Algebras Of Rank Research Articles

A generalized Kac–Moody algebra of rank 14

We construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12. The BRST-cohomology group of this vertex algebra is a new generalized Kac–Moody algebra of rank 14. We determine its root space multiplicities and a set of simple roots.

TEST RANK OF THE LIE ALGEBRA F/[R′, F

Let F be a free Lie algebra of rank n ≥ 2 and R be a fully invariant ideal of F. We show that the test rank of the Lie algebra F/[R′, F] is equal to 1 when n is even and less than or equal to 2 when n is odd.

FILIFORM LIE ALGEBRAS OF DIMENSION 8 AS DEGENERATIONS

For each complex 8-dimensional filiform Lie algebra we find another nonisomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank ≥ 1, only the characteristically nilpotent ones should be considered.

On nilpotent and solvable Lie algebras of derivations

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerKA of all K-derivations of A is an A-module in a natural way, and if R is the quotient field of A then RDerKA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerKA of rank k over R (i.e. such that dimRRL=k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerKA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector fields with polynomial, rational, or formal coefficients. Analogously, if X is an irreducible affine variety of dimension n over an algebraically closed field K of characteristic zero and AX is its coordinate ring, then all nilpotent (solvable) subalgebras of DerKAX have derived length at most n (2n respectively).

Rationality of the Hilbert series of Hopf-invariants of free algebras

It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function whenever the ground field has zero characteristic.

Clifford Algebras of Forms of Higher Degrees

The aim of this lecture is to introduce Clifford algebras of polynomial forms of higher degrees. We recall that these algebras are in general of infinite dimension, and we give a basis depending on a given basis of the underlying vector space. We then show that, though they contain large free associative algebras, we may construct finite dimensional representations of these algebras, also called linearizations of the polynomial form. If the polynomial form is, in a certain sense, non degenerate, the dimensions of these representations are multiples of the degree of the form. In the end, we recall some results known for the special case of a binary cubic form with at least one simple zero, when explicit computations can be done: the Clifford algebra is an Azumaya algebra of rank 9 over its center, which is the algebra of functions over a cubic curve depending on the given cubic form.

Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ϕ on 𝔤 is said to preserve solvability in both directions if ϕ, as well as ϕ−1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ϕ on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.

Centralizer Algebras and Degenerate Affine Hecke Algebras

Let (F, R, k) be a p-modular system, and let denote the centralizer of the symmetric group S ℓ in the group algebra RS n , where ℓ ≤n. We show that the decomposition map of can be determined from that of the degenerate affine Hecke algebra of rank n − ℓ. We use this to determine the blocks of for ℓ =n − 2, n − 3. For each p-core κ, there is an n 0 such that if n > n 0 and E n is a block idempotent of RS n with core κ, then E n E n−ℓ is zero or a block idempotent of , for each block idempotent E n−ℓ of RS n−ℓ.

Cluster algebras of infinite rank

Holm and Jorgensen have shown the existence of a cluster structure on a certain category $D$ that shares many properties with finite type $A$ cluster categories and that can be fruitfully considered as an infinite analogue of these. In this work we determine fully the combinatorics of this cluster structure and show that these are the cluster combinatorics of cluster algebras of infinite rank. That is, the clusters of these algebras contain infinitely many variables, although one is only permitted to make finite sequences of mutations. The cluster combinatorics of the category $D$ are described by triangulations of an $\infty$-gon and we see that these have a natural correspondence with the behaviour of Plucker coordinates in the coordinate ring of a doubly-infinite Grassmannian, and hence the latter is where a concrete realization of these cluster algebra structures may be found. We also give the quantum analogue of these results, generalising work of the first author and Launois. An appendix by Michael Groechenig provides a construction of the coordinate ring of interest here, generalizing the well-known scheme-theoretic constructions for Grassmannians of finite-dimensional vector spaces.

Q-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

We study a relationship between Q-polynomial distance-regular graphs and the double affine Hecke algebra of type (C1∨,C1). Let Γ denote a Q-polynomial distance-regular graph with vertex set X. We assume that Γ has q-Racah type and contains a Delsarte clique C. Fix a vertex x∈C. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a C-vector space W.The universal double affine Hecke algebra of type (C1∨,C1) is the C-algebra Hˆq defined by generators {tn±1}n=03 and relations (i) tntn−1=tn−1tn=1; (ii) tn+tn−1 is central; (iii) t0t1t2t3=q−1/2. In this paper, we display an Hˆq-module structure for W. For this module and up to affine transformation,•t0t1+(t0t1)−1 acts as the adjacency matrix of Γ;•t3t0+(t3t0)−1 acts as the dual adjacency matrix of Γ with respect to C;•t1t2+(t1t2)−1 acts as the dual adjacency matrix of Γ with respect to x. To obtain our results we use the theory of Leonard systems.

Positivity for Cluster Algebras of Rank 3

We prove the positivity conjecture for skew-symmetric coefficient-free cluster algebras of rank 3.

A computational approach to the Kostant–Sekiguchi correspondence

Let g be a real form of a simple complex Lie algebra. Based on ideas of ‐okovi´ c and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant‐Sekiguchi correspondence. Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8. In addition, we consider two real forms g and g 0 of a complex simple Lie algebra g c with Cartan decompositions gD k p and g 0 D k 0 p 0 . We describe an explicit construction of an isomorphism g! g 0 , respecting the given Cartan decompositions, which fails if and only if g and g 0 are not isomorphic. This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realizations of the same algebra.

Menger algebras of n-place interior operations

Algebraic properties of $n$-place opening operations on a fixed set are described. Conditions under which a Menger algebra of rank $n$ can be represented by $n$-place opening operations are found.

Algèbres De Lie Triple Sans Idempotent

Idempotents play an important role in the investigation of nonassociative algebras structure (Albert in Trans Am Math Soc 69:503–527, 1950; Schafer in An introduction to nonassociative algebras Academic Press, New York, 1966; Zhevlakov et al. in Rings that are nearly associative. Academic Press, New York, 1982). However, the existence of such elements is not always guaranteed, specially when dealing with algebras that are defined by polynomial identities. Hence, in many cases, one has to assume the existence of idempotents (Arenas in Comm Algebra 35:675–688, 2007; Osborn in Trans Am Math Soc 16:1114–1120, 1965; Petersson in Math Zeitschr 98:104–118, 1967) in order to investigate algebraic structures via the classical Peirce decomposition. Lie triple algebras appear historically and independently in Osborn (Trans Am Math Soc 16:1114–1120, 1965) and Petersson (Math Zeitschr 97:1–15, 1967). In this paper, we deal mainly with Lie triple algebras which admit no idempotent. We prove that non nil algebras possess either an idempotent or a pseudo-idempotent. In the latter case, it is possible to obtain a Peirce decomposition relatively to the pseudo-idempotent and some relations on Peirce’s components. We notice that these results are analogous to that obtained by Osborn in (Trans Am Math Soc 16:1114–1120, 1965). When a Lie triple algebra has a non-trivial idempotent, we show that its heart is a Jordan subalgebra. Then we give a Lie triple non-conservative algebras classification for dimensions 2 and 3. Finally, we characterize train algebras of rank \(\le 3\) that are Lie triple algebras.

TABLE ALGEBRAS OF RANK 3 AND ITS APPLICATIONS TO STRONGLY REGULAR GRAPHS

A table algebra is called quasi self-dual if there exists a permutation on the set of primitive idempotents under which any Krein parameter is equal to its corresponding structure constants. In this paper we investigate the question of when a table algebra of rank 3 is quasi self-dual. As a direct consequence we find necessary and sufficient conditions for the Bose–Mesner algebra of a given strongly regular graph to be quasi self-dual. In fact, our result generalizes the well-known Delsarte's characterization of a self-duality of the Bose–Mesner algebra of a strongly regular graph given in [P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl.10 (1973) 1–97]. Among our results we determine conditions under which the Krein parameters of an integral table algebra of rank 3 are non-negative rational numbers.

The isomorphic realization of nondegenerate solvable Lie algebras of maximal rank based on Kac-Moody algebras

In this paper, based on Kac-Moody algebra, the isomorphic realization of nondegenerate solvable Lie algebras of maximal rank is given, which in turn revels the closed connections between nondegenerate solvable Lie algebras and Kac-Moody algebras, resulting in some new worthy topics in this area.

Free Fields in Malcev–Neumann Series Rings

It is shown that the skew field of Malcev–Neumann series of an ordered group frequently contains a free field of countable rank, i.e. the universal field of fractions of a free associative algebra of countable rank. This is an application of a criterion on embeddability of free fields on skew fields which are complete with respect to a valuation function, following K. Chiba. Other applications to skew Laurent series rings are discussed. Finally, embeddability questions on free fields of uncountable rank in Malcev–Neumann series rings are also considered.

On the polynomial identities of the algebra [formula omitted

Verbally prime algebras are important in PI theory. They were described by Kemer over a field K of characteristic zero: 0 and K〈T〉 (the trivial ones), Mn(K),Mn(E),Mab(E). Here K〈T〉 is the free associative algebra of infinite rank, with free generators T,E denotes the infinite dimensional Grassmann algebra over K,Mn(K) and Mn(E) are the n×n matrices over K and over E, respectively. The algebras Mab(E) are subalgebras of Ma+b(E), see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of Mn(K) and lots of its properties. Models for the generic algebras of Mn(E) and Mab(E) are also known but their structure remains quite unclear.In this paper we study the generic algebra of M11(E) in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of M11(E). Also we describe the polynomial identities in two variables of the algebra M11(E).

Freudenthal Gauge Theory

AbstractWe present a novel gauge field theory, based on theFreudenthal Triple System(FTS), a ternary algebra with mixed symmetry (not completely symmetric) structure constants. The theory, namedFreudenthal Gauge Theory(FGT), is invariant under two (off-shell) symmetries: the gauge Lie algebra constructed from theFTStriple product and a novelglobalnon-polynomial symmetry, the so-calledFreudenthal duality.Interestingly, a broad class ofFGTgauge algebras is provided by the Lie algebras “of type$ {{\mathfrak{e}}_7} $” which occur as conformal symmetries of Euclidean Jordan algebras of rank 3, and asU-duality algebras of the corresponding (super)gravity theories inD= 4.We prove aNo-Go Theorem, stating the incompatibility of the invariance underFreudenthal dualityand the coupling to space-time vectorand/orspinor fields, thus forbidding non-trivial supersymmetric extensions ofFGT.We also briefly discuss the relation betweenFTSand the triple systems occurring in BLG-type theories, in particular focusing on superconformal Chern-Simons-matter gauge theories inD= 3.

ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

Let L be a free Lie algebra of finite rank over a field K and let Ln denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [Lm, Ln] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [Lm, Ln, Lk] = [[Lm, Ln], Lk]. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field K. For example, the dimension of [L2, L2, L1] over fields of characteristic 2 is different from the dimension over fields of characteristic other than 2. Our main results are formulae for the dimension of [Lm, Ln, Lk]. Under certain conditions on m, n and k they lead to explicit formulae that do not depend on the characteristic of K, and express the dimension of [Lm, Ln, Lk] in terms of Witt's dimension function.