Algebraic Series Research Articles

Bott type periodicity for the higher octonions

We study the series of complex nonassociative algebras On and real nonassociative algebras $O_{p,q}$ introduced in [10]. These algebras generalize the classical algebras of octonions and Clifford algebras. The algebras $O_{n}$ and $O_{p,q}$ with $p + q = n$ have a natural $Z_n^2$-grading, and they are characterized by cubic forms over the field $Z_2$. We establish a periodicity for the algebras $O_{n}$ and $O_{p,q}$ similar to that of the Clifford algebras Cln and $Cl_{p,q}$.

Algebraic time series forecasting and segmentation models

An algebraic segmentation method based on algebraic predictor with internal smoothing is proposed. The concept of the rank of the sequence is proposed for the detection of a base fragment of the sequence. Numerical experiments with a real-world financial time series illustrate the segmentation method.

JORDAN–KRONECKER INVARIANTS OF FINITE-DIMENSIONAL LIE ALGEBRAS

For any finite-dimensional Lie algebra we introduce the notion of Jordan–Kronecker invariants, study their properties, and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko–Fomenko’s argument shift method. We also state a generalised argument shift conjecture and prove it for many series of Lie algebras.

Super 3-Lie Algebras Induced by Super Lie Algebras

We propose a notion of a super n-Lie algebra and construct a super n-Lie algebra with the help of a given binary super Lie algebra which is equipped with an analog of a supertrace. We apply this approach to the super Lie algebra of a Clifford algebra with even number of generators and making use of a matrix representation of this super Lie algebra given by a supermodule of spinors we construct a series of super 3-Lie algebras labeled by positive even integers.

Exponential growth of codimensions of identities of algebras with unity

The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly 1 when an outer unity is adjoined to the original algebra. It is shown that the PI-exponents of unital algebras can take any value greater than 2, and the exponents of finite-dimensional unital algebras form a dense subset in the domain . Bibliography: 34 titles.

On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.

The Hochschild Cohomology Ring for Self-Injective Algebras of Tree Class E 6

The Hochschild cohomology ring for one of the two series of self-injective algebras of tree class E6 is described in terms of generators and relations.

Classification of the Algebras 𝕆 p,q

We study a series of real nonassociative algebras 𝕆 p, q introduced in [5]. These algebras have a natural -grading, where n = p + q, and they are characterized by a cubic form over the field ℤ2. We establish all the possible isomorphisms between the algebras 𝕆 p, q preserving the structure of -graded algebra. The classification table of 𝕆 p, q is quite similar to that of the real Clifford algebras Cl p, q , the main difference is that the algebras 𝕆 n, 0 and 𝕆0, n are exceptional.

Une version relative de la conjecture des périodes de Kontsevich-Zagier

We start with a series F = P r1 fr $ r with indeterminate $ and where the coecients fr = fr(z1;:::;zn) are holomorphic functions dened on an open neighborhood of the closed polydisc D n =f(z1;:::;zn);jzij 1g. Integrating the coecients of this series on the n-dimensional real cube [0; 1] n yields a Laurent series R [0;1]n F . When F is algebraic we say that R [0;1]n F is a series of periods. In this article, our goal is to determine the algebraic series F such that R [0;1]n F is zero. In principle, this gives informations on the transcendence properties of series of periods. Our main result is reminiscent to the Kontsevich-Zagier conjecture on periods in a modied form.

Generalized Calogero system for simple Lie algebra series

We derive equations of motions and solutions for the generalized spin-Calogero models associated for the classical Lie algebra series Bn, Cn BCn, and Dn using the pole expansion.

Jordan-Hölder theorem for finite dimensional Hopf algebras

We show that a Jordan-Hölder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus’ butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Hölder theorem holds as well for lower and upper composition series, even though the factors of such series may not be simple as Hopf algebras.

PI-exponents of some Simple Algebras with Unit

Over a field of characteristic zero we have constructed a discrete series of simple finite-dimensional algebras with unit. Its sequence of dimensions is not bounded, but PI-exponents are different fractional numbers strictly less than 4.

Lie algebras with extremal properties

We present two series of Lie algebras with extremal properties. Each algebra of the first series generates a variety of minimal degree polynomial growth. The algebras of this series belong to the Volichenko variety which is of almost polynomial growth. Each algebra of the second series generates a variety of polynomial growth minimal with respect to the leading coefficient of the polynomial. The algebras of this series belong to the variety N2A of almost polynomial growth.

Hilbert series of the finite dimensional generalized Heckealgebras

It is known from the early results of Coxeter that the generalized Hecke algebras $H(Q_{m},3)$, $m\in\{2,3,4,5\}$, are finite dimensional. In this paper we compute the Hilbert series of these finite-type group algebras.

Groups of Lie Type, Vertex Algebras, and Modular Moonshine

We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $\mathbb{F}_q$ as most of'' the full automorphism group of a vertex algebra over $\mathbb{F}_q$. These finite simple groups are \[ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) \] \[ \text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), \] where $q$ is a prime power. Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{\cdot }19{\cdot} 31$ in $E_8(3)$.

Nontrivial central extensions of 3-Lie algebras

We construct a series of infinite-dimensional 3-Lie algebras which include w∞ and SDiff(T2) 3-Lie algebras as subalgebras, and determine their nontrivial one-dimensional central extensions.

Groups of Lie type, vertex algebras, and modular moonshine

We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $\mathbb{F}_q$ as most of'' the full automorphism group of a vertex algebra over $\mathbb{F}_q$. These finite simple groups are \[ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) \] \[ \text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), \] where $q$ is a prime power. Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{\cdot }19{\cdot} 31$ in $E_8(3)$.

C*-algebraic intertwiners for degenerate principal series of special linear groups

The author constructs unitary intertwiners for degenerate C*-algebraic universal principal series of SL(n + 1) over a local field by explicitely normalizing standard intertwining integrals at the level of Hilbert modules.

An Algebro-geometric Construction of Lower Central Series of Associative Algebras

The lower central series invariants Mk of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the Mk provide a filtration of A. We study the relationship between the geometry of X=Spec Aab and the associated graded components Nk of this filtration. We show that the Nk form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. Under certain freeness assumptions on A, we give an alternative construction of Nk purely in terms of the geometry of X (and in particular, independent of A). Applying a construction of Kapranov, we exhibit the Nk as natural vector bundles on the category of smooth schemes.

Real Milnor Fibres and Puiseux Series

Given a real polynomial function and a point in its zero locus, we defined a set consisting of algebraic real Puiseux series naturally attached to these data. We prove that this set determines the topology and the geometry of the real Milnor fibre of the function at this point. To achieve this goal, we balance between the tameness properties of this set of Puiseux series, considered as a real algebraic object over the field of algebraic Puiseux series, and its behaviour as an infinite dimensional object over the real numbers.