|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In mathematics and [[theoretical physics]], a '''Gerstenhaber algebra''' (sometimes called an '''antibracket algebra''' or '''braid algebra''') is an [[algebraic structure]] discovered by [[Murray Gerstenhaber]] (1963) that combines the structures of a [[supercommutative ring]] and a [[graded Lie superalgebra]]. It is used in the [[Batalin–Vilkovisky formalism]].
| | Hello. Let me [http://www.redbookmag.com/health-wellness/advice/best-delivery-diets introduce] the author. Her title is Emilia Shroyer but it's not the most female title out there. Years in the past we moved to North Dakota and [http://Www.Details.com/health-fitness/diet/201208/diets-that-deliver-meal-delivery-plans-nutrition I adore] each day living right here. In her professional diet meal delivery lifestyle she is a payroll clerk but she's always wanted her own business. One of the issues she enjoys most is to do aerobics and now she is trying to earn money with it.<br><br>Feel free diet [http://www.Personaltrainerfood.com/ meal delivery] to surf to my site :: diet meal delivery ([http://tomjones1190.livejournal.com/ just click the up coming internet site]) |
| | |
| ==Definition==
| |
| A '''Gerstenhaber algebra''' is a differential graded [[commutative algebra]] with a [[Lie algebra|Lie bracket]] of degree -1 satisfying the [[Poisson algebra|Poisson identity]]. Everything is understood to satisfy the usual [[superalgebra]] sign conventions. More precisely, the algebra has two products, one written as ordinary multiplication and one written as [,], and a '''Z'''-grading called '''degree''' (in theoretical physics sometimes called '''ghost number'''). The '''degree''' of an element ''a'' is denoted by |''a''|. These satisfy the identities
| |
| *|''ab''| = |''a''| + |''b''| (The product has degree 0)
| |
| *|[''a'',''b'']| = |''a''| + |''b''| - 1 (The Lie bracket has degree -1)
| |
| *(''ab'')''c'' = ''a''(''bc'') (The product is associative)
| |
| *''ab'' = (−1)<sup>|''a''||''b''|</sup>''ba'' (The product is (super) commutative)
| |
| *[''a'',''bc''] = [''a'',''b'']''c'' + (−1)<sup>(|''a''|-1)|''b''|</sup>''b''[''a'',''c''] (Poisson identity)
| |
| *[''a'',''b''] = −(−1)<sup>(|''a''|-1)(|''b''|-1)</sup> [''b'',''a''] (Antisymmetry of Lie bracket)
| |
| *[''a'',[''b'',''c'']] = [[''a'',''b''],''c''] + (−1)<sup>(|''a''|-1)(|''b''|-1)</sup>[''b'',[''a'',''c'']] (The Jacobi identity for the Lie bracket)
| |
| | |
| Gerstenhaber algebras differ from [[Poisson superalgebra]]s in that the Lie bracket has degree -1 rather than degree 0. The Jacobi identity may also be expressed in a symmetrical form
| |
| :<math>(-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b|-1)}[c,[a,b]] = 0.\,</math>
| |
| | |
| ==Examples==
| |
| *Gerstenhaber showed that the [[Hochschild cohomology]] H<sup>*</sup>(''A'',''A'') of an algebra ''A'' is a Gerstenhaber algebra.
| |
| *A [[Batalin–Vilkovisky algebra]] has an underlying Gerstenhaber algebra if one forgets its second order Δ operator.
| |
| *The [[exterior algebra]] of a [[Lie algebra]] is a Gerstenhaber algebra.
| |
| *The differential forms on a [[Poisson manifold]] form a Gerstenhaber algebra.
| |
| *The multivector fields on a [[manifold]] form a Gerstenhaber algebra using the [[Schouten–Nijenhuis bracket]]
| |
| | |
| ==References==
| |
| *{{Cite journal |last=Gerstenhaber |first=Murray |title=The cohomology structure of an associative ring |jstor=1970343 |journal=[[Annals of Mathematics|Ann. of Math.]] |volume=78 |year=1963 |issue=2 |pages=267–288 |doi=10.2307/1970343 }}
| |
| *{{Cite journal |last=Getzler |first=E. |title=Batalin-Vilkovisky algebras and two-dimensional topological field theories |journal=Communications in Mathematical Physics |volume=159 |issue=2 |year=1994 |pages=265–285 |doi=10.1007/BF02102639 |arxiv = hep-th/9212043 |bibcode = 1994CMaPh.159..265G }}
| |
| *{{springer|id=p/p110170|title=Poisson algebra|author=Kosmann-Schwarzbach, Y.}}
| |
| | |
| [[Category:Algebras]]
| |
| [[Category:Theoretical physics]]
| |
| [[Category:Symplectic geometry]]
| |
Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most female title out there. Years in the past we moved to North Dakota and I adore each day living right here. In her professional diet meal delivery lifestyle she is a payroll clerk but she's always wanted her own business. One of the issues she enjoys most is to do aerobics and now she is trying to earn money with it.
Feel free diet meal delivery to surf to my site :: diet meal delivery (just click the up coming internet site)