Commutative Hilbertian Frobenius algebras are those commutative semigroup objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this adjoint ``comultiplication'' is a bimodule map. In this note we show that the Frobenius relation forces the multiplication operators to be normal. We then prove that these algebras have a strong Wedderburn decomposition where the (ortho)complement of the Jacobson radical or equivalently of the annihilator, is the closure of the linear span of elements which essentially are the non-trivial characters. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In particular every commutative special Hilbertian Frobenius algebra, that is, with a coisometric multiplication, is semisimple. Moreover we characterize from a setting a priori free of an involution, Ambrose's commutative H*-algebras as the underlying algebras of Hilbertian Frobenius algebras. Extending a known result in the finite-dimensional situation, we prove that the structures of such Frobenius algebras on a given Hilbert space are in one-one correspondence with its bounded above orthogonal sets. We show, moreover, that the category of commutative Hilbertian Frobenius algebras is dually equivalent to a category of pointed sets.
The category of internal coalgebras in a cocomplete category C with respect to a variety V is equivalent to the category of left adjoint functors from V into C. This can be seen best when considering such coalgebras as finite coproduct preserving functors from the dual of the Lawvere theory T_V of V, into C: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of the dual of T_V into Alg T. Since S_Mod-coalgebras in the variety R_Mod for rings R and S are nothing but left S-, right R-bimodules, the equivalence above generalizes the Eilenberg-Watts Theorem and all its previous generalizations. Generalizing and strengthening Bergman's completeness result for categories of internal coalgebras in varieties we also prove that the category of coalgebras in a locally presentable category C is locally presentable and comonadic over C and, hence, complete in particular. We show, moreover, that Freyd's canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where V is a commutative variety, are coreflectors from the category Coalg(T,V) into V.
A Hilbertian (co)algebra is defined as a (co)semigroup object in the monoidal category of Hilbert spaces. The carrier Hilbert space of such an algebra splits as an orthogonal direct sum of its Jacobson radical and the closure of the linear span of a special class of elements, the group-like elements of its adjoint coalgebra, which by the Riesz representation, correspond to closed maximal modular ideals. When the coproduct is isometric, semisimplicity is shown to be equivalent to the existence of adjoints in the sense of Ambrose’s H*-algebras. We also prove that the category of semisimple special Hilbertian algebras, that is, semisimple Hilbertian algebras with an isometric coproduct, i.e., essentially the algebras of the form l2(X) with the pointwise product, are dually equivalent to a subcategory of pointed sets and base-point preserving maps.
A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of ``topologically-free'' modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and some coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality.
Any commutative algebra equipped with a derivation may be turned into a Lie algebra under the Wronskian bracket. This provides an entirely new sort of a universal envelope for a Lie algebra, the Wronskian envelope. The main result of this paper is the characterization of those Lie algebras which embed into their Wronskian envelope as Lie algebras of vector fields on a line. As a consequence we show that, in contrast to the classical situation, free Lie algebras almost never embed into their Wronskian envelope.
This contribution mainly focuses on some aspects of Lipschitz groups, i.e., metrizable groups with Lipschitz multiplication and inversion map. In the main result it is proved that metric groups, with a translation-invariant metric, may be characterized as particular group objects in the category of metric spaces and Lipschitz maps. Moreover, up to an adjustment of the metric, any metrizable abelian group also is shown to be a Lipschitz group. Finally we present a result similar to the fact that any topological nilpotent element $x$ in a Banach algebra gives rise to an invertible element $1-x$, in the setting of complete Lipschitz groups.
It is well-known that any associative algebra becomes a Lie algebra under the commutator bracket. This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra. This correspondence may be lifted to the setting of differential (Lie) algebras. In this contribution it is shown that, also in the differential context, there is another, similar, but somewhat different, correspondence. Indeed any commutative differential algebra becomes a Lie algebra under the Wronskian bracket $W(a,b)=ab^{\prime}-a^{\prime}b$. It is proved that this correspondence again is functorial, and that it admits a left adjoint, namely the differential enveloping (commutative) algebra of a Lie algebra. Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.
It is shown that for every monoidal bi-closed category C left and right dualization by means of the unit object not only defines a pair of adjoint functors, but that these functors are monoidal as functors from C^{op}, the dual monoidal category of C into the transposed monoidal category C^t. We, thus, generalize the case of a symmetric monoidal category, where this kind of dualization is a special instance of convolution. We apply this construction to the monoidal category of bimodules over a not necessarily commutative ring R and so obtain various contravariant dual ring functors defined on the category of R-corings. It becomes evident that previous, hitherto apparently unrelated constructions of this kind are all special instances of our construction and, hence, coincide. Finally we show that Sweedler's Dual Coring Theorem is a simple consequence of our approach and that these dual ring constructions are compatible with the processes of (co)freely adjoining (co)units.
Even in spaces of formal power series is required a topology in order to legitimate some operations, in particular to compute infinite summations. In general the topologies considered are just a product of the topology of the base field, an inverse limit topology or a topology induced by a pseudo-valuation. As our main result we prove the following phenomenon: the (left and right) topological dual spaces of formal power series equipped with the product topology with respect to any Hausdorff division ring topology on the base division ring, are all the same, namely just the space of polynomials. As a consequence, this kind of rigidity forces linear maps, continuous with respect to any (and then to all) those topologies, to be defined by very particular infinite matrices similar to row-finite matrices.
The adjunction of a unit to an algebraic structure with a given binary associative operation is discussed by interpreting such structures as semigroups and monoids respectively in a monoidal category. This approach then allows for results on the adjunction of counits to coalgebraic structures with a binary co-associative co-operation as well. Special attention is paid to situations where a given coalgebraic structure induces a ``dual" algebraic one; here the compatibility of adjoining (co)units and dualization is examined. The extension of this process to starred algebraic structures and to monoid actions is discussed as well. Particular emphasis is given to examples from many areas of mathematics.
It is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular character theory and the appropriate Fourier transform for some particular kind of finite Abelian groups. Moreover we introduce the notion of bent functions for finite field valued functions rather than usual complex-valued functions, and we study several of their properties.
On any set $X$ may be defined the free algebra $R\langle X\rangle$ (respectively, free commutative algebra $R[X]$) with coefficients in a ring $R$. It may also be equivalently described as the algebra of the free monoid $X^*$ (respectively, free commutative monoid $\mathpzc{M}(X)$). Furthermore, the algebra of differential polynomials $R\{X\}$ with variables in $X$ may be constructed. The main objective of this contribution is to provide a functorial description of this kind of objects with their relations (including abelianization and unitarization) in the category of differential algebras, and also to introduce new structures such as the differential algebra of a semigroup, of a monoid, or the universal differential envelope of an algebra.
The set of natural integers is fundamental for at least two reasons: it is the free induction algebra over the empty set (and at such allows definitions of maps by primitive recursion) and it is the free monoid over a one-element set, the latter structure being a consequence of the former. In this contribution, we study the corresponding structure in the linear setting, i.e. in the category of modules over a commutative ring rather than in the category of sets, namely the free module generated by the integers. It also provides free structures of induction algebra and of monoid (in the category of modules). Moreover we prove that each of its linear endomorphisms admits a unique normal form, explicitly constructed, as a non-commutative formal power series.
A locally finite category is defined as a category in which every arrow as only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some field may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover both groups are actually pro-affine algebraic groups. In this contribution we study the Hopf algebras of coordinate functions on these groups and show that the semi-direct product structure obtained from the action of reversible series on invertible series by (anti-)automorphisms gives rise to an interaction at the level of their Hopf algebras of coordinate functions under the form of a smash coproduct.
An operator on formal power series of the form $S\mapsto \mu S(\sigma)$, where $\mu$ is an invertible power series, and $\sigma$ is a series of the form $\t+\mathcal{O}(\t^2)$ is called a unipotent substitution with pre-function. Such operators, denoted by a pair $(\mu,\sigma)$, form a group. The objective of this contribution is to show that it is possible to define a generalized powers for such operators, as for instance fractional powers $(\mu,\sigma)^{\frac{a}{b}}$ for every $\frac{a}{b}\in\mathbb{Q}$.
The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope - its universal enveloping algebra - as a sub-Lie algebra for the usual commutator Lie bracket. However there is another functorial way - less known - to associate a Lie algebra to an associative algebra, and inversely. Any commutative algebra equipped with a derivation $a\mapsto a^{\prime}$, i.e., a commutative differential algebra, admits a Wronskian bracket $W(a,b)=ab^{\prime}-a^{\prime}b$ under which it becomes a Lie algebra. Conversely, to any Lie algebra is universally associated a commutative differential algebra, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.
The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial, and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski's theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between non commutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the later is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was even not mention by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a non convergent rewriting system, outside the scope of Brylawski's work.
Perfect nonlinear functions from a finite group $G$ to another one $H$ are those functions $f: G \rightarrow H$ such that for all nonzero $\alpha \in G$, the derivative $d_{\alpha}f: x \mapsto f(\alpha x) f(x)^{-1}$ is balanced. In the case where both $G$ and $H$ are Abelian groups, $f: G \rightarrow H$ is perfect nonlinear if and only if $f$ is bent i.e for all nonprincipal character $\chi$ of $H$, the (discrete) Fourier transform of $\chi \circ f$ has a constant magnitude equals to $|G|$. In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where $G$ and/or $H$ are (finite) non Abelian groups. Thus we extend the concept of bent functions to the framework of non Abelian groups.
The purpose of this paper is to present extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.
Starting with the Heisenberg-Weyl algebra, fundamental to quantum physics, we first show how the ordering of the non-commuting operators intrinsic to that algebra gives rise to generalizations of the classical Stirling Numbers of Combinatorics. These may be expressed in terms of infinite, but row-finite, matrices, which may also be considered as endomorphisms of $\mathbb{C}[[x]]$. This leads us to consider endomorphisms in more general spaces, and these in turn may be expressed in terms of generalizations of the ladder-operators familiar in physics.
The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the Möbius functions of a monoid and its Rees quotient are also provided.
The concern of this paper is a famous combinatorial formula known under the name "exponential formula". It occurs quite naturally in many contexts (physics, mathematics, computer science). Roughly speaking, it expresses that the exponential generating function of a whole structure is equal to the exponential of those of connected substructures. Keeping this descriptive statement as a guideline, we develop a general framework to handle many different situations in which the exponential formula can be applied.
A partial monoid $P$ is a set with a partial multiplication $\times$ (and total identity $1_P$) which satisfies some associativity axiom. The partial monoid $P$ may be embedded in a free monoid $P^*$ and the product $\star$ is simulated by a string rewriting system on $P^*$ that consists in evaluating the concatenation of two letters as a product in $P$, when it is defined, and a letter $1_P$ as the empty word $\epsilon$. In this paper we study the profound relations between confluence for such a system and associativity of the multiplication. Moreover we develop a reduction strategy to ensure confluence and which allows us to define a multiplication on normal forms associative up to a given congruence of $P^*$. Finally we show that this operation is associative if, and only if, the rewriting system under consideration is confluent.
The Riordan group is the semi-direct product of a multiplicative group of invertible series and a group, under substitution, of non units. The Riordan near algebra, as introduced in this paper, is the Cartesian product of the algebra of formal power series and its principal ideal of non units, equipped with a product that extends the multiplication of the Riordan group. The later is naturally embedded as a subgroup of units into the former. In this paper, we prove the existence of a formal calculus on the Riordan algebra. This formal calculus plays a role similar to those of holomorphic calculi in the Banach or Fréchet algebras setting, but without the constraint of a radius of convergence. Using this calculus, we define "en passant" a notion of generalized powers in the Riordan group.
Due to implementation constraints the XOR operation is widely used in order to combine plaintext and key bit-strings in secret-key block ciphers. This choice directly induces the classical version of the differential attack by the use of XOR-kind differences. While very natural, there are many alternatives to the XOR. Each of them inducing a new form for its corresponding differential attack (using the appropriate notion of difference) and therefore block-ciphers need to use S-boxes that are resistant against these nonstandard differential cryptanalysis. In this contribution we study the functions that offer the best resistance against a differential attack based on a finite field multiplication. We also show that in some particular cases, there are robust permutations which offers the best resistant against both multiplication and exponentiation base differential attacks. We call them doubly perfect nonlinear permutations.
The left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been introduced. In this paper we show that this generalized concept of nonlinearity is actually equivalent to a new bentness notion that deals with functions defined on a finite Abelian group G that acts on a finite set X and with values in the finite-dimensional vector space of complex-valued functions defined on X.
A function from a finite Abelian group $G$ and with values in the unit circle $T$ of the complex field is called bent if its Fourier transform (i.e., the decomposition of $f$ in the basis of characters of $G$) has a constant magnitude equal to the number of elements of $G$. In this contribution we define a modulo 2 notion of characters by allowing the characters of an elementary finite Abelian p-group $G$ to take their values in the multiplicative group $GF(2^n)^*$ (with $p = 2^n − 1$) of the roots of the unity in the finite field $GF(2^n$ ) with $2^n$ elements rather than in the complex roots of the unity $T$. We show that this kind of characters forms an orthogonal basis of the $GF(2^n )$-vector space of functions from $G$ to $GF(2^n)$ that permits us to define a modulo 2 version of the Fourier transform (as a decomposition of a vector in this basis of characters). We show that many classical properties of the Fourier transform still hold for this characteristic 2 version. In particular, we can define an appropriate notion of bent functions, called $GF(2^n)$-bent functions, with respect to this Fourier transform. Finally we construct a class of $GF(2^n)$-bent functions and we also study their relations with classical and group action versions of perfect nonlinearity.
Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks.We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both $\Z_2$-valued and $\Z_2^a$-valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from $\Z_2^a$ to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.
The objective of this contribution is to introduce an analogue to the classical secretkey block ciphers, such as DES, IDEA or AES, in the nondenumerable setting, namely where cleartexts, plaintexts and keys are real numbers. The nonlinear part of traditional secretkey block ciphers, the S-boxes, is designed to provide confusion, i.e., to resist to several kind of cryptanalysis such as algebraic, differential or linear attacks. By analogy we construct S-boxes in the uncountable setting which provide the best resistance to a classical or modified version of the differential attack. Since our S-boxes are real-valued functions defined on the real-line, we also need to prevent possible new attacks based on real analysis (such as continuity and derivability), which are ignored since impossible in the finite case: we must hide the topological structure. So we introduce a new kind of discontinuous boxes for this purpose.
Bent or perfect nonlinear Boolean functions represent the best resistance against the so-called linear and differential cryptanalysis. But this kind of cryptographic relevant functions only exists when the number of input bits m is an even integer and is larger than the double of the number of output bits n. Unfortunately the non-existence cases, the odd dimension (m is an odd integer) or the plane dimension ( m = n ), are not illegitimate from a cryptographic point of view and even commonly considered. New notions of bentness and perfect nonlinearity are then needed in those impossible cases for the traditional theory. In this paper, by replacing the usual XOR by another kind of bit-strings combination, we explicitly construct new “bent” Boolean functions in traditionally impossible cases.
We introduce the notion of a bent function on a finite nonabelian group which is a natural generalization of the well-known notion of bentness on a finite abelian group due to Logachev, Salnikov and Yashchenko. Using the theory of linear representations and noncommutative harmonic analysis of finite groups we obtain several properties of such functions similar to the corresponding properties of traditional Abelian bent functions.
The concept of bent functions, originally introduced by Dillon and Rothaus, is very relevant in cryptography because this kind of functions represents the maximal resistance against the so-called linear cryptanalysis. In 1997, Logachev, Salnikov and Yashchenko described a fundamental notion of bentness for functions defined on a finite Abelian group G with values in the unit circle of the complex field. in this paper, by replacing this unit circle by the unit hypersphere S_H(0_H,1) of an arbitrary finite-dimensional Hermitian space H, we develop a generalization of the concept of bentness for S_H(0_H,1)-valued functions defined on G, called multidimensional bent functions.
In a recent paper, we generalized the notion of perfect nonlinearity of boolean functions by replacing the translations of a vector space on $\mathbb{F}_2$ by an Abelian group of fixed-point free involutions acting regularly on this vector space. We now show this generalization to be still valid when considering a finite nonempty set $X$ rather than a vector space on $\mathbb{F}_2$ and a faithful or regular action of a finite Abelian group $G$ on $X$. We also develop a dual characterization of this new concept through the Fourier transform as for the classical notion of perfect nonlinearity. By considering faithful actions we highlight a fundamental concept underlying to perfect nonlinearity that extends the classical notions. In short we integrate the traditional concepts within a more general and primitive framework.
The notions of perfect nonlinearity and bent functions are closely dependent on the action of the group of translations over $\mathbb{F}_2^m$. Extending the idea to more generalized groups of involutions without fixed points gives a larger framework to the previous notions. In this paper we largely develop this concept to define $G$-perfect nonlinearity and $G$-bent functions, where $G$ is an Abelian group of involutions, and to show their equivalence as in the classical case.