80 research outputs found

### q-Witt Algebras, q-Virasoro algebra, q-Lie Algebras, q-Holomorph Structure and Representations

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.Comment: 18 page

### Double-bosonization and Majid's Conjecture, (II): cases of irregular $R$-matrices and type-crossings of $F_4$, $G_2$

The purpose of the paper is to build up the related theory of weakly
quasitriangular dual pairs suitably for non-standard $R$-matrices (irregular),
and establish the generalized double-bosonization construction theorem for
irregular $R$, which generalize Majid's results for regular $R$ in
\cite{majid1}. As an application, the type-crossing construction for the
exceptional quantum groups of types $F_{4}$, $G_{2}$ is obtained. This affirms
the Majid's expectation that the tree structure of nodes diagram associated
with quantum groups can be grown out of the node corresponding to
$U_q(\mathfrak{sl}_2)$ by double-bosonization procedures. Notably from a
representation perspective, we find an effective method to get the minimal
polynomials for the non-standard $R$-matrices involved.Comment: 42 page

### Double-bosonization and Majid's Conjecture, (III): type-crossing and inductions of $E_6$ and $E_7$, $E_8$

Double-bosonization construction in Majid \cite{majid1} is expectedly allowed
to generate a tree of quantum groups. Some main branches of the tree in
\cite{HH1, HH2} have been depicted how to grow up. This paper continues to
elucidate the type-crossing and inductive constructions of exceptional quantum
groups of types $E_6$ and $E_7$, $E_8$, respectively, based on the generalized
double-bosonization Theorem established in \cite{HH2}. Thus the Majid's
expectation for the inductive constructions of $U_q(\mathfrak g)$'s for all
finite-dimensional complex simple Lie algebras is completely achieved.Comment: 24 page

### Double-bosonization and Majid's Conjecture, (IV): Type-Crossings from $A$ to $BCD$

Both in Majid's double-bosonization theory and in Rosso's quantum shuffle
theory, the rank-inductive and type-crossing construction for $U_q(\mathfrak
g)$'s is still a remaining open question. In this paper, working with Majid's
framework, based on our generalized double-bosonization Theorem proved in
\cite{HH2}, we further describe explicitly the type-crossing construction of
$U_q(\mathfrak g)$'s for $(BCD)_n$ series direct from type $A_{n-1}$ via adding
a pair of dual braided groups determined by a pair of $(R, R')$-matrices of
type $A$ derived from the respective suitably chosen representations. %which
generalize the lower rank cases constructed in \cite{HH1}. Combining with our
work in \cite{HH1,HH2,HH3}, this solves Majid's conjecture, that is, any
quantum group $U_q(\mathfrak g)$ associated to a simple Lie algebra $\mathfrak
g$ can be grown out of $U_q({\mathfrak {sl}}_2)$ inductively by a series of
suitably chosen double-bosonization procedures.Comment: 26 pages, 1 figure, Sci. China Ser A (2016) (to appear

### Double-bosonization and Majid's conjecture, (I): rank-induction of $ABCD$

Majid developed in \cite{majid3} his double-bosonization theory to construct
$U_q(\mathfrak g)$ and expected to generate inductively not just a line but a
tree of quantum groups starting from a node. In this paper, the authors confirm
the Majid's first expectation (see p. 178 \cite{majid3}) through giving and
verifying the full details of the inductive constructions of $U_q(\mathfrak g)$
for the classical types, i.e., the $ABCD$ series. Some examples in low ranks
are given to elucidate that any quantum group of classical type can be
constructed from the node corresponding to $U_{q}(\mathfrak{sl}_2)$.Comment: 22 page

### Two-parameter Quantum Group of Exceptional Type G_2 and Lusztig's Symmetries

We give the defining structure of two-parameter quantum group of type G_2
defined over a field {\Bbb Q}(r,s) (with r\ne s), and prove the Drinfel'd
double structure as its upper and lower triangular parts, extending an earlier
result of [BW1] in type A and a recent result of [BGH1] in types B, C, D. We
further discuss the Lusztig's Q-isomorphisms from U_{r,s}(G_2) to its
associated object U_{s^{-1},r^{-1}}(G_2), which give rise to the usual
Lusztig's symmetries defined not only on U_q(G_2) but also on the centralized
quantum group U_q^c(G_2) only when r=s^{-1}=q. (This also reflects the
distinguishing difference between our newly defined two-parameter object and
the standard Drinfel'd-Jimbo quantum groups). Some interesting (r,s)-identities
holding in U_{r,s}(G_2) are derived from this discussion.Comment: 34 pages. Pacific J. Math. (to appear in its simplified version

### Universal Central Extensions of the Matrix Leibniz Superalgebras sl(m, n, A)

The universal central extensions and their extension kernels of the matrix
Lie superalgebra sl(m, n, A), the Steinberg Lie superalgebra st(m, n, A) in
category {\bf SLeib} of Leibniz superalgebras are determined under a weak
assumption (compared with \cite{MP}) using the first Hochschild homology and
the first cyclic homology group.Comment: 8 pages. Communications in Algebra (to appear

### Loewy filtration and quantum de Rham cohomology over quantum divided power algebra

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 realization
model for a class of indecomposable $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.
Meanwhile, the quantum Grassmann algebra $\Omega_q(n)$ over $\mathcal{A}_q(n)$
is constructed, together with the quantum de Rham complex $(\Omega_q(n),
d^\bullet)$ via defining the appropriate $q$-differentials, and its subcomplex
$(\Omega_q(n,\bold m), d^\bullet)$. For the latter, the corresponding quantum
de Rham cohomology modules are decomposed into the direct sum of some
sign-trivial $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.Comment: 26 page

### The Green rings of the 2-rank Taft algebra and its two relatives twisted

In the paper, the representation rings (or the Green rings) for a family of
Hopf algebras of tame type, the 2-rank Taft algebra (at $q=-1$) and its two
relatives twisted by 2-cocycles are explicitly described via a representation
theoretic analysis. It turns out that the Green rings can serve to detect
effectively the twist-equivalent Hopf algebras here.Comment: J. Algebra (to appear), 34 pages (revised the noncommutativity of
Green ring of 2-rank Taft algebra \bar A, rewrote the proofs of Jacobson
radicals of the three Green algebras, added some remarks and updated
references, etc.

### Two-parameter Quantum Affine Algebra of Type ${\mathrm C_n^{(1)}},$ Drinfeld Realization and Vertex Representation

The two-parameter quantum vertex operator representation of level-one is
explicitly constructed for $U_{r,s}(C^{(1)}_n)$ based on its two-parameter
Drinfeld realization we give. This construction will degenerate to the
one-parameter case due to Jing-Koyama-Misra (\cite{JKM2}) when $rs=1$.Comment: 28 pages. arXiv admin note: text overlap with arXiv:math/9802123 by
other author

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