 About 140 results  books.google.com ... and counitary); c) invertible left Acolinear maps A » A (which is unitary/
counitary/unitary and counitary). All of these sets are nonempty. 3.11 REMARK.
Fix such a unitary and counitary isomorphism A £* A 8 B as in (3. 10. 2. Quotient
Theory ... 

 books.google.com ... our usual explicit summation notation for coproducts. Similarly to see that d(x7)
= 1 if dx = 1, and to check the counitarity conditions. We proceed here directly in
terms of linear maps and compute = (7<8>7<8>7)(l<8>xA7~1)(id®A)(l®7xA7~1)
... 

 books.google.com ... of counitarity is satisfied: 34 4. FROBENIUS ALGEBRAS. 

 books.google.com module homomorphisms A and e satisfying the coassociativity and counitary
properties. Define t : H ® H  , H ® H tobe the switch map, r(hi & /i2) = ^2 & hi An /
?bialgebra is an RHopf algebra if there is an ^module homomorphism A : H ' H
... 

 books.google.com 1.2) [/i(i) (twisted module condition) (3.1.3) (A(i) — (/(i) — a))a(h(2), 1(2)) = for any
h,l,m £ B and a e /I. Furthermore, let r: B — > /4 ® /4 ; assume that (counitary
condition) (3.1.4) eB(b)\A = (EA ® id)r(&) = (id ® (cycle condition) (3.1.5) /M^s3 (A
... 

 books.google.com (I ⊗ ∆)∆ =(∆ ⊗ I)∆ (Coassociativity) m(I ⊗ ε)∆ =I = m(ε ⊗ I)∆ (Counitary) m(I ⊗ λ)∆ =ε
= m(λ ⊗ I)∆ (Antipode Property) An augmented Ralgebra that satisfies these
properties is known as a Hopf algebra. We summarize this in the following ... 

 books.google.com (h, ® £«)£«• (c) r,H is counitary, i.e. (CH ® 1 „){« = (!« ® *«)£«= \,i The
multiplication of H will be denoted by p.H '. H ® H —> H, and the canonical
morphism of algebras rjw: A"v H is known as the unit of H. An antipode of a Hopf
algebra H is a ... 

 books.google.com In general, a fcalgebra A together with algebra homomorphisms A : A — > A ® A
and E : A — > k and a linear map S : A — > A is called a Hopf algebra if (2) (A \g\
1) o A = (1 ® A) o A, (coassociativity) (3) (e ® 1) o A = (1 ® e) o A = id, (counitary)
... 

 books.google.com These are required to satisfy the left and right counitary laws, and the
coassociative law, as in [22]. It is wellknown that associated to any cotriple (T, d,
s) on a category C there is a functor W from C to the category of simplicial objects
over C. In ... 

 books.google.com ... fc),g,(C©fc) with (C®C)®CeC&k, the coproduct is given by: (c.X)^(Ac(c).c,c.X)
One can easily verify that this coproduct is coassociative and counitary, with
counit e(c,X) = A.. We also verify straightforward that the projection C > C ((c. X) ^
c) is ... 

 