by Cimprič, J.

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ∗-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime ∗-ideal, that every maximal proper quadratic module in a Noetherian ∗-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schmüdgen’s Strict Positivstellensatz for the Weyl algebra: Let c be an element of the Weyl algebra $\mathcal{W}(d)$ which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements $r_1,\ldots,r_k \in \mathcal{W}(d)$ such that ∑
_j=1

^k

r

_j

c
r

_j

^∗ is a finite sum of hermitian squares. This result is not a proper generalization however because we don’t have the bound k ≤d.

DOI: 10.1007/s10468-007-9076-z
Online Date: 6/28/2007
Print publication date: 3/1/2008
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by Westreich, Sara

We study pointed Hopf algebras of the form U(R

_Q
), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis, vol. I, Academic, Boston, MA, pp. 129–139, 1988; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys., vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991), where R

_Q
is the Yang–Baxter operator associated with the multiparameter deformation of GL

_n
supplied in Artin et al. (Commun. Pure Appl. Math. 44:8–9, 879–895, 1991) and Sudbery (J. Phys. A, 23(15):697–704, 1990). We show that U(R

_Q
) is of type A

_n
in the sense of Andruskiewitsch and Schneider (Adv. Math. 154:1–45, 2000; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series, Cambridge University Press, Cambridge, 2002). We consider the non-negative part of U(R

_Q
) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U(R

_Q
) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(R

_Q
) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(R

_Q
), U(R

_Q′) can be obtained from each other by twisting the comultiplication if and only if $G(U_{\!Q})\cong G(U_{Q’}).$ In the last part we show that U

_Q
is always a quotient of a double crossproduct.

DOI: 10.1007/s10468-007-9079-9
Online Date: 6/28/2007
Print publication date: 3/1/2008
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by Heckenberger, I.

The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators. This has strong consequences for both objects. As an application all rank 2 Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators are determined.

DOI: 10.1007/s10468-007-9060-7
Online Date: 6/28/2007
Print publication date: 4/1/2008
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by Abe, Hiroki; Hoshino, Mitsuo

Let n ≥ 1 be an integer and π a permutation of I = {1, ⋯ ,n}. For any ring R, we provide a systematic construction of rings A which contain R as a subring and enjoy the following properties: (a) 1 = ∑ 
_i ∈ I

e

_i
with the e

_i
orthogonal idempotents; (b) e

_i

x = xe

_i
for all i ∈ I and x ∈ R; (c) e

_i

A
e

_j
 ≠ 0 for all i, j ∈ I; (d) e

_i

A

_A
 ≇ e

_j

A

_A
unless i = j; (e) every e

_i

Ae

_i
is a local ring whenever R is; (f) e

_i

A

_A
 ≅ Hom
_R
(Ae

_π(i),R

_R
) and
_A

Ae

_π(i) ≅ 
_A
Hom
_R
(e

_i

A,
_R

R) for all i ∈ I; and (g) there exists a ring automorphism η ∈ Aut(A) such that η(e

_i
) = e

_π(i) for all i ∈ I. Furthermore, for any nonempty π-stable subset J of I, the mapping cone of the multiplication map $\bigoplus_{i \in J} Ae_{i} \otimes_{R} e_{i}A_{A} \to A_{A}$ is a tilting complex.

DOI: 10.1007/s10468-007-9065-2
Online Date: 6/26/2007
Print publication date: 6/1/2008
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by Kaneda, Masaharu

DOI: 10.1007/s10468-007-9080-3
Online Date: 6/21/2007
Print publication date: 8/1/2008
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by Dibaei, Mohammad T.; Yassemi, Siamak

Let ${\left( {R,\mathfrak{m}} \right)}$ be a commutative Noetherian local ring and let $\mathfrak{a}$ be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to $\mathfrak{a}$. As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module ${\text{H}}^{i}_{\mathfrak{a}} {\left( M \right)}$ are finite if one of the following holds:
(a)
$R \mathord{\left/ {\vphantom {R \mathfrak{a}}} \right. \kern-\nulldelimiterspace} \mathfrak{a} = 1$,(b)
$\mathfrak{a}$ is a principal ideal.

DOI: 10.1007/s10468-007-9072-3
Online Date: 6/20/2007
Print publication date: 6/1/2008
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by Pirashvili, Teimuraz; Redondo, María Julia

We consider a homology theory on a triangulated category with values in an abelian category. If the functor h reflects isomorphisms, is full and is such that for any object x in there is an object X in with an isomorphism between h(X) and x, we prove that is a hereditary abelian category, all idempotents in split and the kernel of h is a square zero ideal which as a bifunctor on is isomorphic to.

DOI: 10.1007/s10468-007-9077-y
Online Date: 6/20/2007
Print publication date: 4/1/2008
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by Baur, Karin; Goodwin, Simon M.

Let G be a simple algebraic group of classical type over an algebraically closed field k. Let P be a parabolic subgroup of G and let ${\mathfrak p} = \text{Lie } P$ be the Lie algebra of P with Levi decomposition ${\mathfrak p} = {\mathfrak l} \oplus {\mathfrak u}$, where ${\mathfrak u}$ is the Lie algebra of the unipotent radical of P and ł is a Levi complement. Thanks to a fundamental theorem of Richardson (Bull. London Math. Soc. 6:21–24, 1974), P acts on ${\mathfrak u}$ with an open dense orbit; this orbit is called the Richardson orbit and its elements are called Richardson elements. Recently Baur (J. Algebra 297(1):168–185, 2006), the first author gave constructions of Richardson elements in the case $k = {\mathbb C}$ for many parabolic subgroups P of G. In this note, we observe that these constructions remain valid for any algebraically closed field k of characteristic not equal to 2 and we give constructions of Richardson elements for the remaining parabolic subgroups.

DOI: 10.1007/s10468-007-9071-4
Online Date: 6/19/2007
Print publication date: 6/1/2008
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by Bazzoni, Silvana; Herbera, Dolors

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one.

DOI: 10.1007/s10468-007-9064-3
Online Date: 6/15/2007
Print publication date: 3/1/2008
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by Montarani, Silvia

In this note we refine the methods of Etingof and Montarani (Represent. Theory, 9: 457–467, 2005) in order to extend the main result of that article to a wider class of finite dimensional representations of wreath product symplectic reflection algebras.

DOI: 10.1007/s10468-007-9066-1
Online Date: 6/15/2007
Print publication date: 10/1/2007
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