Integral Bases for the Universal Enveloping Algebras of Cartan Map Superalgebras

Let ?? be a finite dimensional complex simple Lie superalgebra of Cartan type and A be a commutative, associative algebra with unity over ?. We refer to the Lie superalgebras of the form ?? ? A as Cartan map superalgebras. In this paper, following Bagci and Chamberlin (J. of Pure and Applied Algebra 218(8), 1563–1576, 2014), we define an integral form for the universal enveloping algebra of the Cartan map superalgebras, and exhibit an explicit integral basis for this integral form.     

BCC-algebras and residuated partially-ordered groupoid

The aim of the paper is to investigate the relationship between BCC-algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC-algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group.     

Integral bases for the universal enveloping algebras of map superalgebras

Let g$\mathfrak{g}$ be a finite dimensional complex simple classical Lie superalgebra and A   be a commutative, associative algebra with unity over C$\mathbb{C}$. In this paper we define an integral form for the universal enveloping algebra of the map superalgebra g⊗A$\mathfrak{g}\otimes A$, and exhibit an explicit integral basis for this integral form.     

Generalized Serre relations for Lie algebras associated with positive unit forms

Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebra and a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre’s Theorem [J.-P. Serre, Complex Semisimple Lie Algebras (G.A. Jones, Trans.), Springer-Verlag, New York, 1987] gives then a representation of the given Lie algebra in generators and relations in terms of the Cartan matrix.In this work, we generalize Serre’s Theorem to give an explicit representation in generators and relations for any simply laced semisimple Lie algebra in terms of a positive quasi-Cartan matrix. Such a quasi-Cartan matrix expresses the dual of the Killing form for a Z-base of roots. Here, by a Z-base of roots, we mean a set of linearly independent roots which generate all roots as linear combinations with integral coefficients.     

Levels and sublevels of algebras obtained by the Cayley–Dickson process

In this paper, we generalize the concepts of level and sublevel of a composition algebra to algebras obtained by the Cayley–Dickson process and we will show that, in the case of level for algebras obtained by the Cayley–Dickson process, the situation is the same as for the integral domains, proving that for any positive integer n, there is an algebra A obtained by the Cayley–Dickson process with the norm form anisotropic over a suitable field, which has the level ${n \in \mathbb{N}-\{0\}}$ .     

Représentation de Weil associée à une représentation d'une algèbre de Jordan

In the present Note we construct the Weil representation of the Kantor-Koecher-Tits Lie algebra g associated to a simple real Jordan algebra V. Afterwards we introduce a family of integral operators intertwining the Weil representation with the infinitesimal representations of the degenerate principal series of the conformai group of a Jordan algebra V. We apply this result to the case conformal group SL(2r, ℝ).     

Fonctions holomorphes cliffordiennes

Let be the Clifford algebra of with a quadratic form of negative signature, D = e_i ∂/∂x_i, Δ the ordinary Laplacian. The holomorphic cliffordian functions are solutions of D Δ^mƒ = 0. We study the polynomial and singular solutions, representation integral formulas, and the foundation of the Cliffordian elliptic function theory.     

Invariants of the action of a semisimple Hopf algebra on PI-algebra

We extend several classical results in the theory of invariants of finite groups to the case of action of a finite-dimensional Hopf algebra H on an algebra satisfying a polynomial identity. In particular, we prove that an H-module algebra A over an algebraically closed field k is integral over the subalgebra of invariants, if H is a semisimple and cosemisimple Hopf algebra. We show that for char k > 0, the algebra Z\({\left( A \right)^{{H_0}}}\) is integral over the subalgebra of central invariants Z(A)^H, where Z(A) is the center of algebra A, H⁰ is the coradical of H. This result allowed us to prove that the algebra A is integral over the subalgebra Z(A)^H in some special case. We also construct a counterexample to the integrality of the algebra \({A^{{H_0}}}\) over the subalgebra of invariants A^H for a pointed Hopf algebra over a field of non-zero characteristic.     

Lattice-Ordered Matrix Rings Over Totally Ordered Rings

For an n ×n matrix algebra over a totally ordered integral domain, necessary and sufficient conditions are derived such that the entrywise lattice order on it is the only lattice order (up to an isomorphism) to make it into a lattice-ordered algebra in which the identity matrix is positive. The conditions are then applied to particular integral domains. In the second part of the paper we consider n ×n matrix rings containing a positive n-cycle over totally ordered rings. Finally a characterization of lattice-ordered matrix ring with the entrywise lattice order is given.     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

Presenting Schur Algebras as Quotients of the Universal Enveloping Algebra of gl2

We give a presentation of the Schur algebras S _Q (2,d) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S _Q (2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis.     

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

We consider the C*-algebra generated by multidimensional integral operators with (?n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|^iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.     

Generalized Boolean Algebras in Lattice-Ordered Groups

In this paper, characterizations are given for the free lattice-ordered group over a generalized Boolean algebra and the freel -module of a totally ordered integral domain with unit over a generalized Boolean algebra. Extensions of lattice-ordered groups using generalized Boolean algebras are defined and their properties studied.     

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

The Elements in Crystal Bases Corresponding to Exceptional Modules

According to the Ringel-Green theorem, the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum group, in which the BGP- reflection functors coincide with Lusztig＇s symmetries. It is first asserted that the elements corresponding to exceptional modules lie in the integral generic composition algebra, hence in the integral form of the quantum group. Then it is proved that these elements lie in the crystal basis up to a sign. Eventually, it is shown that the sign can be removed by the geometric method. The results hold for any type of Cartan datum.     

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Let g be a finite-dimensional simple Lie algebra and let Sg be the locally finite part of the algebra of invariants (EndCV⊗Sg(g)) where V is the direct sum of all simple finite-dimensional modules for g and S(g) is the symmetric algebra of g. Given an integral weight ξ, let Ψ=Ψ(ξ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra of Sg and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger's equation

A Banach algebra A of functionals on C[a, b] is introduced and it is proved that the operator-valued Feynman integral recently defined by Cameron and Storvick exists for functionals in A. Two existence theorems of Cameron and Storvick are seen to be special cases of this result; in fact, even in these cases, the present theorem gives improved results.Cameron and Storvick have used their function space integral to give a solution to an integral equation formally equivalent to Schroedinger's equation; using our existence theorem, we give a relatively brief and transparent proof of this result.     

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.     

On Isotypic Decompositions for Non-Semisimple Hopf Algebras

In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra. For a large class of Hopf algebras, these are shown to form a complete set of orthogonal idempotents. We give an example which illustrates that the Chevalley property is crucial.