In this paper, we study representations of hom-Lie algebras. In particular, the adjoint representation and the trivial representation of hom-Lie algebras are studied in detail. Derivations, ...deformations, central extensions and derivation extensions of hom-Lie algebras are also studied as an application.
We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra
g
, a notion first introduced by Slofstra. The associated graded space of this filtration ...on dominant weight spaces of integrable highest weight modules of
g
has Hilbert series coinciding with Lusztig’s
t
-analog of weight multiplicities. For the level 1 vacuum module
L
(Λ
0
) of affine Kac-Moody algebras of type
A
, we show that the Brylinski filtration may be most naturally understood in terms of representations of the corresponding
𝒲
-algebra. We show that the sum of dominant weight spaces of
L
(Λ
0
) in the principal vertex operator realization forms an irreducible Verma module of
𝒲
and that the Brylinski filtration is induced by the Poincaré-Birkhoff-Witt basis of this module. This explicitly determines the subspaces of the Brylinski filtration. Our basis may be viewed as the analog of Feigin-Frenkel’s basis of
𝒲
for the
𝒲
-action on the principal rather than on the homogeneous realization of
L
(Λ
0
).
The aim of this paper is to develop a framework for localization theory of triangulated categories
C
, that is, from a given extension-closed subcategory
N
of
C
, we construct a natural ...extriangulated structure on
C
together with an exact functor
Q
:
C
→
C
~
N
satisfying a suitable universality, which unifies several phenomena. Precisely, a given subcategory
N
is thick if and only if the localization
C
~
N
corresponds to a triangulated category. In this case,
Q
is nothing other than the usual Verdier quotient. Furthermore, it is revealed that
C
~
N
is an exact category if and only if
N
satisfies a generating condition
Cone
(
N
,
N
)
=
C
. Such an (abelian) exact localization
C
~
N
provides a good understanding of some cohomological functors
C
→
Ab
, e.g., the heart of
t
-structures on
C
and the abelian quotient of
C
by a cluster-tilting subcategory
N
.
In order to see the behavior of
ı
canonical bases at
q
=
∞
, we introduce the notion of
ı
crystals associated to an
ı
quantum group of certain quasi-split type. The theory of
ı
crystals clarifies why
...ı
canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of
ı
crystals whose projective limit can be thought of as the
ı
canonical basis of the modified
ı
quantum group at
q
=
∞
.
We investigate the problem when the tensor functor by a bimodule yields a singular equivalence. It turns out that this problem is equivalent to the one when the Hom functor given by the same bimodule ...induces a triangle equivalence between the homotopy categories of acyclic complexes of injective modules. We give conditions on when a bimodule appears in a pair of bimodules, that defines a singular equivalence with level. We construct an explicit bimodule in a combinatorial manner, which yields a singular equivalence between a quadratic monomial algebra and its associated algebra with radical square zero. Under certain conditions which include the Gorenstein cases, the bimodule does appear in a pair of bimodules defining a singular equivalence with level.
We show that Nichols algebras of most simple Yetter–Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We ...introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs $(n,q)$, any finite-dimensional pointed Hopf algebra $H$ with $G(H)\simeq\mathbf {PSL}_{n}(q)$ or $\mathbf {SL}_{n}(q)$ is isomorphic to a group algebra.
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