Gr obner-Shirshov bases method in algebra L.A. Bokut Introduction - PowerPoint PPT Presentation

Gr¨ obner-Shirshov bases method in algebra L.A. Bokut Introduction Composition-Diamond . . . Examples Sobolev Institute of Mathematics, Russia PBW theorems Linear bases of free . . . Normal forms for . . . South China Normal University, China Extensions of groups . . . Embedding algebras Yuqun Chen Home Page South China Normal University, China Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 37 Go Back Novosibirsk, July 21-25, 2014. Full Screen Close Quit

1 Introduction Introduction Composition-Diamond . . . Examples PBW theorems Seminar was organized by the authors in March, 2006. Since Linear bases of free . . . Normal forms for . . . Extensions of groups . . . then, there were some 30 Master Theses and 4 PhD Theses, Embedding algebras about 40 published papers in JA, IJAC, Comm. Algebra, Al- Home Page gebra Coll. and other Journals and Proceedings. There were Title Page organized 2 International Conferences (2007, 2009) with E. ◭◭ ◮◮ ◭ ◮ Zelmanov as Chairman of the Program Committee and sev- Page 2 of 37 eral Workshops. We are going to review some of the papers. Go Back Full Screen Close Quit

Our main topic is Gr¨ obner-Shirshov bases method for dif- Introduction Composition-Diamond . . . ferent varieties (categories) of linear ( Ω -) algebras over a Examples PBW theorems Linear bases of free . . . field k or a commutative algebra K over k : associative al- Normal forms for . . . Extensions of groups . . . gebras (including group (semigroup) algebras), Lie algebras, Embedding algebras dialgebras, conformal algebras, pre-Lie (Vinberg right (left) Home Page symmetric) algebras, Rota-Baxter algebras, metabelian Lie Title Page ◭◭ ◮◮ algebras, L -algebras, semiring algebras, category algebras, ◭ ◮ etc. There are some applications particularly to new proofs Page 3 of 37 of some known theorems. Go Back Full Screen Close Quit

2 Composition-Diamond lemmas As it is well known, Gr¨ obner-Shirshov (GS for short) bases method for a class of algebras based on a Composition-Diamond lemma (CD- Introduction lemma for short) for the class. A general form of a CD-Lemma over a Composition-Diamond . . . field k is as follows. Examples PBW theorems Linear bases of free . . . Composition-Diamond lemma Let M ( X ) be a free algebra of a cat- Normal forms for . . . Extensions of groups . . . egory M of algebras over k , ( N ( X ) , ≤ ) a linear basis (normal words) Embedding algebras of M ( X ) with an ”addmissible” well order and S ⊂ M ( X ) . TFAE Home Page (i) S is a GS basis (i.e. each “composition” of polynomials from S is Title Page “trivial”). ◭◭ ◮◮ (ii) If f ∈ Id ( S ) , then the maximal word of f has a form ¯ f = ◭ ◮ sb ) , s ∈ S, a, b ∈ X ∗ . ( a ¯ Page 4 of 37 Go Back sb ) , s ∈ S, a, b ∈ X ∗ } is a linear (iii) Irr ( S ) = { u ∈ N ( X ) | u � = ( a ¯ Full Screen basis of M ( X | S ) = M ( X ) /Id ( S ) . Close The main property is ( i ) ⇒ ( ii ) . Quit

CD-lemma for associative algebras Introduction Let k � X � be the free associative algebra over a field k gener- Composition-Diamond . . . Examples ated by X and ( X ∗ , < ) a well-ordered free monoid generated PBW theorems Linear bases of free . . . Normal forms for . . . by X , S ⊂ k � X � such that every s ∈ S is monic. Extensions of groups . . . Embedding algebras Let us prove ( i ) ⇒ ( iii ) and define a GS basis. Let f = � n i =1 α i a i s i b i ∈ Id ( S ) where each α i ∈ k, a i , b i ∈ Home Page Title Page X ∗ , s i ∈ S, w i = a i s i b i , w 1 = w 2 = · · · = w l > w l +1 ≥ ◭◭ ◮◮ · · · . ◭ ◮ For l = 1 , it is ok. Page 5 of 37 Go Back For l > 1 , w 1 = a 1 s 1 b 1 = a 2 s 2 b 2 , common multiple of s 1 , s 2 , Full Screen by definition, Close Quit

Introduction w 1 = cwd, w = “ lcm ”( s 1 , s 2 ) , a i s i b i = w | s i �→ s i , i = 1 , 2 , Composition-Diamond . . . Examples PBW theorems Linear bases of free . . . where lcm ( u, v ) ∈ { ucv, c ∈ X ∗ ( a trivial lcm ( u, v )); u = Normal forms for . . . Extensions of groups . . . avb, a, b ∈ X ∗ ( an inclusion lcm ( u, v )); ub = av, a, b ∈ Embedding algebras X ∗ , | ub | < | u | + | v | ( an intersection lcm ( u, v ) } . Home Page Then a 1 s 1 b 1 − a 2 s 2 b 2 = c ( w | s 1 �→ s 1 − w | s 2 �→ s 2 ) d = c ( s 1 , s 2 ) w d . Title Page ◭◭ ◮◮ By definition of GS basis, ( s 1 , s 2 ) w ≡ 0 mod ( S, w ) . So, ◭ ◮ a 1 s 1 b 1 − a 2 s 2 b 2 ≡ 0 mod ( S, w 1 ) . We can decrease l . By Page 6 of 37 induction, ¯ sb, a, b ∈ X ∗ , s ∈ S . f = a ¯ Go Back Full Screen Close Quit

CD-lemma for Lie algebras over a field Introduction Composition-Diamond . . . Examples PBW theorems Let S ⊂ Lie ( X ) ⊂ k � X � be a nonempty set of monic Lie Linear bases of free . . . Normal forms for . . . Extensions of groups . . . polynomials, ( X ∗ , < ) deg-lex order, ¯ s means the maximal Embedding algebras word of s as non-commutative polynomial, Home Page Title Page � s 1 , s 2 � w = [ w ] s 1 | s 1 �→ s 1 − [ w ] s 2 | s 2 �→ s 2 , w ∈ ALSW ( X ) ◭◭ ◮◮ ◭ ◮ associative composition with the special Shirshov bracket- Page 7 of 37 ing. Go Back Full Screen Close Quit

Introduction Composition-Diamond . . . CD-lemma for Lie algebras over a field . TFAE Examples PBW theorems Linear bases of free . . . (i) S is a Lie GS basis in Lie ( X ) (any composition is trivial Normal forms for . . . Extensions of groups . . . Embedding algebras modulo ( S, w ) ). (ii) f ∈ Id Lie ( S ) ⇒ ¯ sb for some s ∈ S and a, b ∈ X ∗ . f = a ¯ Home Page Title Page (iii) Irr ( S ) = { [ u ] ∈ NLSW ( X ) | u � = a ¯ sb, s ∈ S, a, b ∈ ◭◭ ◮◮ X ∗ } is a linear basis for Lie ( X | S ) . ◭ ◮ Page 8 of 37 Go Back Full Screen Close Quit

3 Examples 1. Poincare-Birkhoff-Witt theorem Introduction Composition-Diamond . . . Examples Let L = Lie k ( X | S ) be a Lie algebra over a field k present- PBW theorems Linear bases of free . . . Normal forms for . . . ed by a well-ordered linear basis X = { x i | i ∈ I } and the Extensions of groups . . . multiplication table S = { [ x i x j ] − � α t Embedding algebras ij x t | i > j, i, j ∈ I } , � Home Page U ( L ) = k � X | S ( − ) � , S ( − ) = { x i x j − x j x i − α t ij x t | i > j } Title Page ◭◭ ◮◮ be the universal enveloping associative algebra for L . ◭ ◮ Then with deg-lex order on X ∗ , S ( − ) is a GS basis and hence Page 9 of 37 Go Back following the CD-Lemma for associative algebras a linear Full Screen basis of U ( L ) consists of words x i 1 x i 2 . . . x i n , i 1 ≤ i 2 ≤ · · · ≤ Close i n , n ≥ 0 . Quit

2. Symmetric group S n +1 Introduction Symmetric group S n +1 is isomorphic to the group Composition-Diamond . . . Examples PBW theorems Linear bases of free . . . Coxeter ( A n ) = gp � s 1 , . . . , s n | s 2 i = 1 , Normal forms for . . . Extensions of groups . . . Embedding algebras s i +1 s i s i +1 = s i s i +1 s i , s i s j = s j s i , i − j > 1 � = : gp � Σ | S � Home Page Title Page ◭◭ ◮◮ with an isomorphism s i �→ ( i, i + 1) , 1 ≤ i ≤ n . ◭ ◮ A GS basis of Coxeter ( A n ) is Page 10 of 37 Go Back S ∪{ s i +1 s i s i − 1 . . . s j s i +1 − s i s i +1 s i s i − 1 . . . s j | 1 ≤ j ≤ ( i − 1) } . Full Screen Close Quit

Introduction By CD-Lemma for associative algebras a set of normal forms Composition-Diamond . . . Examples of elements of the group consists of words PBW theorems Linear bases of free . . . Normal forms for . . . Extensions of groups . . . s 1 j 1 . . . s nj n , j 1 ≤ 2 , . . . , j n ≤ n + 1 , Embedding algebras s ij = s i s i − 1 . . . s j , j ≤ i, s i ( i +1) = 1 . Home Page Title Page Hence | Coxeter ( A n ) | = ( n + 1)! and we are done. ◭◭ ◮◮ Analogous results are valid for all finite Coxeter groups (of ◭ ◮ Page 11 of 37 types A n (before), B n , D n , G 2 , F 4 , E 6 , E 7 , E 8 ). Go Back Full Screen Close Quit

3. Lie algebra sl n +1 ( k ) , chark � = 2 Special linear (trace zero) Lie algebra sl n +1 ( k ) over a field Introduction Composition-Diamond . . . Examples k, chark � = 2 is isomorphic to the Lie algebra PBW theorems Linear bases of free . . . Normal forms for . . . Extensions of groups . . . Lie ( A n ) = Lie ( h i , x i , y i , 1 ≤ i ≤ n | [ h i h j ] = 0 , Embedding algebras [ x i y j ] = δ ij h i , [ h i x j ] = 2 δ ij x i , [ h i y j ] = − 2 δ ij y i , Home Page [[ x i +1 [ x i +1 x i ]] = 0 , [ x j x i ] = 0 , Title Page ◭◭ ◮◮ [[ y i +1 [ y i +1 y i ]] = 0 , [ y j y i ] = 0 , j � = i + 1) ◭ ◮ Page 12 of 37 with the isomorphism Go Back Full Screen h i �→ e ii − e i +1 i +1 , x i �→ e ii +1 , y i �→ e i +1 i , 1 ≤ i ≤ n. Close Quit