We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov-Lauda and Rouquier also lead to braid group actions. As an example, we construct an action of Artin’s...

Let G be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let g be its Lie algebra. First we correct and generalise a well-known result about the Picard group of G. Then we prove that, if the derived group is simply connected and g satisfies a mild condition, the algebra K[G] of regular functions on G that are invariant under the act...

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution o...

that g* has the same derived algebra as g itself and that every ideal in g is also an ideal in t*. Let g be any algebraic Lie algebra. Denote by b the radical of g (i.e., the largest solvable ideal in g) and by n the largest ideal of g composed only of nilpotent matrices. By Levi's theorem, g is the direct sum of t and of a semi-simple subalgebra J. It can be proved that f is the direct sum of ...

In this paper we are interested in Hochschild cohomology of finite-dimensional algebras; the main motivation is to generalize group cohomology to larger classes of algebras. If suitable finite generation holds, one can define support varieties of modules as introduced by [SS]. Furthermore, when the algebra is self-injective, many of the properties of group representations generalize to this set...

We present some connections between the max-min general fuzzy automaton theory and the hyper structure theory. First, we introduce a hyper BCK-algebra induced by a max-min general fuzzy automaton. Then, we study the properties of this hyper BCK-algebra. Particularly, some theorems and results for hyper BCK-algebra are proved. For example, it is shown that this structure consists of different ty...

First we show that the cosets of a fuzzy ideal μ in a BCK-algebraX form another BCK-algebra  X/μ (called the fuzzy quotient BCK-algebra of X by μ). Also we show thatX/μ is a fuzzy partition of X and we prove several some isomorphism theorems. Moreover we prove that if the associated fuzzy similarity relation of a fuzzy partition P of a commutative BCK-algebra iscompatible, then P is a fuzzy quo...

A deformation of the harmonic oscillator algebra associated with the Morse potential and the SU (2) algebra is derived using the quantum analogue of the anharmonic oscillator. We use the quantum os-cillator algebra or q-boson algebra which is a generalisation of the Heisenberg-Weyl algebra obtained by introducing a deformation parameter q. Further, we present a new algebraic realization of the ...

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent ∆ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie br...

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent ∆ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie br...