Some representations π of b of particular interest [2] are the Verma modules (V, π) = (V , π), h, c ǫ R. They are characterized by the following conditions. (i) There is a vector v = vφ 6= 0 in V such that Lnv = 0, n > 0, L0v = hv, Zv = cv. (ii) Let A be the set of sequences of integers k1,≥ k2 ≥ . . . ≥ kr > 0 of arbitrary length, and if α ǫ A let vα = π(L−k1) . . . π(L−kr)vφ. Then {vα|α ǫ A} ...

Some representations π of b of particular interest [2] are the Verma modules (V, π) = (V , π), h, c R. They are characterized by the following conditions. (i) There is a vector v = vφ = 0 in V such that Lnv = 0, n > 0, L0v = hv,Zv = cv. (ii) Let A be the set of sequences of integers k1,≥ k2 ≥ . . . ≥ kr > 0 of arbitrary length, and if α A let vα = π(L−k1) . . . π(L−kr)vφ. Then {vα|α A} is a bas...

as gl(V1)⊕ gl(V2)⊕ gl(V3)-modules. This yields a well-known reformulation of (2.1). Proposition A.1. If l(α) 6 dimV1 and l(β) 6 dimV2, then the Kronecker coefficient g(αβγ) is the multiplicity of V1(α)⊗ V2(β) in the gl(V1)⊕ gl(V2)-module (V1 ⊗ V2)(γ). If V is m-dimensional, then V (1) is the one-dimensional gl(V )-module carried by the trace map gl(V ) → C. It follows easily that V (n) ∼= V (1)...

Biochemical studies of purified and dissected fungal polyketide synthase and nonribosomal peptide synthetase (PKS-NRPS) hybrid enzymes involved in biosynthesis of pseurotin and aspyridone indicate that one α-methylation step during polyketide synthesis is a prerequisite and a key checkpoint for chain transfer between PKS and NRPS modules. In the absence of the resulting γ-methyl feature, the co...

Let A and B be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence α between A and B such that α is induced by exact functors. We present a nice correspondence between indecomposable modules over A and B. As a consequence, we have the following: (1) If A is a self-injective algebra, then so is B; (2) If A and B are finite dimensional algebras over an algeb...

We have designed a highly twisted small TADF emitter PXZ-α-DK based on an -diketone (α-DK) as strong acceptor and phenoxazine (PXZ) donor to obtain red-shifted emission in comparison the equivalent linked 9,9-dimethyl-9,10-dihydroacridine (DMAC). The shows at 586 nm DMAC-α-DK 548 1,3-bis(N-carbazolyl)benzene (mCP) host 1.5 wt% doping of emitter, with short-delayed lifetimes 6.9 μs for 7.6 DMAC...

let $r$ be a ring and $m$ a right $r$-module with $s=end_r(m)$. a module $m$ is called semi-projective if for any epimorphism $f:mrightarrow n$, where $n$ is a submodule of $m$, and for any homomorphism $g: mrightarrow n$, there exists $h:mrightarrow m$ such that $fh=g$. in this paper, we study sgq-projective and$pi$-semi-projective modules as two generalizations of semi-projective modules. a m...

we consider the class $mathfrak m$ of $bf r$--modules where $bf r$ is an associative ring. let $a$ be a module over a group ring $bf r$$g$, $g$ be a group and let $mathfrak l(g)$ be the set of all proper subgroups of $g$. we suppose that if $h in mathfrak l(g)$ then $a/c_{a}(h)$ belongs to $mathfrak m$. we investigate an $bf r$$g$--module $a$ such that $g not = g'$, $c_{g}(a) = 1$. we stud...