This article looks at koalgebras over bodies. For koalgebras over
comonads see there.
A koalgebra is a vector space that has a structure that is dual to an algebra. This means that instead of a multiplication that maps two elements to their product, there is a multiplication that maps an element to a tensor product , and instead of a neutral element that enables the basic field to be embedded in the algebra, there is a mapping from the coalgebra into the basic body, which is called Koeins .
definition
A koalgebra over a body is a - vector space with vector space homomorphisms , called multiplication , coproduct or also diagonal , and , called koeins , such that
-
(Coassociativity)
-
(Koeins)
A koalgebra homomorphism between two coalgebras C and D is a vector space homomorphism with
-
and .
example
Be the canonical base of . One can refer to a koalgebra structure by means of
and
define.
is co-associative, there
-
,
and there is no one
-
.
The elements of are second order tensors and can therefore be represented as matrices. The multiplication is then
-
.
duality
The multiplication of a (unitary associative) algebra is bilinear, and due to the universal property of the tensor product it can be understood as a mapping from to . The multiplication is associative if and only if the following diagram commutes.
An algebra has a neutral element if and only if there is a vector space homomorphism , so that the following diagram commutes:
In this case .
A koalgebra is an algebra in the category dual to the vector spaces . That is, instead of multiplication there is a mapping so that the following dual diagram commutes:
And instead of a neutral element there is a mapping , so the following dual diagram commutes:
Sweeder notation
In general, the only known information about the coproduct of an element is that it is in , and thus that it is known as
can represent. In the Sweedler notation (after Moss Sweedler ) this is abbreviated by symbolically
writes. In the sumless Sweedler notation you even do without the sum symbol and write
It is important to note that this notation still denotes a sum. The symbols and are meaningless on their own and are not available for certain elements , because the representation of is not clear. For calculations in the Sweedler notation, one reads the best as "suitable and firmly selected for this calculation" elements.
This notation enables the composition of functions other than
to write.
In sumless Sweedler notation, it is Koeins if and only if
-
.
The co- product is co-associative if and only if
-
.
This element is symbolically represented in Sweedler notation as
and humming as
written.
Applying again results in longer tensor products that are written analogously. You may have to increase the "indices" of the rear elements:
-
.
By using , the tensor products are shortened, the "indices" of the rear elements are adjusted accordingly:
-
.
literature
- Christian Kassel: Quantum Groups In: Graduate Texts in Mathematics . Springer-Verlag, ISBN 0-387-94370-6 .