The purpose of this paper is to study the relationships between an
-Hom-Lie algebra and its induced (n+1)-Hom-Lie algebra. We provide an overview of the theory and explore structure properties such ...as ideals, centers, derived series, solvability, nilpotency, central extensions, and the cohomology.
Let X be a normal variety endowed with an algebraic torus action. An additive group action α on X is called vertical if a general orbit of α is contained in the closure of an orbit of the torus ...action and the image of the torus normalizes the image of α in Aut(X). Our first result in this paper is a classification of vertical additive group actions on X under the assumption that X is proper over an affine variety. Then we establish a criterion as to when the infinitesimal generators of a finite collection of additive group actions on X generate a finite-dimensional Lie algebra inside the Lie algebra of derivations of X.
We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural ...linear action induced on the Frechet space KN by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K((x)) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper 18 where the Riordan group was described using inverse sequences of groups of finite matrices.
3D rotations arise in many computer vision, computer graphics, and robotics problems and evaluation of the distance between two 3D rotations is often an essential task. This paper presents a detailed ...analysis of six functions for measuring distance between 3D rotations that have been proposed in the literature. Based on the well-developed theory behind 3D rotations, we demonstrate that five of them are bi-invariant metrics on
SO
(3) but that only four of them are boundedly equivalent to each other. We conclude that it is both spatially and computationally more efficient to use quaternions for 3D rotations. Lastly, by treating the two rotations as a true and an estimated rotation matrix, we illustrate the geometry associated with iso-error measures.
We offer a new approach to large N limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in ...that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-trace correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism.
As another application, we consider the problem of quantization in the large N limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.
Let
V
be a hypersurface with an isolated singularity at the origin defined by the holomorphic function
f
:
(
C
n
,
0
)
→
(
C
,
0
)
. The Yau algebra
L
(
V
) is defined to be the Lie algebra of ...derivations of the moduli algebra
A
(
V
)
:
=
O
n
/
(
f
,
∂
f
∂
x
1
,
⋯
,
∂
f
∂
x
n
)
, i.e.,
L
(
V
)
=
Der
(
A
(
V
)
,
A
(
V
)
)
and plays an important role in singularity theory. It is known that
L
(
V
) is a finite dimensional Lie algebra and its dimension
λ
(
V
)
is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of
k
-th Yau algebras
L
k
(
V
)
which are defined to be the Lie algebras of derivations of the moduli algebras
A
k
(
V
)
=
O
n
/
(
f
,
m
k
J
(
f
)
)
,
k
≥
0
, i.e.,
L
k
(
V
)
=
Der
(
A
k
(
V
)
,
A
k
(
V
)
)
and where
m
is the maximal ideal of
O
n
. In particular, it is Yau algebra when
k
=
0
. The dimension of
L
k
(
V
)
is denoted by
λ
k
(
V
)
. These numbers i.e.,
k
-th Yau numbers
λ
k
(
V
)
, are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras
L
k
(
V
)
and also compute the Lie algebras
L
1
(
V
)
for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the
λ
k
(
V
)
of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of
k
=
1
for large class of singularities.
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