Enveloping algebra is a Yetter–Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra

Given a Hopf algebra [Formula: see text], Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld [Formula: see text]-module algebra [Formula: see text] gives rise to an associative [Formula: see text]-bialgebroid structure on the smash product algebra [Formula: see text]. They also exhibited an antipode map making [Formula: see text] the total algebra of a Lu’s Hopf algebroid over [Formula: see text]. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair [Formula: see text] and [Formula: see text] of braided commutative Yetter–Drinfeld [Formula: see text]-module algebras, and output is a symmetric Hopf algebroid [Formula: see text] over [Formula: see text]. This construction does not require that the antipode of [Formula: see text] is invertible.

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

On actions of Drinfel'd doubles on finite dimensional algebras

Let L be a finite-dimensional Lie algebra over an algebraically closed field F of characteristic 0, let H(L) be the Hopf algebra of representative functions of L, and let B(L) be the Hochschild basic group B(L) of L. By using Hochschild theory of H(L), we show that two such Lie algebras have the same Hopf algebra if and only if they have the same basic group, or equivalently, they have the same basic Lie algebra (the Lie algebra of the basic group). This is shown by first obtaining the following characterization of B(L). If G(L) is the pro-affine algebraic group associated with H(L), then B(L) is the quotient of G(L) by the intersection of the radical of G(L) with the reductive part of the center of G(L). We also show that the basic Lie algebra of L can be constructed, up to isomorphism, directly from the adjoint representation of L. Finally, we apply the theory of basic groups to obtain an intrinsic characterization of the Hopf algebras (over F ) that are isomorphic to H(L) for some Lie algebra L. Some applications to algebraic Lie algebras are also considered.

In Artin algebra representation theory there is an important result which states that when the order of G is invertible in Λ then gl.dim(ΛG)=gl.dim(Λ). With the development of Hopf algebra theory, this result is generalized to smash product algebra. As known, weak Hopf algebra is an important generalization of Hopf algebra. In this paper we give the more general result, that is the relation of homological dimension between an algebra A and weak smash product algebra A#H, where H is a finite dimensional weak Hopf algebra over a field k and A is an H-module algebra.

We prove that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). This shows that Kitaev models are a special case of the older and more general combinatorial models. This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections. We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories. We first prove that the triangle operators of a Kitaev model form a module algebra over a Hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism. Both algebras can be viewed as the algebra of functions on gauge fields in a Hopf algebra gauge theory. The isomorphism between them induces an algebra isomorphism between their subalgebras of invariants, which are interpreted as gauge invariant functions or observables. It also relates the curvatures in the two models, which are given as holonomies around the faces of the lattice. This yields an isomorphism between the subalgebras obtained by projecting out curvatures, which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models.

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra A that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of A-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e. untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that A is finitely generated and the global Weyl module is a finitely generated A-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in a previous paper. Finally, we show that A is the algebra of coinvariants of the analogous algebra in the untwisted case.

The universal enveloping algebra of a Lie algebra is the analogue of the usual group algebra of a group. It has the analogous function of exhibiting the category of Lie algebra modules as a category of modules for an associative algebra. This becomes more than an analogy when the universal enveloping algebra is viewed with its full Hopf algebra structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible algebraic group over a field of characteristic 0, contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion. This theme is developed in Section 3.

Blocks and modules for Whittaker pairs

Characters of Hopf algebras

Let A be a finite dimensional Hopf algebra and (H, R) a quasitriangular bialgebra. Denote by H^*_R a certain deformation of the multiplication of H^* via R. We prove that H^*_R is a quantum commutative left H\otimes H^{op cop}-module algebra. If H is the Drinfel'd double of A then H^*_R is the Heisenberg double of A. We study the relation between H^*_R and Majid's covariantised product. We give a formula for the canonical element of the Heisenberg double of A, solution to the pentagon equation, in terms of the R-matrix of the Drinfel'd double of A. We generalize a theorem of Jiang-Hua Lu on quantum groupoids and using this and the above we obtain an example of a quantum groupoid having the Heisenberg double of A as base. If, in Richard Borcherds' concept of a we allow the ring of singular functions to be noncommutative, we prove that if A is a finite dimensional cocommutative Hopf algebra then the Heisenberg double of A is a vertex group over A. The construction and properties of H^*_R are given also for quasi-bialgebras and a definition for the Heisenberg double of a finite dimensional quasi-Hopf algebra is proposed.

Formal groups and unipotent affine groups in non-categorical symmetry

It is known that irreducible noncommutative differential structures over $\mathbb {F}_{p}[x]$ are classified by irreducible monics m. We show that the cohomology $H_{\text {dR}}^{0}(\mathbb {F}_{p}[x]; m)=\mathbb {F}_{p}[g_{d}]$ if and only if Trace(m)≠ 0, where $g_{d}=x^{p^{d}}-x$ and d is the degree of m. This implies that there are ${\frac {p-1}{pd}}{\sum }_{k|d, p\nmid k}\mu _{M}(k)p^{\frac {d}{k}}$ such noncommutative differential structures (μM the Mobius function). Motivated by killing this zero’th cohomology, we consider the directed system of finite-dimensional Hopf algebras $A_{d}=\mathbb {F}_{p}[x]/(g_{d})$ as well as their inherited bicovariant differential calculi Ω(Ad;m). We show that Ad = Cd ⊗χA1 is a cocycle extension where $C_{d}=A_{d}^{\psi }$ is the subalgebra of elements fixed under ψ(x) = x + 1. We also have a Frobenius-fixed subalgebra Bd of dimension $\frac {1}{d} {\sum }_{k | d} \phi (k) p^{\frac {d}{k}}$ (ϕ the Euler totient function), generalising Boolean algebras when p = 2. As special cases, $A_{1}\cong \mathbb {F}_{p}(\mathbb {Z}/p\mathbb {Z})$, the algebra of functions on the finite group $\mathbb {Z}/p\mathbb {Z}$, and we show dually that $\mathbb {F}_{p}\mathbb {Z}/p\mathbb {Z}\cong \mathbb {F}_{p}[L]/(L^{p})$ for a ‘Lie algebra’ generator L with eL group-like, using a truncated exponential. By contrast, A2 over $\mathbb {F}_{2}$ is a cocycle modification of $\mathbb {F}_{2}((\mathbb {Z}/2\mathbb {Z})^{2})$ and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry.

Split abelian chief factors and first degree cohomology for Lie algebras