Logarithmic intertwining operators and W(2,2p-1)-algebras

For every $p \geq 2$, we obtained an explicit construction of a family of $\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual $\mathcal{W}(2,2p-1)$-modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic $\mathcal{W}(2,2p-1)$-modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as "logarithmic conformal field theory" of central charge $c_{p,1}=1-\frac{6(p-1)^2}{p}, p \geq 2$. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra $\mathcal{W}(2,(2p-1)^3)$ and other logarithmic models.Comment: 22 pages; v2: misprints corrected, other minor changes. Final version to appear in Journal of Math. Phy

An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}

By extending the methods used in our earlier work, in this paper, we present an explicit realization of logarithmic \mathcal{W}_{p,p'}-modules that have L(0) nilpotent rank three. This was achieved by combining the techniques developed in \cite{AdM-2009} with the theory of local systems of vertex operators \cite{LL}. In addition, we also construct a new type of extension of $\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$. Our results confirm several claims in the physics literature regarding the structure of projective covers of certain irreducible representations in the principal block. This approach can be applied to other models defined via a pair screenings.Comment: 18 pages, v2: one reference added, other minor change

Representations of vertex algebras

In this paper we present some results on the representation theory of vertex operator (super) algebras associated to affine Lie algebras and Neveu-Schwarz algebra

Fusion rules and complete reducibility of certain modules for affine Lie algebras

We develop a new method for obtaining branching rules for affine Kac-Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in Adamovi\'c and O. Per\v{s}e (2008) is closed under fusion. Then we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type $A_{\ell-1}^{(1)}$, obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for $\ell \ge 3$. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type $C_{\ell}^{(1)}$. Next we notice that the category of $D_{2 \ell -1}^{(1)}$ modules at level $- 2 \ell +3$ obtained in Per\v{s}e (2012) has the isomorphic fusion algebra. This enables us to decompose certain $E_6 ^{(1)}$ and $F_4 ^{(1)}$--modules at negative levels.Comment: 18 pages; final version, to appear in Journal of Algebra and Its Application

The N=1 triplet vertex operator superalgebras

We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras SW(m), $m \geq 1$, which are natural super analogs of the triplet vertex algebra family W(p), $p \geq 2$, important in logarithmic conformal field theory. We classify irreducible SW(m)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible SW(m) characters. Finally, we contemplate possible connections between the category of SW(m)-modules and the category of modules for the quantum group U^{small}_q(sl_2), q=e^{\frac{2 \pi i}{2m+1}}, by focusing primarily on properties of characters and the Zhu's algebra A(SW(m)). This paper is a continuation of arXiv:0707.1857.Comment: 53 pages; v2: references added; v3: a few changes; v4: final version, to appear in CM

Lie superalgebras and irreducibility of A_1^(1)-modules at the critical level

We introduce the infinite-dimensional Lie superalgebra ${\mathcal A}$ and construct a family of mappings from certain category of ${\mathcal A}$-modules to the category of A_1^(1)-modules of critical level. Using this approach, we prove the irreducibility of a family of A_1^(1)-modules at the critical level. As a consequence, we present a new proof of irreducibility of certain Wakimoto modules. We also give a natural realizations of irreducible quotients of relaxed Verma modules and calculate characters of these representations.Comment: 21 pages, Late

Wear Behaviour of Hard Cr Coatings for Cold Forming Tools Under Dry Sliding Conditions

Cr hard coatings are largely used in industry in metal cutting and cold forming processes; This work on quantitative way represents improvement, in terms of wear resistance, which is obtained by depositing Cr hard coating on foundation material. Wear testing is done on tribometer with block –on –disc contact geometry at sliding contact of Cr hard coated sample with steel disc. Testing was performed in conditions without lubrication at variable value of contact parameters (normal load, sliding speed). Cr hard coatings in all contact conditions show smaller values of wear rate

On Free Field Realizations of W(2,2)-Modules

The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra H at level zero as modules for the W(2,2)-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight H-module is irreducible as W(2,2)-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly W(2,2) vertex algebra

Morfometrijske i mehaničke osobine kostiju nogu kod pilića autohtonih rasa gološijana u Srbiji

The purpose of this study was to estimate the morphometric and mechanical parameters of femur and tibiotarsal bone in male and female chickens of three Serbian autochthonous naked neck breeds (white, black and gray) and compared these values with chickens of commercial naked neck hybrid Farm Q. Chickens were reared in extensive system and fattening lasted 98 days (14 weeks) The bone length, weight, cross sectional diaphyseal geometry (total area, medullar area, cortical area) and bone breaking force were determined. Between the varieties of autochthonous naked neck breeds, the presence of significant difference was not established. Average bone mass and length, of femur in male chickens were 13.6 g and 8.2 cm and for tibiotarsal bone, 19.5 g and 11.7 cm. In a female chickens those values were 10.6 g and 7.8 cm for femur and 15.9 g and 11.0 cm for tibiotarsus. In a male chickens average breaking force of femur (36.1 kg) and tibiotarsus (31.6 kg) were higher than those in a female chickens (27.0; 29.6 kg, respectively). In comparison with chickens of commercial naked neck hybrid (Farm Q), chickens of three Serbian autochthonous naked neck breeds have had significantly lower (P (lt) 0.05) bone mass, cross sectional diaphyseal area and cross sectional medullar area. Bone length and cross sectional cortical area were not significantly differed. However, tibiotarsal strength, expressed as bone breaking force, were signifficantly (P (lt) 0.05) higher in a chickens of autochthonous naked neck breeds.Cilj ispitivanja je bio da se ustanove morfometrijske i mehaničke osobine kostiju nogu (butna kost i golenjača) pilića tri autohtona varijeteta gološijana koji se gaje u Srbiji (beli, crni i sivi) i poređenje tih rezultata sa parametrima golenjače komercijalnog hibrida gološijana Farm Q. Pilići su gajeni u ekstenzivnim uslovima a period tova je trajao 98 dana (14 nedelja). Na desnoj butnoj kosti i golenjači određivani su masa, dužina, parametri geometrije preseka dijafize (površina preseka dijafize, površina preseka medularne šupljine i površina preseka korteksa) i sila loma. Između varijeteta autohtonih pilića gološijana nije ustanovljeno postojanje značajnih razlika u osobinama kostiju ali su muški pilići ispoljili veće vrednosti u odnosu na ženke. Prosečne vrednosti mase i dužine butne kosti, kod muških pilića su iznosile 13.6 g i 8.2 cm a golenjače 19.5 g i 11.7 cm. Kod ženki ove vrednosti su prosečno iznosile, za butnu kost 10.6 g i 7.8 cm a za golenjaču 15.9 g i 11.0 cm. Vrednosti sile loma ispitivanih kostiju su pokazale da autohtoni gološijani imaju dobru čvrstoću kostiju. Kod mužjaka, prosečna sila loma butne kosti (36.1 kg) i golenjače (31.6 kg) su bile veće nego kod ženki (20.0 kg - butna kost; 29.6 kg - golenjača). U poređenju sa pilićima komercijalnog hibrida glošijana Farm Q, pilići autohtonih varijeteta su imali značajno (p (lt) 0.05) manju masu golenjače, površinu preseka dijafize i medularne šupljine. Dužina kosti i površina preseka korteksa se nisu značajno razlikovale. Međutim, čvrstoća golenjače, izražena kroz silu loma, bila je značajno (P (lt) 0.05) veća kod pilića autohtonih varijeteta gološijana

Quantum-sl(2) action on a divided-power quantum plane at even roots of unity

We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{i\pi/p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$ by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $U_q sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane.Comment: 18 pages, amsart++, xy, times. V2: a reference and related comments adde