All simple weak twisted modules over the Heisenberg-Virasoro vertex operator algebras, and all simple restricted modules over the mirror Heisenberg-Virasoro algebra will be given in this talk.
Tag - Non-associative rings
Classical free field realizations of affine Kac-Moody algebras (introduced by M.Wakimoto, B.Feigin and E.Frenkel) play an important role in quantum field theory. B.Cox initiated the study of free field realizations for the non-standard Borel subalgebras which led to an important class of intermediate (or parabolic) Wakimoto modules. A uniform construction of such realizations will be discussed based on a joint work with L.Krizka and P.Somberg.
This lecture is a continuation of the general talk given at the Drensky conference last month, on axial algebras, which are (not necessarily commutative, not necessarily associative) algebras generated by semisimple idempotents. After a review of the definitions, we investigate the key question, being, "Under what conditions must an axial algebra be finite-dimensional?" Krupnik showed that 3 idempotents can generate arbitrarily large dimensional associative algebras (and thus infinite-dimensional algebras via an ultraproduct argument), so some restriction is needed. We consider 'primitive' axes, in which the left and right eigenspaces having eigenvalue 1 are one-dimensional.
Hall, Rehren, Shpectorov solves obtained a positive answer for commutative axial algebras of 'Jordan type' λ ≠ 1/2, although the proof relies on the classification of simple groups and the given bound of the dimension is rather high. Gorshkov and Staroletov provided a sharp bound for 3-generated commutative axial algebras of 'Jordan type'. Our objective in this project is give a non-commutative version and indicate how to investigate 4-generated commutative axial algebras of 'Jordan type', in terms of the regular representation.
Our method is to build an associative algebra from the adjoint algebra of A, which has a strictly larger dimension which nevertheless also is finite-dimensional.
In a joint work with D. López N. and L.-F. Préville-Ratelle in 2015 we introduce a family of non-symmetric operads Dyckm, which satisfies that:
1. Dyck0 is the operad of associative algebras,
2. Dyck1 is the operad Dend of dendriform algebras, introduced by J.-L. Loday,
3. the vector space spanned by the set of m-Dyck paths has a natural structure of free Dyckm algebra over one element,
4. for any k ≥ 1, there exist degeneracy operators si : Dyckm → Dyckm-1 and face operators dj: Dyckm → Dyckm+1, which defines a simplicial complex in the category of non-symmetric operads.
The main examples of Dyckm algebra are the vector spaces spanned by the m-simplices of certain combinatorial Hopf algebras, like the Malvenuto-Reutenauer algebras and the algebra of packed words.
A well-known result on associative algebras states that, as an 𝒮-module, the operad of Ass of associative algebras is the composition Ass = Com ∘ Lie, where Com is the operad of commutative algebras and Lie is the operad of Lie algebras. The version of this result for dendriform algebras is that Dend = Ass ∘ Brace, where Brace is the operad of brace algebras.
Our goal is to introduce the notion of m-brace algebra, for m ≥ 2, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context, stating that Dyckm = Ass ∘ m-Brace.
I intend to overview classifications of simple Lie (super)algebras of finite dimension and of polynomial growth. Various properties of complex Lie superalgebras resemble same of modular Lie algebras. I will encourage to consider these classifications without fanaticism: certain non-simple Lie (super)algebras, "close" to simple ones, are often "better" for us than simple ones.
Interesting features of deformations: semi-trivial deformations and (in super setting) odd parameters.
I'll formulate classification of finite-dimensional simple complex Lie superalgebras, odd parameters including.
I'll formulate a definition of Lie superalgebra suitable for any characteristic and classification of simple (finite-dimensional) Lie superalgebras over algebraically closed fields of characteristic 2. With a catch: modulo (a) classification of simple (finite-dimensional) Lie superalgebras (over the same field) and (b) classification of their gradings modulo 2. I'll mention conjectures on classification of modular Lie algebras and superalgebras.
Is it feasible to classify simple filtered Lie (super)algebras of polynomial growth? Interesting examples: deforms of the Poisson Lie (super)algebras, Lie (super)algebras of "matrices of complex size", etc.
Examples. Double extensions of simple Lie (super)algebras are definitely "more interesting" than the simple objects they extend.
Let p be a polynomial in several non-commuting variables with coefficients in an algebraically closed field K of arbitrary characteristic. It has been conjectured that for any n, for p multilinear, the image of p evaluated on the set Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). In this talk we will discuss the generalization of this result for non-associative algebras such as Cayley-Dickson algebra (i.e. algebra of octonions), pure (scalar free) octonion Malcev algebra and basic low rank Jordan algebras.
Gradings by abelian groups have played an important role in the theory of Lie algebras since its beginning: the best known example is the root space decomposition of a semisimple complex Lie algebra, which is a grading by a free abelian group (the root lattice). Involutive automorphisms or, equivalently, gradings by the cyclic group of order 2, appear in the classification of real forms of these Lie algebras. Gradings by all cyclic groups were classified by V. Kac in the late 1960s and applied to the study of symmetric spaces and affine Kac-Moody Lie algebras.
In the past two decades there has been considerable interest in classifying gradings by arbitrary groups on algebras of different varieties including associative, Lie and Jordan. Of particular importance are the so-called fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism, although not in a unique way. If the ground field is algebraically closed and of characteristic 0, then the classification of fine abelian group gradings on an algebra (up to equivalence) is the same as the classification of maximal quasitori in the algebraic group of automorphisms (up to conjugation). Such a classification is now known for all finite-dimensional simple complex Lie algebras.
In this talk I will review the above mentioned classification and present a recent joint work with A. Elduque and A. Rodrigo-Escudero in which we classify fine gradings on classical simple real Lie algebras.
The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GLn(K) (K a field, n ≥ 3) which is not contained in the centre, contains SLn(K). A. Rosenberg gave description of normal subgroups of GL(V), where V is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the centre and the group of linear transformations g such that g − idV has finite-dimensional range the proof is not complete. We fill this gap for countable-dimensional V giving a description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field. Similar results for Lie algebras of matrices will be surveyed.
The class of non-associative axial algebras was introduced in 2015 as a broad generalization of Majorana algebras of Ivanov that were modelled after the properties of the Griess algebra, the algebra whose automorphism group is the Monster sporadic simple group. Sakuma's theorem classifies 2-generated Majorana algebras, which in axial terms correspond to algebras of Monster type (1/4,1/32). The quest to classify all 2-generated algebras of arbitrary Monster type (α,β) was started by Rehren who proved an upper bound on the dimension and generalised the Norton-Sakuma algebras to arbitrary (α,β). Recently, new results emerged from the work of Franchi, Mainardis and the speaker, and independently, of Yabe, who classified symmetric 2-generated algebras of Monster type. Several new classes of algebras have been found.
The free nonassociative algebra provides a simple combinatorial context to extend some constructions from the associative setting. In this talk, based on joint work with J. Mostovoy and I. P. Shestakov, I will briefly discuss three of them related to nonassociative Lie theory: the embedding of the free loop as nonassociative formal power series, a nonassociative extension of the Baker-Campbell-Hausdorff formula and a nonassociative version of Solomon's descent algebra.
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