The Etingof-Kazhdan construction of Lie bialgebra deformations.
... Manin triple with Casimir element Ω, and M the Drinfeld category associated to g. Consider the functor F :M→A F (V ) = Hom(U g, V ) which is naturally isomorphic to the forgetful functor. We wish to equip F with a tensor structure: a functorial isomorphism JV W : F (V ) ⊗ F (W ) → F (V ⊗ W ) such th ...
... Manin triple with Casimir element Ω, and M the Drinfeld category associated to g. Consider the functor F :M→A F (V ) = Hom(U g, V ) which is naturally isomorphic to the forgetful functor. We wish to equip F with a tensor structure: a functorial isomorphism JV W : F (V ) ⊗ F (W ) → F (V ⊗ W ) such th ...
Trivial remarks about tori.
... φ is a map GL1 → T , and one evaluates it at a uniformiser; the resulting element of T (F )/T (O) is well-defined. As a consequence we have X ∗ (Tb) = T (F )/T (O). ...
... φ is a map GL1 → T , and one evaluates it at a uniformiser; the resulting element of T (F )/T (O) is well-defined. As a consequence we have X ∗ (Tb) = T (F )/T (O). ...
Algebra in Coding
... 1. (a) Write down the addition and multiplication tables for GF(5) and GF(7). (b) Write down the addition and mulitplication tables for GF(4). 2. Construct GF(16) in three different ways by defining operations modulo the irreducible polynomials x4 +x+1, x4 +x3 +1, and x4 +x3 +x2 +x+1. Find isomorphi ...
... 1. (a) Write down the addition and multiplication tables for GF(5) and GF(7). (b) Write down the addition and mulitplication tables for GF(4). 2. Construct GF(16) in three different ways by defining operations modulo the irreducible polynomials x4 +x+1, x4 +x3 +1, and x4 +x3 +x2 +x+1. Find isomorphi ...
Set 2
... such that ( T π−T ) = 1 and aπ 6= u2 − (T 3 − T )v 2 for any a ∈ F× p and u, v ∈ Fp [T ]. (This can be done with deg π ≤ 2, in fact with deg π = 1 for p > 3 if you are clever enough.) 15. Let f ∈ F2 [T ] not be ℘(g) = g 2 + g for any g ∈ F2 [T ], and π be irreducible in F2 [T ]. a) Prove: if π = u2 ...
... such that ( T π−T ) = 1 and aπ 6= u2 − (T 3 − T )v 2 for any a ∈ F× p and u, v ∈ Fp [T ]. (This can be done with deg π ≤ 2, in fact with deg π = 1 for p > 3 if you are clever enough.) 15. Let f ∈ F2 [T ] not be ℘(g) = g 2 + g for any g ∈ F2 [T ], and π be irreducible in F2 [T ]. a) Prove: if π = u2 ...
Math. 5363, exam 1, solutions 1. Prove that every finitely generated
... Let G be a non-abelian group of order 6. Since G is not abelian, it does not contain any element of order 6. Also, it can’t happen that every element other than 1 is of order 2. Therefore, there is element a ∈ G of order 3. This element generates the subgroup H = {1, a, a2 } ⊆ G of index 2. In parti ...
... Let G be a non-abelian group of order 6. Since G is not abelian, it does not contain any element of order 6. Also, it can’t happen that every element other than 1 is of order 2. Therefore, there is element a ∈ G of order 3. This element generates the subgroup H = {1, a, a2 } ⊆ G of index 2. In parti ...
linear representations as modules for the group ring
... Fix a commutative ring k with identity. It is called the base ring or just the base. 4.1. Associative algebras with identity. A k-module A is a k-algebra if we are given a k-linear map µ : A ⊗k A → A, or, what amounts to the same, a k-bilinear map A × A → A. We call µ the multiplication. It is assoc ...
... Fix a commutative ring k with identity. It is called the base ring or just the base. 4.1. Associative algebras with identity. A k-module A is a k-algebra if we are given a k-linear map µ : A ⊗k A → A, or, what amounts to the same, a k-bilinear map A × A → A. We call µ the multiplication. It is assoc ...
COCOMMUTATIVE HOPF ALGEBRAS WITH ANTIPODE We shall
... "like" that of a universal enveloping algebra. If p = 0 the second factor actually is a universal enveloping algebra. For p>0, we generalize the Birkhoff-Witt theorem by introducing the notion of divided powers. These also play a role in the theory of algebraic groups where certain sequences of divi ...
... "like" that of a universal enveloping algebra. If p = 0 the second factor actually is a universal enveloping algebra. For p>0, we generalize the Birkhoff-Witt theorem by introducing the notion of divided powers. These also play a role in the theory of algebraic groups where certain sequences of divi ...
LIE-ADMISSIBLE ALGEBRAS AND THE VIRASORO
... Let A be an (nonassociative) algebra with multiplication x y over a field F, and denote by A− the algebra with multiplication [x, y] = x y − yx defined on the vector space A. If A− is a Lie algebra, then A is called Lie-admissible. Lie-admissible algebras arise in various topics, including geometry ...
... Let A be an (nonassociative) algebra with multiplication x y over a field F, and denote by A− the algebra with multiplication [x, y] = x y − yx defined on the vector space A. If A− is a Lie algebra, then A is called Lie-admissible. Lie-admissible algebras arise in various topics, including geometry ...
Ring Theory (MA 416) 2006-2007 Problem Sheet 2 Solutions 1
... The 8th roots of unity in C are either primitive 1st roots of unity (1, the root in C of x − 1), primitive 2nd roots of unity (−1, the root in C of x + 1), primitive 4th roots of unity (i and −i, the roots in C of x2 + 1), or primitive 8th roots if unity (the roots in C of x4 + 1). So the irreducibl ...
... The 8th roots of unity in C are either primitive 1st roots of unity (1, the root in C of x − 1), primitive 2nd roots of unity (−1, the root in C of x + 1), primitive 4th roots of unity (i and −i, the roots in C of x2 + 1), or primitive 8th roots if unity (the roots in C of x4 + 1). So the irreducibl ...
Open problems on Cherednik algebras, symplectic reflection
... deformation for concrete algebras A, coming from quantized algebraic surfaces, and to give meaning to this deformation for non-formal (i.e., numerical) values of parameters. One expects that the “spherical subalgebra” eHn,k (Au)e (where e ∈ C[Sn] is the Young symmetrizer) will then be a quantizatio ...
... deformation for concrete algebras A, coming from quantized algebraic surfaces, and to give meaning to this deformation for non-formal (i.e., numerical) values of parameters. One expects that the “spherical subalgebra” eHn,k (Au)e (where e ∈ C[Sn] is the Young symmetrizer) will then be a quantizatio ...
Review Problems
... possible rational roots of the second factor are 1 and −1, and these do not work. (It is important to note that since the degree of the polynomial is greater than 3, the fact that it has not roots in Q does not mean that it is irreducible over Q.) Since the polynomial has no linear factors, the only ...
... possible rational roots of the second factor are 1 and −1, and these do not work. (It is important to note that since the degree of the polynomial is greater than 3, the fact that it has not roots in Q does not mean that it is irreducible over Q.) Since the polynomial has no linear factors, the only ...
Endomorphisms The endomorphism ring of the abelian group Z/nZ
... The endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Z/nZ that maps each element to the sum of r copies of it. This is a bijection if and only if r is coprime with n, so the automorphism group ...
... The endomorphism ring of the abelian group Z/nZ is isomorphic to Z/nZ itself as a ring. Under this isomorphism, the number r corresponds to the endomorphism of Z/nZ that maps each element to the sum of r copies of it. This is a bijection if and only if r is coprime with n, so the automorphism group ...
