Poincare-Birkhoff-Witt Research Articles

Cluster realisations of ıquantum$\imath {\rm quantum}$ groups of type AI

AbstractThe group of type is a coideal subalgebra of the quantum group , associated with the symmetric pair . In this paper, we give a cluster realisation of the algebra . Under such a realisation, we give cluster interpretations of some fundamental constructions of , including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of , which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form.

Laurent family of simple modules over quiver Hecke algebras

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring $\mathcal {A}_q(\mathfrak {n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition vector of a simple module $X$ and categorical interpretation of (co)degree of $[X]$. As applications of such $\mathbb {Z}\mspace {1mu}$-vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $\Lambda$-invariants of $R$-matrices in the quiver Hecke algebra theory.

From quantum loop superalgebras to super Yangians

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra glM|N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra Uq(LglM|N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of Uq(LglM|2n) with respect to the comultiplication.

Lie pairs

Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``$\preceq$-morphisms'' preserving a surpassing relation $\preceq$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.

The Generators and Relations for Little q-Schur Algebra uk(3,5)

In this paper, we investigate the presentation for little q-Schur algebra [see formula in PDF] at odd roots. The Poincaré-Birkhoff-Witt (PBW) basis of [see formula in PDF] is obtained through the basis of q-Schur algebra [see formula in PDF]. Then we present a new set for generators and relations for [see formula in PDF] by PBW basis.

Filtrations on block subalgebras of reduced enveloping algebras

We study the interaction between the block decompositions of reduced enveloping algebras in positive characteristic, the Poincaré-Birkhoff-Witt (PBW) filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra [Formula: see text] of the restricted universal enveloping algebra [Formula: see text] and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighborhood of [Formula: see text] in the nilpotent cone and the coinvariants algebra corresponding to [Formula: see text]. In the case of [Formula: see text] in characteristic [Formula: see text] we determine the associated graded algebras of these filtrations on block subalgebras of [Formula: see text]. We also apply this to determine the structure of the adjoint representation of [Formula: see text].

The alternating presentation of [formula omitted] from Freidel-Maillet algebras

An infinite dimensional algebra denoted A¯q that is isomorphic to a central extension of Uq+ - the positive part of Uq(sl2ˆ) - has been recently proposed by Paul Terwilliger. It provides an ‘alternating’ Poincaré-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of A¯q in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for A¯q are constructed; (b) explicit isomorphisms from A¯q to certain Drinfeld type ‘alternating’ subalgebras of Uq(gl2ˆ) are obtained; (c) the image in Uq+ of all the generators of A¯q in terms of Damiani's root vectors is obtained. A new tensor product decomposition for Uq(sl2ˆ) in terms of Drinfeld type ‘alternating’ subalgebras follows. The specialization q→1 of A¯q is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra.

Gröbner-Shirshov basis of derived Hall algebra of type $\boldsymbol~B_{\boldsymbol2}$

In the Ringel-Hall algebra of Dynkin type, the set of all skew commutator relations among the iso-classes of indecomposable modules forms aminimal Gröbner-Shirshov basis and the corresponding irreducible elements form a PBW (Poincaré-Birkhoff-Witt) type basis of the Ringel-Hall algebra. We aim to generalize this result to the derived Hall algebra $DH(B_2)$. First, we compute all the skew commutator relations among the iso-classes of indecomposable objects in the bounded derived category $D^b(B_2)$, and then we prove that all the possible compositions among these skew commutator relations are trivial. Finally, we construct a PBW type basis of $DH(B_2)$.

THE PBW THEOREM FOR AFFINE YANGIANS

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.

Endofunctors and Poincaré–Birkhoff–Witt Theorems

AbstractWe determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a PBW property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.

Free Gelfand–Dorfman–Novikov superalgebras and a Poincaré–Birkhoff–Witt type theorem

We construct linear bases of free Gelfand–Dorfman–Novikov (GDN) superalgebras. As applications, we prove a Poincaré–Birkhoff–Witt (PBW) type theorem, that is, every GDN superalgebra can be embedded into its universal enveloping associative differential supercommuative algebra. An Engel theorem is given.

Skew Clifford algebras

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the Z2-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.

Weighted PBW degenerations and tropical flag varieties

We study algebraic, combinatorial and geometric aspects of weighted Poincaré–Birkhoff–Witt (PBW)-type degenerations of (partial) flag varieties in type [Formula: see text]. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone, which can be identified with a maximal cone in the tropical flag variety. Varying the degree function in the cone, we recover, for example, the classical flag variety, its abelian PBW degeneration, some of its linear degenerations and a particular toric degeneration.

PBW deformations of quadratic monomial algebras

A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincaré-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked. Several examples are also provided. Finally, with these tools, we show that each quadratic monomial algebra admits a nontrivial PBW deformation.

THE CURVED A∞-COALGEBRA OF THE KOSZUL CODUAL OF A FILTERED DG ALGEBRA

AbstractThe goal of this article is to study the coaugmented curvedA∞-coalgebra structure of the Koszul codual of a filtered dg algebra over a fieldk. More precisely, we first extend one result of B. Keller that allowed to compute theA∞-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term isk. We then show how to compute the coaugmented curvedA∞-coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of anN-Koszul algebra.

Factorization of Graded Traces on Nichols Algebras

A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize.

Comprehensive Gröbner systems in PBW algebras, Bernstein–Sato ideals and holonomic D-modules

A computation method of comprehensive Gröbner systems is introduced in Poincaré–Birkhoff–Witt (PBW) algebras and applications to Bernstein–Sato ideals, holonomic D-modules for parametric cases are considered. The proposed method provides in particular holonomic D-modules associated with primary components of Bernstein–Sato ideals. Furthermore, with our implementation, effective methods are illustrated for computing b-functions of μ-constant deformations of hypersurface isolated singularities by utilizing holonomic D-modules. High versatility of the proposed method is also illustrated by several examples.

Non-commutative algebraic geometry of semi-graded rings

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

On Dunkl angular momenta algebra

We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.