Let g be an algebra over K with a bilinear operation ⋅,⋅:g×g→g not necessarily associative. For A⊆g, let Ak be the set of elements of g written combining k elements of A via + and ⋅,⋅.
We show a ...“sum-bracket theorem” for simple Lie algebras over K of the form g=sln,son,sp2n,e6,e7,e8,f4,g2: if char(K) is not too small, we have growth of the form |Ak|≥|A|1+ε for all generating symmetric sets A away from subfields of K. Over Fp in particular, we have a diameter bound matching the best analogous bounds for groups of Lie type 2.
As an independent intermediate result, we prove also an estimate of the form |A∩V|≤|Ak|dim(V)/dim(g) for linear affine subspaces V of g. This estimate is valid for all simple algebras, and k is especially small for a large class of them including associative, Lie, and Mal'cev algebras, and Lie superalgebras.
Abstract The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality ...between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon’s Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach.
For the associative algebra A(g) of an infinite-dimensional Lie algebra g, we introduce twisted fiber bundles over arbitrary compact topological spaces. Fibers of such bundles are given by elements ...of algebraic completion of the space of formal series in complex parameters, sections are provided by rational functions with prescribed analytic properties. Homotopical invariance as well as covariance in terms of trivial bundles of twisted A(g)-bundles is proven. Further applications of the paper's results useful for studies of the cohomology of infinite-dimensional Lie algebras on smooth manifolds, K-theory, as well as for purposes of conformal field theory, deformation theory 10,14,15, and the theory of foliations are mentioned.
In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible ...associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.
For any vertex operator algebra V and any finite automorphism g of V, we prove that the associative algebra Ag,n(V) is isomorphic to some (sub)quotient of the universal enveloping algebra of V with ...respect to g.
We investigate the ε-Lie structure of K and K,K; here K denotes the skew-symmetric elements of an (ε,G)-Lie color algebra (obtained from an associated algebra A) with an ε-involution. The ...relationship with the (associative) ideals of A is also explored.
In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily ...associative) algebras over any commutative ring k with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent pre-endomorphisms are related to semidirect-product decompositions of the sub-adjacent anticommutative algebra.
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