Representations of Skew Polynomial Rings Zongzhu Lin Kansas State - - PDF document

Representations of Skew Polynomial Rings

Zongzhu Lin Kansas State University

Conference on Geometric Methods in Representation Theory

University of Missouri November 24, 2013

• I. Linear Algebra

Given a field k, describe the conjugacy classes in Mn(k)

• f n × n-matrices, describe the equivalence classes of

bilinear forms on n-dimensional vector spaces, i.e., GLn(k)-orbits in Mn(k) with respect to the following actions: g ∈ GLn(k) and A ∈ Mn(k) (a). g · A = gAg−1. (b). g · A = gAgtr. Let τ : k → k is field automorphism, then define the new actions (a’) g · A = gAτ(g)−1, (b’) g · A = gAτ(g). Here τ((gij)) = (τ(gij)). In particular when k is a perfect field of characteristics p > 0 and τ(α) = αp.

Why?

• Let g be an n-dim Lie algebra /k.

A restricted structure on g is a map [p] : g → g such that: (αX)[p] = αpX[p] and (X + Y )[p] = X[p] + Y [p] +

p−1

i=1 si(X,Y ) i

where si(X, Y ) is the coefficient of ti−1 in the formal expression ad(tX + Y )p−1(X). Each [p] is determined by a matrix in Mn(k) under a basis and Aut(g) ⊂ GL(g) acts on Mn(k) by (a′). Determined the isomorphism classes. If g = kn is a commutative Lie algebra, then Aut(g) = GLn(k).

• Let kτ[x] be the skew polynomial algebra with

xα = τ(α)x, then the GLn(k)-orbits are iso classes

• f n-dim representations of kτ[x].
• If ( , ) : kn × kn → k is a τ-sesquilinear form:

(αx, βy) = ατ(β)(x, y). Then the iso. classes of τ- sesquilinear forms are the GLn(k)-orbits in Mn(k) under the action (b’).

• If G is an algebraic group and X is a G × G-variety

with a dense B × B-orbit. Let F : G → G be a Fronenius homomorphism. Consider the group homomorphism G (1,F) → G × G. Then G-orbits in X has special interests. For example X = ¯ G is the wonderful compactification. see the work of Springer, Lusztig, He.

• II. Orbit classifications

Theorem

• 1. Let k be an algebraically closed and

τ(α) = αq (q = pr).The GLn(k)-orbits in Mn(k), under either action (a’) or (b’), are in one-to-one correspon- dence to the pairs (r, λ) with r being a nonnegative integer not exceeding n and λ is a partition of n − r. The pair (r, λ) corresponds to the matrix with a block decomposition

• Ir

Jλ

• .

Example 1. n = 1: α·A = α1−qA (or α1+qA). There are only two orbits either A = 0 or A = 1. The main reason of this finiteness condition is the following: Theorem 2 (Lang). If G is a connected algebraic group and F : G → G is an endomorphism of the alge- braic group such with finitely many fixed points, then the map: L : G → G (L(g) = g−1F(g)) is surjective. In particular, under both (a’) and (b’) actions, GLn(k) is one dense orbit in Mn(k). Remark 1. Jacobson studied the isomorphism classes

• f restricted commutative Lie algebras over perfect

field and classified all semi simple commutative re- stricted Lie algebras.

• III. Stabilizers groups

Theorem 3. For a G-orbit (r, λ) Stab(r, λ) ∼ = GLr(Fq) × (L(λ) ⋉ U(λ)) where L(λ) is a connected reductive algebraic group such that CGL|λ|(k)(Jλ) = L(λ) ⋉ Radu(CGL|λ|(k)(Jλ)) and U(λ) ∼ = Radu(CGL|λ|(k)(Jλ)) as algebraic varieties (only). Corollary

• 1. The identity connected component of

Stab(r, λ) is isomorphic to CGL|λ|(k)(Jλ) (as algebraic variety) and the component group A(r, λ) = Stab(r, λ)/(Stab(r, λ))0 ∼ = GLr(Fq). Corollary 2. The isomorphism classes of simple GLn(k)- equivariant perverse sheaves on Mn(k) is in one-to-one correspondence to irreducible characters of GLr(Fq).

• IV. The orbit closures

Let (r, λ) and (s, µ) be two pairs such that r + |λ| = s + |µ|. We define a partial ordering (r, λ) ≥ (s, µ) ⇔ r +

m

• i=1

λi ≥ r′ +

m

• i=1

µi for all m ≥ 0. Note that (r, λ) ≥ (r, µ) if and only if λ ≥ µ under the dominance order of partitions of n − r. For a partition, λ and a ∈ N, we define λ(a) to be the partition such that λ(a)

1

= λ1 + a and λ(a)

i

= λi for all i > 1. If r > 1, one can easily check that (r, λ) ≥ (r − 1, λ(1)) (by considering m ≥ λ1 or m ≥ λ1 + 1). Lemma

• 1. (r, λ) ≥ (s, µ) if and only if either r = s

and λ ≥ µ or r > s and λ(r−s) ≥ µ. Theorem 4. For any two orbits O(r,λ) and O(s,µ) in Mn(k), O(s,µ) ⊆ ¯ O(r,λ) if and only if (s, µ) ≤ (r, λ).

• V. The category of kτ[x]-mod

Theorem 5. If k is an algebraically closed of positive characteristic p and τ : k → k is defined by τ(α) = αq for all α ∈ k (q is a fixed power of p), Then the fol- lowing is a complete list of non-isomorphic finite di- mensional indecomposable representations of the skew polynomial ring kτ[x]: {(k, 1), (kn, Jn) | n = 1, 2, · · · }. Proposition 1. The following are true (a) Ext1

kτ[x](k1, k1) = 0 ;

(b) Ext1

kτ[x](k1, Jn) = Ext1 kτ[x](Jn, k1) = 0 ;

(c) Homkτ[x](k1, Jn) = Homkτ[x](Jn, k1) = 0 ; The category of nilpotent modules of kτ[x] is almost equivalent to that of k[x].

• VI. Representations of weighted quivers
• Q = (Q0, Q1), where w : Q1 → Z is a function

(weights of arrows). (Q, w) is a weighted quiver.

• (k, τ) fixed, a representation of (Q, w) is similar to

representation of the quiver Q over k except that the arrow α ∈ Q1 is τw(α)-semilinear .

• Weighted path algebra kτ(Q, w) (not a k-algebra!)
• If the underline graph of Q has not circuits (i.e, Q

is tree) then for any w the representation type of (Q, w) is the same as that of Q (independent of w!)

• If Q has circuits, and all different paths between

any two fixed vertices have the same weight, then the representation type is independent of w.

• The category of finite dimensional representations

is abelian and each hom space is a k-vector spaces, but not k-linear.

• VI. Various Hall algebras of weighted

quivers

Note that the result depends on the field being algebraically closed (Lang’s Theorem). One can define the Hall algebras over finite, but the classi- fication of orbits will be different. Over algebraically closed fields: Motive hall alge- bras and Hall algebras defined in terms of perverse sheaves. For example, for the skew polynomial algebra kτ[x], the vector spaces attached to each dimension vec- tor is class function algebras of various GLr(Fq) for r ≤ n and for each r there are P(n−r) many copies

• f the character ring.

H =

• n

n

• r=0

Char(GLr(Fq))P(n−r). The multiplication is more completed using the

• rbit closure relation.

THANK YOU!