EXAMPLES OF HOMOTOPY LIE ALGEBRAS. Klaus Bering and Tom Lada

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1 ARCHIVUM MATHEMATICUM BRNO) Tomus ), EXAMPLES OF HOMOTOPY LIE ALGEBRAS Klaus Bering and Tom Lada Abstract. We look at two examples of homotopy Lie algebras also known as L algebras) in detail from two points of view. We will exhibit the algebraic point of view in which the generalized Jacobi expressions are verified by using degree arguments and combinatorics. A second approach using the nilpotency of Grassmann-odd differential operators to verify the homotopy Lie data is shown to produce the same results. 1. Introduction Homotopy Lie algebras, or L algebras, have been a topic of great interest to both mathematical physicists and to algebraists. By considering two different points of view, one can hope to gain a deeper understanding of these structures. In this note, we provide notations and definitions used by both communities, and hopefully illuminate both perspectives. On one hand, the second author and his collaborators [4, 5] have algebraicly constructed two concrete finite dimensional examples of homotopy Lie algebras from first principles. On the other hand, the first author and his collaborators have developed a generalization of the Batalin-Vilkovisky formalism [1] in which a nilpotent, Grassmann-odd, differential operator may be used to identify L structures, cf. Lemma in Sec. 2.3 of Ref. [3], and Theorem 3.6 in Ref. [2]. This method is here applied to rederive the two examples of the second author and his collaborators. 2. Homotopy Lie algebras We begin by recalling the definition of an L algebra [7], [6]. Definition 1. An L algebra structure on a Z graded vector space V is a collection of graded skew symmetric linear maps l n : V n V of degree 2 n that satisfy generalized Jacobi identities 2.1) eσ) 1) σ 1) ij 1) i+j=n+1 σ l j li v σ1),..., v σi) ), v σi+1),..., v σn) ) = 0, 2000 Mathematics Subject Classification: primary 18G55. Key words and phrases: homotopy Lie algebras, generalized Batalin-Vilkovisky algebras, Koszul brackets, higher antibrackets.

2 266 K. BERING AND T. LADA where 1) σ is the sign of the permutation, eσ) is the Koszul sign which is equal to 1 raised to the product of the degrees of the permuted elements, and σ is taken over all i, n i) unshuffles. This is the cochain complex point of view; for chain complexes, require the maps l n to have degree n Desuspension. We will require an equivalent way to describe homotopy Lie algebra data that will be compatible with the operator approach. Definition 2. Let S c W ) be the cofree cocommutative coassociative coalgebra on the graded vector space W. Then an L algebra structure on W is a coderivation D : S c W ) S c W ) of degree +1 such that D 2 = 0. Given an L algebra structure V, l i ) as in Definition 1, we may desuspend V to obtain the graded vector space W = V, where W n = V n+1 and is the desuspension operator. Define D : S c W ) S c W ) by D = ˆl 0 + ˆl 1 + ˆl , where each ˆl n is a degree +1 symmetric map given by 2.2) ˆln = 1) nn 1) 2 l n n : S c W ) W, and then extended to a coderivation in the usual fashion. We will demonstrate this construction explicitly in the examples. The examples that we consider will be structures on relatively small graded vector spaces: V = V 0 V 1, where each V i is finite dimensional. When we desuspend V, we will consider the graded vector space W = W 1 W 0. We now describe the operator approach. 3. The operator approach 3.1. Vector space W with two Fermions. To be concrete, we let dimw 1 ) = 2. We use Greek indices α, β,... {1, 2} for a Fermionic basis θ α W 1 with Grassmann parity εθ α ) = 1. On the other hand, it will be useful to allow W 0 in the beginning to have infinitely many dimensions, and only at the very end perform a consistent truncation to a finite dimensional subspace. We use roman indices i, j,... {1, 2,...} for the infinitely many Bosonic/even variables x i W 0 with Grassmann parity εx i ) = 0. Hence, we are given a super) vector space 3.1) W := W 1 W 0, W 1 := span θ 1, θ 2, W 0 := span x 1, x 2,..., x i,.... We will for simplicity here only consider one kind of grading, although it is easy to generalize to several Z 2 and Z gradings. In Section 2 we introduced a Z grading, called the degree. From an operational point of view, only a Z 2 grading, the so-called Grassmann parity ε, is needed. We shall start by only considering the Z 2 grading ε, and only later implement the full Z grading. This will lead to selection rules, i.e., further restrictions.

3 EXAMPLES OF HOMOTOPY LIE ALGEBRAS Algebra. For an operational point of view, we use the fact the cocommutative coalgebra S c W ) has the same underlying vector space as the super) symmetric algebra A := Sym W ), where 3.2) x i x j = x j x i, x i θ α = θ α x i, θ α θ β = θ β θ α, or 3.3) z w = 1) εz)εw) w z for short, where z, w W Bracket hierarchy Φ ˆl. The family of maps ˆl on W will be denoted by Φ to conform with notation used in Ref. [3] and Ref. [2]. We shall not always write super) symmetric tensor symbol explicitly. The sign convention is as follows: ε Φ n z 1 z 2 z n )) = 1 + εz 1 ) + εz 2 ) εz n ), Φ n z k z k+1... ) = 1) εz k)εz k+1 ) Φ n z k+1 z k... ), 3.4) Φ n λz 1 z 2 z n ) = 1) ελ) λφ n z 1 z 2 z n ), Φ n z 1 z n λ) = Φ n z 1 z n )λ, z k λ = 1) εz k)ελ) λz k. Here λ is a super number. We shall use multi-index notation 3.5) m = m 1, m 2,..., m i,... ), m = m i, m! = m i!, i=1 x m = x m1 1 x m2 2 x mi i.... i=1 The most general bracket hierarchy Φ on W is 3.6) 3.7) 3.8) Φ m x m ) = c α mθ α, Φ m +1 θ α x m ) = b i αmx i, Φ m +2 θ α θ β x m ) = ɛ αβ a γ mθ γ, where a α m, b i αm and c γ m are coefficients, and where 3.9) ɛ αβ = ɛ βα, ɛ αβ ɛ βγ = δ α γ, ɛ 12 = 1 = ɛ The operator. Define generating functions 3.10) f α p) := m a α p m m m!, gi αp) := m b i p m αm m!, hα p) := m c α p m m m!.

4 268 K. BERING AND T. LADA Define operator 3.11) 3.12) 3.13) := , 2 := 1 2 θ γf γ x )ɛ αβ, θ β θ α 1 := x i gα i x ), θ α 3.14) 0 := θ α h α x ). We will from now on not always write the for Koszul bracket hierarchy Φ. Define Koszul brackets hierarchy Φ as 3.15) 3.16) where x dependence explicitly in the formula Φ n z 1 z n ) := [[... [, L z1 ],... ], L zn ] 1, }{{} n commutators Φ 0 := 1) θ αc α 0, 3.17) L z w) := zw is the left multiplication operator with algebra element z. It is easy to check that the Φ Koszul brackets hierarchy 3.15)-3.16) reproduces the original Φ bracket hierarchy 3.6)-3.8): 3.18) Φ = Φ L structure and nilpotency conditions. A consequence of Lemma in Section 2.3 of [3], or alternatively Theorem 3.6 in [2], is that Φ forms a homotopy Lie algebra if and only if is nilpotent of order two), i.e., squares to zero, 3.19) 2 1 [, ] = 0. 2 We calculate: 3.20) 3.21) 3.22) 3.23) 3.24) 3.25) [ 2, 2 ] = 0, [ 2, 1 ] = 1 2 x igγf i γ ɛ αβ, θ β θ α [ 1, 1 ] = 2x i gα,jg i j β θ α θ β = x igα,jɛ i αβ g j β ɛ γδ θ δ θ γ, [ 2, 0 ] = θ γ f γ h α ɛ αβ, θ β [ 1, 0 ] = θ α h α,igβ i θ β + x igαh i α, [ 0, 0 ] = 0.

5 EXAMPLES OF HOMOTOPY LIE ALGEBRAS 269 For instance, eq. 3.21) is proved as follows. Write shorthand 2 = θ γ D γ, where 3.26) D γ := 1 2 f γ x )ɛ αβ. θ β θ α Then 3.27) [ 2, 1 ] = θ γ [D γ, 1 ] + [θ γ, 1 ]D γ = θ γ [D γ, 1 ] + [ 1, θ γ ]D γ = θ γ [D γ, x i gδ i ] + [x i g i α, θ γ ]D γ θ δ θ α = θ γ [D γ, x i ]gδ i + x i g i θ α[, θ γ ]D γ δ θ α = 1 2 θ γf γ,iɛ αβ θ β g i δ + x i g θ α θ γd i γ. δ Note that the first term on the right hand side of 3.27) must vanish because it contains three Fermionic derivatives, but there are only two different Fermions. The second term yields the result 3.21). Altogether, the nilpotency condition 2 = 0 read 3.28) 3.29) 3.30) g i γf γ + g i α,jɛ αβ g j β = 0, f α h γ ɛ γβ + h α,ig i β = 0, g i αh α = Special cases. Let us now discuss special cases. Let us assume h α 0. Then the two last nilpotency conditions are satisfied, and only the first of the three nilpotency conditions 3.28) 3.30) remains. Notice that we can explain h α 0 as a selection rule from the degree Z grading, where x i W 0 have degree 0; θ α W 1 have degree 1; the brackets Φ have degree +1; and the operator has degree +1. Then c α m 0, h α 0, and 0 0. Let us assume only one Bosonic/even variable x x 1, i.e., 0 = x 2 = x 3 =.... Then the first nilpotency condition 3.28) reads: 3.31) where 3.32) g γ f γ + W g 1, g 2 ) = 0, W g 1, g 2 ) := g αɛ αβ g β g 1g 2 g 1 g 2 is the Wronskian. Let us assume that g 1 is given with g 1 p = 0) b α=1,m=0 0. Then we can interpret the inverse 1/g 1 as a formal power series.

6 270 K. BERING AND T. LADA If there is also given g 2, then we can e.g., choose 3.33) f 1 = W g 1, g 2 ) g 1, f 2 = 0. Or if there instead is also given f 1, then we can e.g., choose g ) = dp f 1, f 2 = 0. g 1 g 1 4. First example 4.1. Algebra approach. The following L algebra was studied in [5]. Let V = V 0 V 1 be the graded vector space where V 0 has basis v 1, v 2 and V 1 has basis w. Define l n : V n V by 4.1) l 1 v 1 ) = l 1 v 2 ) = w, l 2 v 1 v 2 ) = v 1, l 2 v 1 w) = w, l n v2 w n 1) = C n w for n 3, and all other sectors are zero, and where C n = 1) n 2)n 3) 2 n 3)!. To verify the L relations 2.1), the summands in the L relation can be calculated as follows. The first summand reads 4.2) l 1 l n v1 v 2 w n 2) = 0. The next summand reads 4.3) For all 3 k n 3 we have l 2 l n 1 v1 v 2 w n 2) = 1) n 1 l 2 ln 1 v 2 w n 2 ) v 1 ) = 1) n 1 C n 1 l 2 w v 1 ) = 1) n C n 1 w. 4.4) l k l n k+1 v1 v 2 w n 2) = 0, because each summand in this expansion contains the term l k v 1 w k 1 ) = 0. The second-last summand reads 4.5) l n 1 l 2 v1 v 2 w n 2) = l n 1 l2 v 1 v 2 w n 2) n 2)l n 1 l2 v 1 w) v 2 w n 3) = l n 1 v 1 w n 2 ) n 2)l n 1 w v2 w n 3) = 0 + n 2)l n 1 v 2 w n 2 ) = n 2)C n 2 w. The last summand reads 4.6) l n l 1 v 1 v 2 w n 2 ) = l n w v 2 w n 2 ) l n w v 1 w n 2 ) = C n w.

7 EXAMPLES OF HOMOTOPY LIE ALGEBRAS 271 Consequently, the nth Jacobi expression is satisfied if and only if n 1) pn p) l n p+1 l p v1 v 2 w n 2) = 0 p=1 1) n 1)1 1) n C n 1 w + 1) 2n 2) n 2)C n 1 w + 1) 1n 1) 1)C n w = 0 4.7) 1)C n 1 + n 2)C n 1 + 1) n C n = 0 C n = 1) n 1 n 3)C n 1. One can check that C n must equal 1) n 2)n 3) 2 n 3)! Desuspension. We next desuspend V to obtain W = W 1 W 0 where W 1 has basis θ 1, θ 2 and W 0 has basis x and then rewrite the L data in terms of degree +1 maps 4.8) ˆl1 θ 1 ) = ˆl 1 θ 2 ) = x, ˆl2 θ 1 θ 2 ) = θ 1, ˆln θ2 x n 1) = 1) n n 3)!x, and all other sectors are zero. The ˆl n s will correspond to the Φ n s in the next section operator approach. The algebra of [5] has only one Bosonic generator x x 1, and is given as Φ 2 θ 1 θ 2 ) = θ 1, Φ 1 θ 1 ) = x, Φ 2 θ 1 x) = x, 4.9) Φ m+1 θ 2 x m) x for m = 0, = 0 for m = 1, 1) m m 2)! x for m 2, and all other sectors are zero. Thus the coefficients are a 1 m = δ 0 m, a 2 m = 0, b 1m = δ 0 m + δ 1 m, 1 for m = 0, 4.10) b 2m = 0 for m = 1, 1) m m 2)! for m 2. The generating functions become 4.11) g 2 p) = 1 f 1 p) = 1, f 2 p) = 0, g 1 p) = 1 + p, m=2 p) m = 1 + p)[1 ln1 + p)]. mm 1) It is easy to check that the nilpotency condition 3.31) is satisfied. Alternatively, g 2 could have been predicted from eq. 3.34).

8 272 K. BERING AND T. LADA 5. Second example 5.1. Algebra approach. This next example was constructed by M. Daily [4]. Let V = V 0 V 1 with dimv 1 ) dimv 0 ). Denote the basis for V 0 by v 1,..., v i and the basis for V 1 by w 1,..., w j. Define l 1 v i ) = w i, l 2 v i v j ) = 0, l 2 v i w j ) = w i + w j, 5.1) l n v i v j w-terms) = 0, l n v i w-terms) = C n w i for n 3, and all other sectors are zero. We begin verification of the L algebra relations 2.1) with 5.2) l 1 l 2 v i v j ) l 2 l 1 v i v j ) = l 1 0) [l 2 l 1 v i ) v j ) l 2 l 1 v j ) v i )] = 0 l 2 w i v j ) + l 2 w j v i ) = w j + w i w i + w j ) = 0. We next consider the generalized Jacobi expression evaluated on v i v j w k. The first summand reads 5.3) l 1 l 3 v i v j w k ) = 0. The next summand reads 5.4) l 2 l 2 v i v j w k ) = l 2 l 2 v i v j ) w k ) l 2 l 2 v i w k ) v j ) + l 2 l 2 v j w k ) v i ) = 0 l 2 w i + w k ) v j ) + l 2 w j + w k ) v i ) = w j + w i ) + w j + w k ) w i + w j ) w i + w k ) = w i + w j. The last summand reads 5.5) l 3 l 1 v i v j w k ) = l 3 w i v j w k ) l 3 w j v i w k ) = C 3 w j + C 3 w i. Thus, the generalized Jacobi expression 5.6) l 1 l 3 + l 2 l 2 + l 3 l 1 )v i v j w k ) = 0 w i + w j C 3 w j + C 3 w i = 0 C 3 = 1. For n 4, we compute 5.7) n 1) pn p) l n p+1 l p v i v j w k1 w kn 2 ). p=1

9 EXAMPLES OF HOMOTOPY LIE ALGEBRAS 273 The first summand with p = 1 reads l n l 1 v i v j w k1 w kn 2 ) = l n w i v j w k1 w kn 2 ) l n w j v i w k1 w kn 2 ) 5.8) = C n w j + C n w i = C n w i w j ). The next summand with p = 2 reads l n 1 l 2 v i v j w k1 w kn 2 ) = l n 1 l 2 v i w kα ) v j w-terms) α + α = α + α l n 1 l 2 v j w kα ) v i w-terms) l n 1 w i + w kα ) v j w-terms) l n 1 w j + w kα ) v i w-terms) 5.9) For 3 p n 2, we have = 2n 2)C n 1 w j w i ). l n p+1 l p v i v j w k1 w kn 2 ) ) n 2 = 1) p 1 l n p+1 l p v i w terms) v j w-terms) ) n 2 1) p 1 l n p+1 l p v j w-terms) v i w-terms) ) n 2 = 1) p 1 l n p+1 C p w i v j w-terms) ) n 2 1) p 1 l n p+1 C p w j v i w-terms) ) ) n 2 n 2 = 1) p C n p+1 C p w j 1) p C n p+1 C p w i ) n ) = 1) p+1 C n p+1 C p w i w j ). The second-last summand with p = n 1 reads 5.11) l 2 l n 1 v i v j w k1 w kn 2 ) = 1) n 2 l 2 l n 1 v i w-terms) v j ) 1) n 2 l 2 l n 1 v j w-terms) v i ) = 1) n 2 l 2 C n 1 w i v j ) 1) n 2 l 2 C n 1 w j v i ) = 1) n 1 C n 1 w j + w i ) 1) n 1 C n 1 w i + w j ) = 0.

10 274 K. BERING AND T. LADA The last summand with p = n reads 5.12) l 1 l n v i v j w k1 w kn 2 ) = 0. We add together all of the above summands with p = 1, 2,..., n to obtain 5.13) So, 5.14) n 1) pn p) l n p+1 l p v i v j w k1 w kn 2 ) p=1 = 1) n 1 C n w i w j ) 2n 2)C n 1 w i w j ) n 2 ) n 2 + 1) pn p) 1) p+1 C n p+1 C p w i w j ) p=3 n 1) pn p) l n p+1 l p v i v j w k1 w kn 2 ) = 0 p=1 1) n 1 C n 2n 2)C n 1 n 2 ) n 2 + 1) pn+1 C n p+1 C p = 0. p=3 One can then solve for 5.15) C n = 1) n[ n 2 n 2 ] 2n 2)C n 1 + 1) )C pn+1 n p+1 C p with C 3 = Desuspension. As before, we desuspend the vector space to obtain W = W 1 W 0 and convert the l n s to degree +1 symmetric maps and end up with the homotopy Lie algebra structure given by p=3 ˆl1 θ i ) = x i, ˆl2 θ i θ j ) = 0, ˆl2 θ i x j ) = x i + x j, ˆln θ i θ j x-terms) = 0, 5.16) ˆln θ i x-terms) = 1) nn 1) 2 1) n 1 C n x i for n 3, and all other sectors are zero. The last equation may be rewritten as 5.17) ˆln θ i x-terms) = 2 n) n 2 x i for n operator approach. In the following we let dimw 1 ) = 2, to conform with the theory developed in Section 3. Moreover, it is practical to let W 0 have infinitely many Bosonic generators

11 EXAMPLES OF HOMOTOPY LIE ALGEBRAS 275 x i. It will be consistent to truncate the tail 0 = x N+1 = x N+2 =... to reduce to only finitely many generators x 1,..., x N.) Then the second example is of the form 5.18) Φ 1 θ α ) = B 0 x α, Φ 2 θ α x i ) = B 1 x α + x i, Φ m +1 θ α x m ) = B m x α for m 2, and all other sectors are zero, and where B 0, B 1, B 2,... are complex numbers with B 0 0. By scaling 5.19) x i = B 0 x i, θ α = θ α, Φ = Φ, B M = B 0 ) M 1 B M for M = 0, 1, 2,..., and by dropping the primes again afterwards) we will from now on always assume the initial condition 5.20) B 0 = 1. We will below prove the following Proposition 3. Proposition 3. The Φ bracket hierarchy 5.18) with initial condition 5.20) is a homotopy Lie algebra if and only if 5.21) B M = 1 M) M 1 for M = 0, 1, 2,... with the convention that 0 0 := 1). Proof. The bracket coefficients are in this example a α m = 0, 5.22) b i αm = δαb i m + δm 0 1 δm δm 0 i 1 δm 1 i δm 0 i The generating functions become 5.23) where 5.24) and 5.25) GP ) = m f α p) = 0, The initial condition 5.20) becomes g i αp) = δ i αgp ) + p i, B m p m m! = P := p i. i=1 5.26) GP =0) = 1. The nilpotency condition 3.28) reads 5.27) M=0 B M P M M!, α β) = g i α,jg j β = δi αg P ) + δ i j)δ j β GP ) + pj ) = δ i αg P )GP ) + P ) + δ i βgp ) + p i.

12 276 K. BERING AND T. LADA This is equivalent to the ODE 5.28) 5.29) G P )GP ) + P ) = GP ) dp dg = 1 + P G d [ ] P = 1 dg G G P = LnG) + constant. G We deduce from the initial condition 5.26) that the inverse function P = P G) is 1 G) n 5.30) P G) = GLnG) = 1 G) + nn 1). Let us now recall the Lambert function W = W P ), whose inverse function P = P W ) is 5.31) P W ) = W e W. Hopefully, the reader will not be confused by the fact that we denote two different function P = P G) and P = P W ) and in fact also the momentum variable P itself) with the same symbol P. It should be clear from the context which is which.) Note that the Lambert function W = W P ) has a zero in P = ) W P =0) = 0. By comparing eqs. 5.30) and 5.31) we deduce that the sought-for function G = GP ) is just the exponential of the Lambert function n=2 5.33) GP ) = e W P ) = P W P ). The Taylor expansion for the Lambert function W = W P ) is 5.34) W P ) = n) n 1 P n n!. n=1 The Taylor coefficients with n 1 follow from Lagrange s inversion formula, or simply by calculating W n) P =0) = 1 dp W P ) n! 0 2πi P n = 1 dw 1 n! 0 2πi P W ) n = 1 dw e nw n! 2πi W n = dn 1 e nw dw n ) 0 = n) n 1. W=0 Similarly, the Taylor coefficients for the function G = GP ) with n 1 are B n = G n) P =0) = 1 dp G P ) n! 0 2πi P n = 1 dp W P )e W P ) n! 0 2πi P n = 1 dw e W n! 2πi P W ) n = 1 dw e 1 n)w n! 2πi W n = dn 1 e1 n)w dw n ) 0 = 1 n) n 1. 0 W=0

13 EXAMPLES OF HOMOTOPY LIE ALGEBRAS 277 The Taylor expansion for the function G = GP ) is 5.37) GP ) = 1 n) n 1 P n n!. n=0 Both Taylor series 5.34) and 5.37) have radius of convergence equal to 1/e, as may be seen by the ratio test. This completes the proof of Proposition 3. Acknowledgement. The work of K. B. is supported by the Ministry of Education of the Czech Republic under the project MSM References [1] Batalin, I. A., Vilkovisky, G. A., Gauge algebra and quantization, Phys. Lett. 102B 1981), [2] Bering, K., Non-commutative Batalin-Vilkovisky algebras, homotopy Lie algebras and the Courant bracket, Comm. Math. Phys ), [3] Bering, K., Damgaard, P. H., Alfaro, J., Algebra of higher antibrackets, Nuclear Phys. B ), [4] Daily, M., Examples of L m and L structures on V 0 V 1, unpublished notes. [5] Daily, M., Lada, T., A finite dimensional L algebra example in gauge theory, Homotopy, Homology and Applications ), [6] Lada, T., Markl, M., Strongly homotopy Lie algebras, Comm. Algebra ), [7] Lada, T., Stasheff, J. D., Introduction to sh Lie algebras for physicists, Internat. J. Theoret. Phys ), Institute for Theoretical Physics & Astrophysics Masaryk University, Kotlářská Brno, Czech Republic bering@physics.muni.cz Department of Mathematics North Carolina State University Raleigh NC lada@math.ncsu.edu