One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category Vect. In the past this problem has been resolved by working with a weaker structure called a ‘twisted’ bialgebra. In this paper we solve the problem differently by first switching to a d...

of generators T (s) ij with s = 1, 2 , . . . and i , j = 1 , . . . , N is given by (1.1), (1.2). Comultiplication ∆ : Y(glN ) → Y(glN )⊗Y(glN ) is defined by (1.1), (1.16). The algebra Y(glN ) admits an alternative definition in terms of the ascending chain U(gl1) ⊂ U(gl2) ⊂ . . . of classical universal enveloping algebras [O]. For any non-negative integer M consider the commutant in U(glM+N ) ...

This equation was introduced in [Agu00, Agu01] and independently in [Pol00]. The algebraic meaning of this equation, explained in [Agu00, Agu01], is as follows. An associative algebra A is called an infinitesimal bialgebra if it is equipped with a coassociative coproduct which is a derivation, i.e. ∆(ab) = (a⊗ 1)∆(b)+∆(a)(1⊗b). This notion was introduced by Joni and Rota [RJ79] and is useful in...

The double Schur functions form a distinguished basis of the ring Λ(x ||a) which is a multiparameter generalization of the ring of symmetric functions Λ(x). The canonical comultiplication on Λ(x) is extended to Λ(x ||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood–Richardson coefficients in two different ways thus pr...

One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category Vect. In the past this problem has been resolved by working with a weaker structure called a ‘twisted’ bialgebra. In this paper we solve the problem differently by first switching to a d...

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. this work, we extend classification to noncommutative monoid structures on surfaces. We prove that monoids are toric. also show how find all a toric surface. Every such structure is induced by comultiplication formula involving Demazure roots. give descriptions of opposite monoids, quotient b...