The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum $n$-space. A kind of braided category $\Cal {GB}$ of $\La$-graded $þ$-commutative associative algebras over a field $k$ is established. The quantum divided power algebra over $k$ related to the quantum $n$-space is introduced and described as a braided Hopf algebra in $\Cal {GB}$ (in terms of its 2-cocycle structure), over which the so called special $q$-derivatives are defined so that several new interesting quantum groups, especially, the quantized polynomial algebra in $n$ variables (as the quantized universal enveloping algebra of the abelian Lie algebra of dimension $n$), and the quantum group associated to the quantum $n$-space, are derived from our approach independently of using the $R$-matrix. As a verification of its validity of our discussion, the quantum divided power algebra is equipped with a structure of $U_q(\frak {sl}_n)$-module algebra via a certain $q$-differential operators realization. Particularly, one of the four kinds of roots vectors of $U_q(\frak {sl}_n)$ in the sense of Lusztig can be specified precisely under the realization.
The paper explores the indecomposable submodule structures of quantum divided power algebra $\mathcal{A}_q(n)$ defined in \cite{HU} and its truncated objects $\mathcal{A}_q(n, \bold m)$. An "intertwinedly-lifting" method is established to prove the indecomposability of a module when its socle is non-simple. The Loewy filtrations are described for all homogeneous subspaces $\mathcal{A}^{(s)}_q(n)$ or $\mathcal{A}_q^{(s)}(n, \bold m)$, the Loewy layers and dimensions are determined. The rigidity of these indecomposable modules is proved. An interesting combinatorial identity is derived from our …
In this note we prove that the explicit realization of arbitrary complex powers of generators of quantum group $U_{q}(\mathfrak{sl}(2))$ satisfies all the commutation relations of the algebra of complex powers, including the generalized Kac's identity which was announced in our previous paper. It turns out that the latter identity in this realization is equivalent to $6-9$ integral identity on quantum dilogarithm.
We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhauser to describe triangular decompositions.
In this paper we study the integral form of the lattice vertex algebra $V_L$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $\mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.
Let $\mathcal{B}_\mathfrak{q}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q}$. We consider the graded dual $\mathcal{L}_{\mathfrak{q}}$ of the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}_{\mathfrak{q}}$ from [A3] and the divided powers algebra $\mathcal{U}_{\mathfrak{q}}$, a suitable Drinfeld double of $\mathcal{L}_{\mathfrak{q}} # \mathbf{k} \mathbb{Z}^θ$. We provide basis and presentations by generators and relations of $\mathcal{L}_{\mathfrak{q}}$ and $\mathcal{U}_{\mathfrak{q}}$, and prove that they are noetherian and have f…
Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $α\in R$, $v,w\in W$, we denote by $\partial_α$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew divided difference operators acting on the coinvariant algebra of $W$. Generalizing the work of Liu, we prove that $\partial_{w/v}$ can be written as a polynomial with nonnegative coefficients in $\partial_α$ where $α\in R^+$. In fact, we prove the stronger and analogous statement in the Nichols-Woronowicz algebra model for Schubert calculus on $W$ after Bazlov. We draw …
We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type $A,$ we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated…
We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{iπ/p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$ by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $U_q sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane.
In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig's quantum group of divided powers. We do so by writing down non-obvious primitive elements with the correct adjoint action. As application we explain how this gives a realization of the unrolled quantum group as operators on a conformal feld theory and match some calculations on this side. In particular our results explain a prominent weight shift that appears in [FT10]. Our result extends to other Nichols algebras of diagonal type, including super Lie algebras.
We clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the RSK construction for quiver Hecke algebras of type A, that was recently imported from the setting of representations of p-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules, that categorifies the Berenstein-Zelevinsky strings framework on quantum groups, and generalizes a graded variant of the classical Bernstein-Zelevinsky derivatives from the p-adic setting. Graded cyclotomic decomposition numbers are shown to be a special s…
The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are re…
The aim of the paper is twofold. First, we introduce analogs of (partial) derivatives on certain Noncommutative algebras, including some enveloping algebras and their "braided counterparts", namely, the so-called modified Reflection Equation algebras. By using these derivatives we construct an analog of the de Rham complex on these algebras. Second, we discuss deformation property of some quantum algebras and show that contrary to a commonly held view, in the so-called q-Witt algebra there is no analog of the PBW theorem. In this connection, we discuss different forms of the Jacobi condition r…
For q generic or a primitive l-th root of unity, q-Witt algebras are described by means of q-divided power algebras. The structure of the universal q-central extension of the q-Witt algebra, the q-Virasoro algebra, is also determined. q-Lie algebras are investigated and the q-PBW theorem for the universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of the q-Witt algebras is given. Based on it, the q-holomorph structure for the q-Witt algebras is constructed, which interprets the realization in the context of representation theory.
Consider any representation $ϕ$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $Λ(g)$. We show that the representation $ϕ$ extends canonically to the representation $\tildeϕ$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncomm…
Two approaches to the tangent space of a noncommutative space whose coordinate algebra is the enveloping algebra of a Lie algebra are known: the Heisenberg double construction and the approach via deformed derivatives, usually defined by procedures involving orderings among noncommutative coordinates or equivalently involving realizations via formal differential operators. In an earlier work, we rephrased the deformed derivative approach introducing certain smash product algebra twisting a semicompleted Weyl algebra. We show here that the Heisenberg double in the Lie algebra case, is isomorphi…
We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra $S$ (the Landweber-Novikov algebra) acting on its dual Hopf algebra $S^*$ with a distinguished ``topologically integral'' part $Λ$ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of $S$. We introduce one-dimensional representations of a Hopf algebra. We give series of examples of such representati…
Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $ U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley generators over a suitable subring $ R $ of $ K(q) \, $. We prove a Poincaré-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $, yielding a basis over $ R $ made of ordered products of $ q $-divided powers of suitable quantum root vectors.
Quantum calculus based on the right invertible divided difference operator $D_σ^τ$ is proposed here in context of algebraic analysis \cite{DPR}. The linear operator $D_σ^τ$, specified with the help of two fixed maps $σ\;, τ\colon M\rightarrow M$, generalizes the quantum derivative operator used in $h$- or $q$-calculus \cite{kac}. In the domain of $D_σ^τ$ there are special elements defined as $D_σ^τ$-polynomials and the corresponding Taylor formula is proved.
We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over…
Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series over it. This gives another proof of the theorem that the algebra of quasi-symmetric functions is free over the integers.
We develop a calculus of variations for functionals on certain spaces of conformal maps. Such a space Ω is composed of all maps that are conformal on domains containing a fix compact annular set of the Riemann sphere, and that are "sense-preserving". The calculus of variations is based on describing infinitesimal variations of such maps using vector fields. We show that derivatives of all orders with respect to such conformal maps, upon conjugation by an appropriate functional, give rise to the structure of the Virasoro vertex operator algebra. Our construction proceeds in three steps. We firs…
