This work is an investigation into the structure and properties of supersymmetric hypermatrix Lie algebra generated by elements of the dihedral group D3. It is based on previous work on the subject of supersymmetric Lie algebra (Schreiber, 2012). In preview work I used several new algebraic tools; namely cubic hypermatrices (including special arrangements of such hypermatrices) and I obtained a...

In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic p > 2 if the Hopf algebra is also cosemisimple. In fact we show a more general version for any finite-dimensional semisimple algebra with an involution; this more general r...

Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly ...

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra an...

Leibniz algebras are a non-anticommutative version of Lie algebras. They play an important role in different areas mathematics and physics have attracted much attention over the last thirty years. In this paper we investigate whether conditions such as being algebra, cyclic, simple, semisimple, solvable, supersolvable or nilpotent algebra preserved by lattice isomorphisms.

Let K be a field with char(K) 6= 2. Let us fix an algebraic closure Ka of K. Let us put Gal(K) := Aut(Ka/K). If X is an abelian variety of positive dimension over Ka then we write End(X) for the ring of all its Ka-endomorphisms and End (X) for the corresponding (semisimple finite-dimensional) Q-algebra End(X)⊗ Q. We write EndK(X) for the ring of all K-endomorphisms of X and End 0 K(X) for the c...

Given a complex projective algebraic variety, write H•(X,C) for its cohomology with complex coefficients and IH •(X,C) for its Intersection cohomology. We first show that under some fairly general conditions the canonical map H•(X,C) → IH •(X,C) is injective. Now let Gr := G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. ...

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend open-closed two-dimensional topological field theories to nonorientable surfac...

The relations between the radical of crossed product R# σ H and algebra R are obtained. Using this theory, the author shows that if H is a finite-dimensional semisimple, cosemisimle, and either commutative or cocommutative Hopf algebra, then R is H-semiprime iff R is semiprime iff R# σ H is semiprime.

Let p be a prime number. It is known that any non-semisimple Hopf algebra of dimension p over an algebraically closed field of characteristic 0 is isomorphic to a Taft algebra. In this exposition, we will give a more direct alternative proof to this result.