominating Effect Algebras

A special type of effect algebra called anS-dominating effect algebra is introduced. It is shownthat an S-dominating effect algebra P has a naturallydefined Brouwer-complementation that gives P thestructure of a Brouwer–Zadeh poset. This enables usto prove that the sharp elements of P form anorthomodular lattice. We then show that a standardHilbert space effect algebra is S-dominating. Weconclude that S-dominating effect algebras may be usefulabstract models for sets of quantum effects in physicalsystems.     

Transitive Novikov Algebras on Four-Dimensional Nilpotent Lie Algebras

Novikov algebras were introduced in connection with the Poisson brackets (of hydrodynamic type) and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra, and the radical of a finite-dimensional Novikov algebra is transitive. In this paper, we give a classification of transitive Novikov algebras on four-dimensional nilpotent Lie algebras based on Kim (1986, Journal of Differential Geometry 24, 373–394).