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 M n ( k ) of n × n -matrices, describe the equivalence classes of bilinear forms on n -dimensional vector spaces, i.e., GL n ( k )-orbits in M n ( k ) with respect to the following actions: g ∈ GL n ( k ) and A ∈ M n ( k ) (a). g · A = gAg − 1 . (b). g · A = gAg tr . 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 (( g ij )) = ( τ ( g ij )). In particular when k is a perfect field of characteristics p > 0 and τ ( α ) = α p .

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 ] = α p X [ p ] and ( X + Y ) [ p ] = X [ p ] + Y [ p ] + � p − 1 s i ( X,Y ) where s i ( X, Y ) is the coefficient of i =1 i t i − 1 in the formal expression ad( tX + Y ) p − 1 ( X ). Each [ p ] is determined by a matrix in M n ( k ) under a basis and Aut( g ) ⊂ GL ( g ) acts on M n ( k ) by ( a ′ ). Determined the isomorphism classes. If g = k n is a commutative Lie algebra, then Aut( g ) = GL n ( k ). • Let k τ [ x ] be the skew polynomial algebra with xα = τ ( α ) x , then the GL n ( k )-orbits are iso classes of n -dim representations of k τ [ x ]. • If ( , ) : k n × k n → k is a τ -sesquilinear form: ( αx, βy ) = ατ ( β )( x, y ). Then the iso. classes of τ - sesquilinear forms are the GL n ( k )-orbits in M n ( 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 For example X = ¯ X has special interests. G is the wonderful compactification. see the work of Springer, Lusztig, He.

II. Orbit classifications 1. Let k be an algebraically closed and Theorem τ ( α ) = α q ( q = p r ).The GL n ( k ) -orbits in M n ( 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 � � I r 0 . 0 J λ Example 1. n = 1: α · A = α 1 − q A (or α 1+ q A ). There are only two orbits either A = 0 or A = 1. The main reason of this finiteness condition is the following: 2 (Lang) . If G is a connected algebraic Theorem 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 − 1 F ( g ) ) is surjective. In particular, under both (a’) and (b’) actions, GL n ( k ) is one dense orbit in M n ( k ) . Remark 1. Jacobson studied the isomorphism classes of 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 ) ∼ = GL r ( F q ) × ( L ( λ ) ⋉ U ( λ )) where L ( λ ) is a connected reductive algebraic group such that C GL | λ | ( k ) ( J λ ) = L ( λ ) ⋉ Rad u ( C GL | λ | ( k ) ( J λ )) and U ( λ ) ∼ = Rad u ( C GL | λ | ( k ) ( J λ )) as algebraic varieties (only). 1. The identity connected component of Corollary Stab( r, λ ) is isomorphic to C GL | λ | ( k ) ( J λ ) (as algebraic variety) and the component group A ( r, λ ) = Stab( r, λ ) / (Stab( r, λ )) 0 ∼ = GL r ( F q ) . Corollary 2. The isomorphism classes of simple GL n ( k ) - equivariant perverse sheaves on M n ( k ) is in one-to-one correspondence to irreducible characters of GL r ( F q ) .

IV. The orbit closures Let ( r, λ ) and ( s, µ ) be two pairs such that r + | λ | = s + | µ | . We define a partial ordering m m λ i ≥ r ′ + � � ( r, λ ) ≥ ( s, µ ) ⇔ r + µ i for all m ≥ 0 . i =1 i =1 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 ) = λ 1 + a and λ ( a ) = λ i for 1 i all i > 1. If r > 1, one can easily check that ( r, λ ) ≥ ( r − 1 , λ (1) ) (by considering m ≥ λ 1 or m ≥ λ 1 + 1). 1. ( r, λ ) ≥ ( s, µ ) if and only if either r = s Lemma 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 M n ( k ) , O ( s,µ ) ⊆ ¯ O ( r,λ ) if and only if ( s, µ ) ≤ ( r, λ ) .

V. The category of k τ [ x ] -mod Theorem 5. If k is an algebraically closed of positive characteristic p and τ : k → k is defined by τ ( α ) = α q 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 k τ [ x ] : { ( k, 1) , ( k n , J n ) | n = 1 , 2 , · · · } . Proposition 1. The following are true (a) Ext 1 k τ [ x ] ( k 1 , k 1 ) = 0 ; (b) Ext 1 k τ [ x ] ( k 1 , J n ) = Ext 1 k τ [ x ] ( J n , k 1 ) = 0 ; (c) Hom k τ [ x ] ( k 1 , J n ) = Hom k τ [ x ] ( J n , k 1 ) = 0 ; The category of nilpotent modules of k τ [ x ] is almost equivalent to that of k [ x ].

VI. Representations of weighted quivers • Q = ( Q 0 , Q 1 ), where w : Q 1 → 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 ∈ Q 1 is τ w ( α ) -semilinear . • Weighted path algebra k τ ( 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 k τ [ x ], the vector spaces attached to each dimension vec- tor is class function algebras of various GL r ( F q ) for r ≤ n and for each r there are P ( n − r ) many copies of the character ring. n Char( GL r ( F q )) P ( n − r ) . � � H = n r =0 The multiplication is more completed using the orbit closure relation.

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