Leibniz algebra deformations of a Lie algebra

Leibniz algebra deformations of a Lie a lgebra Alice Fialo wski ∗ and Ashis Mandal No v e m ber 4, 2018 Abstract In this note w e compute Lei bniz algebra deformations of the 3-dimensional nilp otent Lie algebra n 3 and compare it with its Lie deformations. It turns out t hat there are 3 extra Leibniz deformations. W e also describe the ver- sal Leibniz deformation of n 3 with the versal b ase. Keyw ords: Nilp otent Lie algebra , Leibniz algebra , cohomolo gy , infinitesi- mal, versal deformatio n. Mathematics Sub ject Classifications (2000): 1 3D10, 14 D15, 1 3D03. 1 In tro duction Since a Lie alg e bra is a lso a Leibniz alge br a, a natural que s tion ar is es. If we consider a Lie algebr a as a Leibniz a lg ebra and c ompute its Leibniz alg e bra deformations, is it true that w e can get more Leibniz alg ebra deformatio ns , than just the Lie deformations of the original Lie algebra ? In this note we will demonstra te the pro blem on a three dimensiona l Lie algebra ex a mple fo r which we completely descr ibe its versal Lie defor mation and versal Leibniz deformation. It turns out that b eside the Lie deformations we g et three non-equiv alent L eibniz deforma tio ns which are not Lie a lg ebras. Our example is the following. Consider a three dimensional v ector space L spanned by { e 1 , e 2 , e 3 } ov er C . Define the Heisenberg Lie algebr a n 3 on it with the br ack et matrix A =    0 0 1 0 0 0 0 0 0    . ∗ The wo rk wa s partially supp orted by gran ts fr om INSA India and HAS Hungary . 1 Here the columns are the Lie brack ets [ e 1 , e 2 ] , [ e 1 , e 3 ] and [ e 2 , e 3 ]. That means [ e 2 , e 3 ] = e 1 and all the other brackets are zero (except o f cour se [ e 3 , e 2 ] = − e 1 ). This is the only nilp otent thr ee dimensional Lie alg ebra. W e compute infinites- imal Lie a nd Leibniz deforma tions and show that there a re three a dditional Leibniz co cycles beside the five Lie co cycles . W e will show that all these in- finitesimal deformations are extendable without an y obstr uc tio ns. W e a lso de- scrib e the v ersal Leibniz deformation. The str ucture of the pa p er is as follows. In Section 1 we recall the necessa ry preliminaries ab out Lie and Leibniz cohomo logy and deformations. In Section 2 we r e call the cla s sification of three dimensio nal Lie algebr as and descr ib e all no n- equiv alent deformations of the nilp otent Lie a lgebra n 3 . In Section 3 we give the classifica tion of three dimensional nilp otent Leibniz a lg ebras. Among those n 3 is the only nontrivial Lie a lgebra. Then we compute Leibniz co ho mology and give explicitly all non-e q uiv alent infinitesimal deformations. In Sectio n 4 we show that all infinitesimal deforma tions are extendable as a ll the Mas sey squares turn o ut to b e zero. W e identify our deforma tions with the classified ob jects. Finally w e show that the versal Leibniz defor mation is the universal infinitesimal one, and describ e the ba s e of the versal deformation. 2 Preliminaries Let us recall firs t the Lie algebra cohomolog y . Definition 2.1. Su pp ose g is a Lie algebr a and A is a mo dule over g . T hen a q -dimensional c o chain of the Lie algebr a g with c o efficients in A is a skew -symmetric q - line ar map on g with values in A ; t he sp ac e of a l l such c o chains is denote d by C q ( g ; A ) . Thus, C q ( g ; A ) = H om (Λ q g , A ) ; this last r epr esentation tr ansforms C q ( g ; A ) int o a g -mo dule. The differ ential d = d q : C q ( g ; A ) − → C q +1 ( g ; A ) is define d by the formula dc ( g 1 , · · · , g q +1 ) = X 1 ≤ s . References [1] Alb everio, S. and O mirov, B. A. and Rakhimo v , I. S. 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Math. 7 (2), (2005), 1 4 5-16 5 , math.R T/051 2354. 14 [7] Lo day J.- L , Une version c on-c ommutative des algebr es de Lie: L es algebr es de L eibniz , E nseign. Ma th., 39, No.3-4 (1993), 269-293. [8] Lo day J .-L, Overview on L eibniz algebr as, dialgebr as and their homolo gy , Fields Institut e Co mmu nications, 1 7 (19 97), 91- 102. [9] Lo day J.-L and Pir ashvili, T. Universal envelop ing algebr as of L eibniz al ge- br as and (c o)homolo gy , Math.Ann., 296 (1993),1 39-15 8. [10] Manda l, A. An Example of Constructing V ersal D eformation for L eibniz Alge br as , T o app ear in Co mm. Alg ., arXiv : math.QA/ 071 212.2 0 96v1 1 3 Dec 2 007. [11] Schlessinger, M. F unctors of Artin rings , T r a ns. Amer. Math. So c. 130 (1968), 20 8-222 . Alice Fialowski E ¨ o tv ¨ o s Lor´ a nd Universit y , Budap est, Hunga ry . e-mail: fialowsk@cs.elte.hu Ashis Mandal Indian Statistical Institute, Ko lk ata, India. e-mail: as his r@isical.a c.in 15