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Given a symmetric operad |$\mathcal{P}$| and a |$\mathcal{P}$| -algebra |$V$|⁠ , the associative universal enveloping algebra |${\textsf{U}_{\mathcal{P}}}$| is an associative algebra whose category of modules is isomorphic to the abelian category of |$V$| -modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case |$\mathcal{P}$| is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for |$\mathcal{P}$| is discovered. Moreover, given any symmetric operad |$\mathcal{P}$|⁠ , together with a Gröbner basis |$G$|⁠ , a condition is given in terms of the structure of the underlying trees associated with leading monomials of |$G$|⁠ , sufficient for the PBW property to hold. Examples are provided. [ABSTRACT FROM AUTHOR]

(so3) is demonstrated. The approach presented here is successful in other cases of quantum algebras and superalgebras.

In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for nonhereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

By applying the Poincaré—Birkhoff—Witt property and the Gröbner—Shirshov bases technique, we find the linear basis of the associative universal enveloping algebra in the sense of Ginzburg and Kapranov of a pair of compatible Lie brackets. We state that the growth rate of this universal enveloping over n -dimensional compatible Lie algebra equals  n + 1. [ABSTRACT FROM AUTHOR]

We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras. [ABSTRACT FROM AUTHOR]

We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld in 1986. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over C with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic 0 with an orthogonal module. Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call Barbasch-Sahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology. This theorem is a common generalization of a result called "Vogan's conjecture" for connected semisimple Lie groups which was proved by Huang and Pandˇzić; analogous results for graded affine Hecke algebras were proved by Barbasch, Ciubotaru and Trapa, and for symplectic reflection algebras and Drinfeld Hecke algebras by Ciubotaru. [ABSTRACT FROM AUTHOR]

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature. [ABSTRACT FROM AUTHOR]

We construct the basic representation of the double affine Hecke algebra at critical level q = 1 associated to an irreducible reduced affine root system R with a reduced gradient root system. For R of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights. [ABSTRACT FROM AUTHOR]

We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeld-type deformation. We fit all these deformations into a general theory, connecting Poincaré-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define a pair of symplectic Dirac operators (D+, D–) in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of ℤ/2-graded quadratic Lie algebras 픤 = 픨 + 픭 and of graded affine Hecke algebras ℍ. In these contexts, we show analogues of the Parthasarathy's formula for [D+, D–] and certain generalisations of the Casimir inequality. [ABSTRACT FROM AUTHOR]

A meson algebra is involved in the Duffin wave equation for mesons in the same way as a Clifford algebra is involved in the Dirac wave equation for electrons. Therefore meson algebras too should have geometrical properties after the manner of Grassmann. Actually it is possible to define interior multiplications with similar properties, and deformations too. Every meson algebra is a deformation of a neutral meson algebra, in the same way as (almost) every Clifford algebra is a deformation of an exterior algebra. Some applications follow: the PBW-property is proved for all meson algebras, the injectiveness of Jacobson’s diagonal morphism is proved with the minimal hypothesis, and the existence of Lipschitz monoids is established at least for meson algebras over fields. [ABSTRACT FROM AUTHOR]

Abstract: Let Σ be a set of n×n matrices with entries from a field, for n>1, and let c(Σ) be the maximum length of products in Σ necessary to linearly span the algebra it generates. Bounds for c(Σ) have been given by Paz and Pappacena, and Paz conjectures a bound of 2n−2 for any set of matrices. In this paper we present a proof of Paz’s conjecture for sets of matrices obeying a modified Poincaré–Birkhoff–Witt (PBW) property, applicable to finite dimensional representations of Lie algebras and quantum groups. A representation of the quantum plane establishes the sharpness of this bound, and we prove a bound of 2n−3 for sets of matrices with this modified PBW property which do not generate the full algebra of all n×n matrices. This bound of 2n−3 also holds for representations of Lie algebras, although we do not know whether it is sharp in this case. [Copyright &y& Elsevier]

Ginzburg, Bocklandt, and Le Bruyn defined an infinite-dimensional Lie algebra canonically associated to any quiver. Following suggestions of V. Turaev, P. Etingof, and Ginzburg, we define a cobracket and prove that it defines a Lie bialgebra structure. We then present a Hopf algebra quantizing this Lie bialgebra and prove that it is a Hopf algebra satisfying the Poincaré-Birkhoff-Witt (PBW) property. We also define representations to differential operators quantizing the trace representations of the Lie algebra defined by Ginzburg. [ABSTRACT FROM AUTHOR]

In this article, we develop the theory of commutative and cocommutative Hopf algerbas in the Verlinde category and prove that the category of affine group schemes of finite type in the Verlinde category is equivalent to the category of Harish-Chandra pairs in the Verlinde category. Subsequently, we extend this equivalence to an equivalence between corresponding representation categories. [ABSTRACT FROM AUTHOR]

We develop Gröbner–Shirshov bases technique for pre-associative (dendriform) algebras and prove a version of composition-diamond lemma. [ABSTRACT FROM PUBLISHER]

We introduce a class of hyperplane arrangements A in C n that generalise the locus configurations of Chalykh, Feigin and Veselov. To such an arrangement we associate a second order partial differential operator of Calogero–Moser type and prove that this operator is completely integrable (in the sense that its centraliser in D (C n \ A) contains a maximal commutative subalgebra of Krull dimension n). Our approach is based on the study of shift operators and associated ideals in spherical Cherednik algebras that may be of independent interest. Examples include all known completely integrable deformations of Calogero–Moser operators with rational potentials. In addition, we construct new families of examples, including a BC-type generalisation of the deformed Calogero-Moser operators recently found by Gaiotto and Rapčák. We describe these examples in a unified representation-theoretic framework of rational Cherednik algebras. [ABSTRACT FROM AUTHOR]

In the classical Batalin–Vilkovisky formalism, the BV operator Δ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by Δ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin–Vilkovisky algebras where the operator Δ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever Δ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s. [ABSTRACT FROM AUTHOR]

In this paper, we give an alternative construction of a certain class of deformed double current algebras. These algebras are deformations of |$ U(\textrm {End}(\Bbbk ^r)[x,y]) $| and they were initially defined and studied by N. Guay in his papers. Here, we construct them as algebras of endomorphisms in Deligne category. We do this by taking an ultraproduct of spherical subalgebras of the extended Cherednik algebras of finite rank. [ABSTRACT FROM AUTHOR]

We propose a new general formalism that allows us to study Poincaré–Birkhoff–Witt type phenomena for universal enveloping algebras in the differential graded context. Using it, we prove a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras. In particular, our results imply that all the previously known constructions of universal envelopes of L ∞ -algebras (due to Baranovsky, Lada and Markl, and Moreno-Fernández) represent the same object of the homotopy category of differential graded associative algebras. We also extend Quillen’s classical quasi-isomorphism C ⟶ B U from differential graded Lie algebras to L ∞ -algebras; this confirms a conjecture of Moreno-Fernández. [ABSTRACT FROM AUTHOR]

Using the theory of Macdonald [ Symmetric functions and Hall polynomials , The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q=t. We generalize this to restricted rational Cherednik algebras of wreath product groups C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [ Combinatorics, symmetric functions, and Hilbert schemes , Int. Press, Somerville, MA, 2003, pp. 39–111]. [ABSTRACT FROM AUTHOR]

Quantum Drinfeld orbifold algebras are the generalizations of Drinfeld orbifold algebras, which are obtained by replacing polynomial rings by quantum polynomial rings. In [6], the authors give necessary and sufficient conditions on a defining parameters to obtain Drinfeld orbifold algebras. In this article we generalize their result. It also simultaneously generalizes the result of [5] about quantum Drinfeld Hecke algebras. [ABSTRACT FROM AUTHOR]