Abstract
The theory of Galois orders was introduced by Futorny and Ovsienko 9. We introduce the notion of $\mathcal {H}$-Galois $\Lambda $-orders. These are certain noncommutative orders $F$ in a ...smash product of the fraction field of a noetherian integral domain $\Lambda $ by a Hopf algebra ${\mathcal {H}}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster’s 25 principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author 12. The main results are (1) for every maximal ideal $\mathfrak {m}$ of $\Lambda $ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak {m}$; (2) for every character of $\Lambda $, we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; and (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.
We construct weak (i.e. nongraded) modules over the vertex operator algebra M(1)^+, which is the fixed-point subalgebra of the higher rank free bosonic (Heisenberg) vertex operator algebra with ...respect to the -1 automorphism. These weak modules are constructed from Whittaker modules for the higher rank Heisenberg algebra. We prove that the modules are simple as weak modules over M(1)^+ and calculate their Whittaker type when regarded as modules for the Virasoro Lie algebra. Lastly, we show that any Whittaker module for the Virasoro Lie algebra occurs in this way. These results are a higher rank generalization of some results by Tanabe Proc. Amer. Math. Soc. 145 (2017), no. 10, pp. 4127-4140.
Twisted generalized Weyl algebras (TGWAs)
A
(
R
,
σ
,
t
) are defined over a base ring
R
by parameters
σ
and
t
, where
σ
is an
n
-tuple of automorphisms, and
t
is an
n
-tuple of elements in the ...center of
R
. We show that, for fixed
R
and
σ
, there is a natural algebra map
A
(
R
,
σ
,
t
t
′
)
→
A
(
R
,
σ
,
t
)
⊗
R
A
(
R
,
σ
,
t
′
)
. This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all
t
) of the Grothendieck groups of the categories of weight modules for
A
(
R
,
σ
,
t
). We give presentations of these Grothendieck rings for
n
= 1,2, when
R
=
ℂ
z
. As a consequence, for
n
= 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over
s
l
2
is a tensor product of two Weyl algebra modules.
In this note we compute the leading term with respect to the De Concini–Kac filtration of Uq(gln) of a generating set for the quantum Gelfand–Tsetlin subalgebra.
For any complex reflection group G = G(m,p,n), we prove that the G-invariants of the division ring of fractions of the n:th tensor power of the quantum plane is a quantum Weyl field and give explicit ...parameters for this quantum Weyl field. This shows that the q-difference Noether problem has a positive solution for such groups, generalizing previous work by Futorny and the author
10
. Moreover, the new result is simultaneously a q-deformation of the classical commutative case and of the Weyl algebra case recently obtained by Eshmatov et al.
8
.
Second, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk
18
originating in the classical Gelfand-Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of
n
and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko
11
, with symmetry group being a direct product of complex reflection groups G(m,p,r
k
).
Finally, using these results, we prove that the quantum OGZ algebras satisfy the quantum Gelfand-Kirillov conjecture by explicitly computing their division ring of fractions.
In this article we develop an approach to deformations of the Witt and Virasoro algebras based on
σ-derivations. We show that
σ-twisted Jacobi type identity holds for generators of such deformations. ...For the
σ-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the
q-deformations of the Witt and Virasoro algebras associated to
q-difference operators, providing also corresponding
q-deformed Jacobi identities.
We report on an improved test of the Universality of Free Fall using a rubidium-potassium dual-species matter wave interferometer. We describe our apparatus and detail challenges and solutions ...relevant when operating a potassium interferometer, as well as systematic effects affecting our measurement. Our determination of the Eötvös ratio yields
η
Rb,K
= −1.9 × 10
−7
with a combined standard uncertainty of
σ
η
= 3.2 × 10
−7
.
Graphical abstract
We construct a family of twisted generalized Weyl algebras which includes Weyl–Clifford superalgebras and quotients of the enveloping algebras of
gl
m
n
and
osp
m
2
n
. We give a condition for when a ...canonical representation by differential operators is faithful. Lastly, we give a description of the graded support of these algebras in terms of pattern-avoiding vector compositions.
We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coefficient ring
R
), which is assumed to carry an involution of the ...form
X
∗
=
Y
,
R
∗
⊆
R
. We prove that a weight module
V
is pseudo-unitarizable iff it is isomorphic to its finitistic dual
V
♯
. Using the classification of weight modules by Drozd, Guzner and Ovsienko, we obtain necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be pseudo-unitarizable. Some examples are given, including
U
q
(
s
l
2
)
for
q
a root of unity.
We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and ...we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order.
As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type A, and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative.
Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if q is not a root of unity, the Gelfand-Zeitlin subalgebra of Uq(gln) is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.
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