The Koszul dual of the polynomial ring $\mathbb Q$ [ $x$ ] is the exterior algebra on one odd generator. This is the enveloping algebra of the free Lie algebra on one odd generator $a$ modulo [$a$,$a$].
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The Ext-algebra of $L$ is $Ext_{UL}(k,k)$, where $k$ is the coefficient field of $L$. It may be obtained using extAlgebra. A vector space basis for the Ext-algebra in positive degrees is obtained using generators(ExtAlgebra). This basis originates from the Lie generators in the minimal model, minimalModel, for which the homological degree have been raised by 1 and the signs changed.
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The product in the Ext-algebra, ExtElement ExtElement, is derived by the program from the quadratic part of the differential in the minimal model. The Ext-algebra is a skew-commutative algebra. In case $L$ is the Koszul dual of a skew-commutative Koszul algebra $R$, the Ext-algebra of $L$ is equal to $R$.
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Observe that the first row of the matrix dims(4,E) gives the dimensions of $E$ in degree 1 to 5 and homological degree 1.
Here is the first known example of a non-Koszul algebra, due to Christer Lech. It is the polynomial algebra in four variables modulo five general quadratic forms, which may be specialized as follows.
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The minimal model may also be used to compute a minimal presentation of a Lie algebra, see minimalPresentation(ZZ,LieAlgebra). Below is an example of computing a minimal presentation of the Lie algebra of strictly upper triangular 5x5-matrices. The Lie algebra is presented by means of the multiplication table of the natural basis \{$ekn;\ 1\ \le\ k\ <\ n \le\ 5$\}. The degree of $ekn$ is $n-k$. The relation [ $e14$, $e15$ ] is of degree 7 in the free Lie algebra $F$ on the basis, and the dimension of $F$ in degree 7 is 7596. To avoid a computation of the normal form of [ $e14$, $e15$ ] one uses "formal" operators. The symbol $@$ is used as formal Lie multiplication and formal multiplication by scalars, ++ is used as formal addition, and / is used as formal subtraction. Observe that $@$, like SPACE, is right associative, while / is left associative, so $a/b/c$ means $a-b-c$ and not $a-b+c$. Here is an example of a formal expression, whose normal form is 0. The normal form may be obtained by applying normalForm.
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Here is the computation of the matrix example.
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Below is a differential Lie algebra, which is non-free, and where the linear part of the differential is non-zero.
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The homology in homological degree 0 is concentrated in first degree 1 and 2. In the general case, for a differential Lie algebra $L$, the function minimalPresentation(ZZ,LieAlgebra) gives a minimal presentation of the Lie algebra $H_0(L)$.
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We now check that the homology of the minimal model $M$ is the same as for $L$.
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The quasi-isomorphism \ $f:\ M\ \to\ L$ from the minimal model $M$ of $L$ to $L$ is obtained as map(M). If $L$ has no differential, then \ $f$ \ is surjective, but in general this is not true as is shown by the example below. Another example is obtained letting $L$ be a non-zero Lie algebra with zero homology, see Differential Lie algebra tutorial.
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We check below that $H(f)$ is iso in degree (5,1).
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