• Introduction

Introduction

We start our discussion of PBW-algebras by recalling their definition. Let $K$ be a field. Given a set of indeterminates $x=\{x_1, \ldots, x_n\},$ we write ${\left\langle {x}\right\rangle}:=\langle x_{1},\ldots, x_{n} \rangle$ for the free monoid on $x$. That is, the elements of $\langle x \rangle$ are the words in the finite alphabet $x$, multiplication means concatination of words, and the identity element is the empty word. The free associative $K$-algebra generated by $x_{1},\dots, x_{n}$ is the corresponding monoid algebra

\[K \langle {x}\rangle:= K \langle x_{1},\dots, x_{n} \rangle.\]

We consider quotients of type $A = K\langle x_1, \dots, x_n \rangle/J$, for some $n$ and some two-sided ideal $J$ of $K\langle x_1, \dots, x_n \rangle$. In case $J$ is given by a finite set of two-sided generators $g_1, \dots, g_r$, we say that $A$ is generated by $x_1, \dots, x_n$ subject to the relations $g_1, \dots, g_r$, and write

\[A = K\langle x_1, \dots , x_n \mid g_k=0, \ 1\leq k \leq r \rangle.\]

Working with Gröbner bases requires that we take monomial orderings into account (see the section on Gröbner bases in the commutative algebra chapter for monomial orderings). In our context here, we use the following notation. A standard monomial in $x$ is a word of type $x^\alpha=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$ where $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$. A standard polynomial in $K \langle x \rangle$ is a $K$-linear combination of standard monomials. Identifying the set of standard monomials in $x$ with the usual set of monomials $\text{Mon}_n(x)$, we may think of each global monomial ordering $>$ on the latter set as a (total) well-ordering $>$ on the former one. Given a standard polynomial $0\neq f \in K \langle x \rangle,$ it makes, then, sense to speak of the leading monomial $\text{LM}_>(f)$ of $f$.

Definition. $\;$ Let $A$ be a $K$-algebra of type

\[A = K\langle x_1, \dots , x_n \mid x_jx_i = c_{ij} \cdot x_ix_j+d_{ij}, \ 1\leq i<j \leq n \rangle,\]

where the $c_{ij}\in K$ are nonzero scalars and the $d_{ij}\in K\langle x_1, \dots , x_n\rangle$ are standard polynomials. Then $A$ is called a $PBW$-algebra if the following two conditions hold:

(1) $\;$ There exists a global monomial ordering $>$ on $\text{Mon}_n(x)$ such that

\[d_{ij}=0\ \text{ or }\ x_ix_j> \text{LM}_>(d_{ij})\ \text{ for all }\ 1\leq i<j \leq n.\]

(2) $\;$ The standard monomials represent a $K$-basis for $A$. We then refer to this basis as a PBW-basis for $A$.

Every ordering as in (1) is called admissible for $A$.

Given a PBW-algebra $A$ as above, we sometimes abuse our notation by denoting the class of the standard monomial $x^{\alpha}$ in $A$ also by $x^{\alpha}$, and refer to $x^{\alpha}$ as a standard monomial in $A$. As these monomials form a $K$-basis for $A$, every element $0\neq f\in A$ has a unique representation

\[f=\sum c_{\alpha}x^{\alpha}, \; \text{ with nonzero coefficients } \; c_{\alpha}\in K.\]

We refer to this representation as the standard representation of $f$, with coefficients $c_{\alpha}$, and exponents $\alpha$.

Example. $\;$ The $n$-th Weyl algebra over $K$ is the PBW-algebra

\[D_n(K)=K \langle x_1,\ldots, x_n, \partial _1,\dots \partial _n \mid \partial_i x_i=x_i\partial _i +1, \partial _i x_j=x_j \partial _i \ \text { for }\ i\neq j\rangle.\]

Here, we tacitly assume that $x_j x_i=x_i x _j$ and $\partial _j \partial_i=\partial_i \partial _j$ for all $i,j$. Note that any global monomial ordering on $\text{Mon}_{2n}(x, \partial)$ is admissible for $D_n(K)$.

Proposition. $\;$ Let $A$ be a PBW-algebra. Then:

• $A$ is an integral domain,
• $A$ is (left and right) Noetherian.