On coset vertex algebras with central charge 1

We present a coset realization of the vertex operator algebra $V_L ^+$ with central charge $1$. We investigate the vertex operator algebra $V_{Z sqrt{2n}} ^+$ (resp. $V_{2Z sqrt{2n+1}} ^+$) as a vertex subalgebra of $L_{D_n ^{(1)}}(Lambda _0) otimes L_{D_n ^{(1)}}(Lambda _0)$ (resp. $L_{B_n ^{(1)}}(Lambda _0) otimes L_{B_n ^{(1)}}(Lambda _0)$). Our construction is based on the boson-fermion correspondence and certain conformal embeddings

Conformal embeddings in affine vertex superalgebras

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$ is a basic classical simple Lie superalgebras. Let $\mathcal V_k (\mathfrak g_{\bar 0})$ be the subalgebra of $V_k(\mathfrak g)$ generated by $\mathfrak g_{\bar 0}$. We first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak g_{\bar 0})$ in $V_k(\mathfrak g)$ is conformal. Next we prove that, for a large family of such conformal levels, $V_k(\mathfrak g)$ is a completely reducible $\mathcal V_k (\mathfrak g_{\bar 0})$--module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of $V_{-2} (osp(2n +8 \vert 2n))$ as a finite, non simple current extension of $V_{-2} (D_{n+4}) \otimes V_1 (C_n)$. This decomposition uses our previous work [10] on the representation theory of $V_{-2} (D_{n+4})$.Comment: Latex file, 45 pages, to appear in Advances in Mathematic

Cross product on (mathbb{R}^{n}), normed algebras and H–spaces

U ovom preglednom radu prezentiramo konstrukciju vektorskog produkta na (mathbb{R}^{n}), danu u radu P. F. McLoughlin, arXiv:1212.3515. Pokazujemo da vektorski produkt, definiran na prirodan način, postoji samo za n = 0, 1, 3, 7 (pritom s (mathbb{R}^{0}) označavamo nulprostor nad (mathbb{R}) ). Proučavamo vezu vektorskog produkta s Hurwitzovim teoremom o postojanju normiranih algebri samo za dimenzije n = 1, 2, 4, 8, te s Adamsovim teoremom o neprekidnim množenjima na sferi. Također, proučavamo mogućnosti generalizacije vektorskog produkta na ( mathbb{R}^{n}) , s funkcije dvije varijable na funkciju više varijabli.In this survey article, we present a construction of the cross product on (mathbb{R}^{n}) , following the paper by P. F. McLoughlin, arXiv:1212.3515. We show that a naturally defined cross product exists only for n = 0, 1, 3, 7 (here (mathbb{R}^{0}) denotes the zero vector space over (mathbb{R}) ). We study the relationship between the cross product and Hurwitz’s theorem on the existence of normed algebras only for dimensions n = 1, 2, 4, 8, and the relationship with Adams’ theorem on continuous multiplications on spheres. Furthermore, we consider the possibilities of a generalization of the cross product on (mathbb{R}^{0}) , from the function of two variables to the function of several variables

Alternativna definicija limesa funkcije

Autori prezentiraju jednu alternativnu definiciju limesa

Riemannova, Darbouxova i Cauchyjeva integrabilnost

U ovom preglednom radu prezentiramo dokaz iz S. Schneider, International Mathematical Forum 9 (2014) da se Riemannova i Cauchyjeva definicija integrabilnosti podudaraju. Također, diskutiramo odnos Riemannove i Darbouxove integrabilnosti

Representations of certain non-rational vertex operator algebras of affine type

In this paper we study a series of vertex operator algebras of integer level associated to the affine Lie algebra $A_{\ell}^{(1)}$. These vertex operator algebras are constructed by using the explicit construction of certain singular vectors in the universal affine vertex operator algebra $N(n-2,0)$ at the integer level. In the case $n=1$ or $l=2$, we explicitly determine Zhu's algebras and classify all irreducible modules in the category $\mathcal{O}$. In the case $l=2$, we show that the vertex operator algebra $N(n-2,0)$ contains two linearly independent singular vectors of the same conformal weight.Comment: 15 pages, LaTeX; final version, to appear in J. Algebr

Riemannova, Darbouxova i Cauchyjeva integrabilnost

U ovom preglednom radu prezentiramo dokaz iz S. Schneider, International Mathematical Forum 9 (2014) da se Riemannova i Cauchyjeva definicija integrabilnosti podudaraju. Također, diskutiramo odnos Riemannove i Darbouxove integrabilnosti

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