Extending ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective Z2-graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups....

Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing via of endomorphism an abelian group and set all homomorphisms modules form [1] along with Ch. 2 set. 1 [2]. The formalized items are shown below list notations: M ab for Abelian suffix “ ” without is used ring. 1. denoted by End ( ). 2. A pair homomorphism <m:math xmlns:m="ht...

Graph homomorphism has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function is defined as

There exists a natural filtration on the module freely generated by knots (or links). This filtration is called the Vassiliev filtration and has many nice properties. In particular every quotient of this filtration is finite dimensional. A knot invariant which vanishes on some module of this filtration is called a Vassiliev invariant. Almost every knot invariant defined in algebraic term can be...

An explicit character formula is established for any strongly generic finite-dimensional irreducible g-module, g being an arbitrary finite-dimensional complex Lie superalgebra. This character formula had been conjectured earlier by Vera Serganova and the author for any generic irreducible finite-dimensional g-module, i.e. such that its highest weight is far enough from the walls of the Weyl cha...

An explicit character formula is established for any strongly generic nite-dimensional irreducible g-module, g being an arbitrary nite-dimensional complex Lie superalgebra. This character formula had been conjectured earlier by Vera Serganova and the author for any generic irreducible nite-dimensional g-module, i.e. such that its highest weight is far enough from the walls of the Weyl chambers....